On the Structure of Equidistant Foliations of Euclidean Space
aa r X i v : . [ m a t h . DG ] D ec On the Structure of EquidistantFoliations of Euclidean Space
Dissertationzur Erlangung des Doktorgrades an derMathematisch-Naturwissenschaftlichen Fakul¨atder Universit¨at Augsburgvorgelegt vonChristian BoltnerAugsburg, Juni 2007 i Erstgutachter: Prof. Dr. Ernst HeintzeZweitgutachter: Prof. Dr. Jost-Hinrich EschenburgTag der m¨undlichen Pr¨ufung: 04. September 2007 ontents
Introduction 1Chapter 1. Preliminaries 51.1. Alexandrov Spaces 51.2. Submetries 71.3. Equidistant Foliations 11Chapter 2. Existence of an Affine Leaf 172.1. A Soul Construction 172.2. Submetries onto Compact Alexandrov Spaces 20Chapter 3. The Induced Foliation in the Horizontal Layers 273.1. The Homogeneous Case 283.2. The Induced Foliation in each Horizontal Layer is Equidistant 323.3. The Induced Foliations in distinct Horizontal Layers are Isometric 323.4. Equidistance of the Leaves in distinct Horizontal Layers 353.5. Isometries of the Induced Foliation 39Chapter 4. Reducibility of Equidistant Foliations 414.1. Invariant Subspaces 414.2. The Non-compact Case 43Chapter 5. Homogeneity Results 495.1. Factorizing the Submetry 495.2. New Examples from Old 505.3. Homogeneity 52Bibliography 55 iii ntroduction
The aim of this thesis is the study of equidistant foliations of Euclidean space,in particular answering the question whether they are homogeneous.An equidistant foliation of R n is a partition F into complete, smooth, connected,properly embedded submanifolds of R n such that for any two leaves F, G ∈ F and p ∈ F the distance d G ( p ) does not depend on the choice of p ∈ F . Such a foliationmay be singular , i.e. the leaves of F may have different dimensions.We point out that this is a more restrictive version of the definition of singularRiemannian foliations as given by [Mol88]. Their leaves only need to be immersedand equidistance is therefore only demanded locally.The advantage of our more restrictive definition is that the space of leaves B := R n / F bears a natural metric — it is even a nonnegatively curved Alexandrovspace (cf. [BBI01]) — and the canonical projection is a submetry. Indeed we makeheavy usage throughout this work of the Alexandrov space structure of B and relyon the rich theory of submetries as found in [Lyt02].The most prominent examples of equidistant foliations are the orbit foliationsof isometric Lie group actions. So the natural question is whether all equidistantfoliations of R n are homogeneous or at least which conditions imply homogeneity.A huge and well studied class of equidistant foliations are those given by isopara-metric submanifolds and their parallel manifolds. Homogeneity of these foliationswas shown by Thorbergsson in [Tho91] if the isoparametric submanifold has codi-mension ≥
3. However, there are inhomogeneous examples — found by Ferus,Karcher and M¨unzner and presented in [FKM81] — if the isoparamtric submani-fold has codimension 2, i.e. if it is a hypersurface in a sphere.To our knowledge these and the Hopf fibration of S (with totally geodesicfibres, isometric to S ) are the only inhomogeneous examples of equidistant foli-ations known today. We point out that all of these inhomogeneous foliations arecompact, i.e. they have compact leaves.On the other hand Gromoll and Walschap examine regular equidistant foliations— which are necessarily noncompact — in [GW97] and [GW01]. They show thatsuch a foliation always has an affine leaf, which they use to prove that the foliationis homogeneous; in fact it is given by a generalized screw motion around the affineleaf.As all inhomogeneous examples are compact it seems reasonable to concentrateon noncompact foliations. Generalizing Gromoll and Walschap’s result we showin this thesis that an equidistant foliation of R n always has an affine leaf andmay be described by a compact equidistant foliation in one normal space of theaffine leaf together with a (not necessarily homogeneous) screw motion around thatleaf. We give conditions for homogeneity and also construct new (noncompact)inhomogeneous examples. A more detailed summery of this work follows:In
Chapter 1 we introduce the concepts of Alexandrov spaces, submetriesand their derivatives and we define equidistant foliations. We present several basicresults concerning these concepts — among others we show that the regular leavesof equidistant foliations are equifocal.In analogy to Gromoll and Walschap’s result we show in
Chapter 2 thatequidistant foliations always have an affine leaf F . Using essentially Cheeger-Gromoll’s soul construction (cf. [CE75]) we prove that even in the singular case B has a soul and its preimage is an affine space. Then an affine leaf exists if this soulis a single point. To show this we cannot follow [GW97, Sect. 2] as the topologicalresults used there rely on F being a fibration. Instead we give a geometrical proof(which also gives a new proof for the regular setting).For any p ∈ F the intersection of the leaves of F with the horizontal layers L p := p + ν p F yields a partition of L p which we call ˜ F p and all of the ˜ F p togethergive us a partition ˜ F of R n . Chapter 3 is dedicated to studying this inducedfoliation, in particular we show that each ˜ F p is an equidistant foliation of L p .We prove that in the homogeneous case F is given by the orbits of G × R k with G a compact Lie group and R k acting on R k + n by generalized screw motionsaround the axis F and we conclude that the induced foliation ˜ F is equidistant.In the remainder of this chapter we give a characterization of when ˜ F is equidis-tant and we show that — provided each ˜ F p is homogeneous — the ˜ F p are isometricto each other and F can be described by two data: any one of the ˜ F p and ageneralized (possibly inhomogeneous) screw motion around F . Chapter 4 deals with questions of reducibility. We show that — as in thecase of homogeneous representation — existence of a non-full regular leaf impliesthat the minimal affine subspace containing it is invariant under F . Moreover, weexamine under which conditions F splits off a Euclidean factor.Finally, in Chapter 5 we address homogeneity of F . First, we consider thequotient A = R k + n / ˜ F and show that — provided ˜ F is equidistant — the imageof F under the natural projection is an equidistant foliation of A and is describedby the same screw motion map as F . Reversing this construction we give new inhomogeneous equidistant foliations of R n .We close with a homogeneity result for F if ˜ F p (for one and hence all p ∈ F )is homogeneous and if its isometry group fulfills certain conditions, e.g. if it issufficiently small. In particular F is homogeneous if ˜ F p is given by either • the orbits of an irreducible representation of real or complex type, • the orbits of an irreducible polar action, • the Hopf fibration of S or S . Acknowledgements.
This work could not have been accomplished withoutthe help of several people.First and foremost I would like to thank my advisor Prof. Dr. Ernst Heintze forhis constant encouragement and many fruitful discussions during the last years. Iwould also like to thank Prof. Dr. Carlos Olmos for his hospitality and friendlysupport during my stay in C´ordoba in 2004 and many useful discussions. To
NTRODUCTION 3
Dr. Alexander Lytchak I am indebted for many helpful suggestions on the topicof submetries and I would like to thank Prof. Dr. Burkhard Wilking for his sugges-tions concerning the existence of an affine leaf. Further thanks go to Prof. Dr. Jost-Hinrich Eschenburg, Dr. habil. Andreas Kollross and Dr. Kerstin Weinl for manyhelpful discussions and to Dipl. Math. Walter Freyn for proofreading this thesis —any remaining errors are, of course, my own.HAPTER 1
Preliminaries
In this chapter we introduce the concepts of
Alexandrov spaces , submetries and equidistant foliations , that form the basis this thesis is built on. We present severalresults arising from these concepts that will be used throughout this work. Manyof these are citations from literature, sometimes equipped with a more accessibleproof, but original work is included as well. The concept of Alexandrov spaces is a generalization of Riemannian manifolds.We only give a brief outline of what an Alexandrov space is and present someproperties relevant to this work. For a more detailed discussion of Alexandrovspaces we refer the reader to [BBI01].A metric space X is called a length space if the distance between any twopoints is given by the infimum of the length of curves connecting these two points.Consequently a curve whose length equals the distance between its endpoints iscalled a shortest curve and a locally shortest curve is called a geodesic . If we doexplicitely say anything else we always assume a geodesic to be parametrized byarc length.An Alexandrov space is a length space with a lower curvature bound κ . Thismeans that small geodesic triangles are always thicker (i.e. points on any side areat a greater or equal distance from the opposite vertex) than a comparison trianglewith the same side lengths in the model space M κ , which is the 2-dimensional spaceform of constant curvature κ .This implies an abundance of properties (some immediate from the definition,others requiring rather sophisticated theory) showing that Alexandrov spaces areindeed very similar to Riemannian manifolds. Some useful results about Alexandrov spaces.
We present a short list of resultsabout the geometry of Alexeandrov spaces, which will be used throughout thisthesis. • Geodesics in Alexandrov spaces do not branch (otherwise this would resultin “thin” triangles, cf. [BBI01, Chap. 4]). • The Hausdorff dimension of an Alexandrov space is either an integer orinfinity (cf. [BBI01, Chap. 10]). • Finite dimensional complete Alexandrov spaces are proper (i.e. closedbounded subsets are compact) and geodesic (i.e. any two points can beconnected by a shortest curve).Moreover, an analogue of the Hopf-Rinow theorem holds (cf. [BBI01,Thm. 2.5.28]). • Any n -dimensional Alexandrov space contains an open dense subset whichis an n -dimensional manifold (cf. [BBI01, Chap. 10]). Remark.
Henceforth, if we talk about an Alexandrov space we will always as-sume it to be complete and finite dimensional .In geodesic spaces we commonly use the notation | xy | for the distance betweentwo points instead of d ( x, y ).For a subset A of a metric space X we denote by d A : X → R +0 the distancefunction d A ( p ) = dist ( A, p ) relative to A . Tangent Cones.
Let X be an Alexandrov space and consider two geodesics α and β emanating at some point p ∈ X . An immediate consequence of the lowercurvature bound is that the angle formed by α and β at p is well defined.We consider the set ˜Σ p of equivalence classes of geodesics emanating from p where two geodesics are identified if they form a zero angle. Definition 1.1.
The space of directions Σ p at p is the completion of ˜Σ p withrespect to the angle metric.The tangent cone T p X of X at p is the metric cone C Σ p over Σ p . Remark.
The space of directions of an n -dimensional Alexandrov space is acompact ( n − ≥
1. Consequently T p X is an n -dimensional Alexandrov space of nonnegative curvature.Note that in general there may be directions at p not represented by any geo-desic. Definition 1.2.
We call a point x in an n -dimensional Alexandrov space X regular if the space of directions Σ x at x is isometric to the Euclidean standardsphere S n − , or equivalently if T x X is isometric to R n . Remark.
Geodesics ending at a regular point x can be extended beyond x andfor any ξ ∈ Σ x there is a geodesic starting at x with direction ξ .Thus at regular points x we can define the exponential map exp x : U ⊂ T x X → X in the same way as for Riemannian manifolds.We point out that the set of regular points of X contains a set which is openand dense in X (cf. [BBI01, Chap. 10]).Remember that the metric cone over Σ p is the topological cone over Σ p , i.e.the set [0 , ∞ ) × Σ p / ∼ where we have identified all points of the form (0 , ξ ), ξ ∈ Σ p ,equipped with the metric | ( t, ξ )( s, η ) | = t + s − h ξ, η i where h ξ, η i = cos ∡ ( ξ, η ). This places an isometric copy of Σ p at distance 1 fromthe apex 0.We present some further notation:For v = ( t, ξ ) ∈ T p X and s ≥ sv the vector ( st, ξ ) ∈ T p X .We usually write | v | as a shorthand for the distance | v | between v and theapex 0 of the cone.Let ξ, η ∈ Σ p be directions which enclose an angle < π and let γ be a shortestcurve in Σ p connecting them. Then the cone over γ can be embedded isometricallyinto R , via φ , say. Thus for v = tξ and w = sη we define v + w := φ − ( φ ( v ) + φ ( w )) . .2. SUBMETRIES 7 Of course this depends on the choice of γ and is only useful if γ is unique. Note,however, that we get the usual relation | v + w | = | v | + | w | + 2 h v, w i where h v, w i := ts h ξ, η i .Finally if A is a subset of Σ p we call (cid:8) ξ ∈ Σ p (cid:12)(cid:12) dist ( ξ, A ) ≥ π (cid:9) the polar set of A . Submetries are a generalization of the notion of linear projections and Rie-mannian submersions to metric spaces.
Definition 1.3.
Let f : X → Y be a mapping between metric spaces. Then f is called a submetry if it maps metric balls in X to metric balls of the same radiusin Y .This simple property turns out to be rather rigid at least for submetries betweenAlexandrov spaces. And we present in the following some interesting results aboutsubmetries relevant to this thesis. We refer the reader to [Lyt02] for a detaileddiscussion.First note that we can characterize submetries by looking at the distance func-tion of fibres (cf. [Lyt02, Lem. 4.3]): Lemma 1.4.
A mapping f : X → Y between metric spaces is a submetry ifand only if for any subset A (possibly a single point) of Y the equality d f − ( A ) = d A ◦ f holds. We call a point p ∈ X near to x ∈ X (with respect to f ) if | xp | = dist ( F x , p )where F x is the fibre of f passing through x . We denote the set of points near to x by N x .A geodesic γ emanating at x will be called horizontal if its image under f is ageodesic of the same length. Thus a shortest curve is horizontal if and only if itsstart and endpoint are near to each other.It should be noticed that many topological and geometric properties are inher-ited by the base space of a submetry (cf. [Lyt02, Prop. 4.4]). We present only afew: Proposition 1.5.
Let f : X → Y be a submetry between metric spaces. Then Y is complete or connected or is a length space or has curvature bounded below by κ or has dimension ≤ n if X has the respective property. Finally, we mention the following factorization property of submetries, whichis an immediate consequence of the definition (cf. [Lyt02, Lem. 4.1]):
Lemma 1.6.
Let
X, Y, Z be metric spaces and f : X → Y , g : Y → Z be mapsbetween them. Suppose that f and h := g ◦ f are submetries then so is g . Proof.
Let B r ( y ) be some metric ball in Y , which is the image under f of someball B r ( x ) in X since f is a submetry. Then g ( B r ( y )) = h ( B r ( x )) = B r ( h ( x )). (cid:3)
1. PRELIMINARIES
With Riemannian submersions p : M → N it is possible to liftgeodesics in the base N to horizontal geodesics in M . This follows easily from theconditions posed on the differential of the submersion.However, this can be shown in a purely geometrical way as is done e.g. in[BG00]. Using essentially the same arguments we see that these lifts exist in thecase of submetries as well: Lemma 1.7.
Let f : X → ¯ X be a submetry between Alexandrov spaces and let ¯ γ : [0 , l ] → ¯ X be a shortest path of length l between two points ¯ p and ¯ q .(a) Let p ∈ f − (¯ p ) then there exists a horizontal lift γ of ¯ γ to p , i.e. a shortestpath γ : [0 , l ] → X of the same length such that γ (0) = p and f ◦ γ = ¯ γ .(b) If ¯ γ can be extended beyond ¯ p as a shortest path then the horizontal lift isunique. Proof.
Assume for now that ¯ γ can be extended beyond ¯ p .(a) Since f is a submetry dist (cid:0) f − (¯ p ) , f − (¯ q ) (cid:1) = | ¯ p ¯ q | and since f − (¯ p ) and f − (¯ q )are closed there is a point q ∈ f − (¯ q ) such that | pq | = | ¯ p ¯ q | , i.e. q is near to p .Let γ : [0 , l ′ ] → X be a shortest path connecting p and q . Then L ( γ ) = l ′ = | pq | = | ¯ p ¯ q | = l and consequently f ◦ γ is a curve of length at most l connecting¯ p and ¯ q . Hence it is a shortest curve. Remember that since ¯ γ is extendible it isthe unique shortest path connecting those two points and so has to agree with f ◦ γ .(b) Suppose there are two different lifts γ and γ to p .Let ¯ α : [ − ε, l ] → ¯ X be an extension as a shortest path of ¯ γ and let ¯ r be thepoint ¯ α ( − ε ).We can now lift ¯ α | [ − ε, to p . Let us call this lift β and its starting point r .Then r is near to p so | ¯ r ¯ q | = | ¯ r ¯ p | + | ¯ p ¯ q | = | rp | + | pq | ≥ | rq | ≥ | ¯ r ¯ q | where the last inequality holds because f does not increase distances.So, continuing β by either γ or γ yields a shortest path between r and q which agrees with the other at least up to p . But then the γ i have to agree aswell since in Alexandrov spaces geodesics do not branch.To show (a) in general just choose some point ¯ x in the interior of ¯ γ , take x ∈ f − (¯ x )near to p and lift ¯ γ to x . This lift then has p as one endpoint. (cid:3) Remark 1.8.
Of course Lemma 1.7 also holds for geodesics instead of shortestpaths. Since geodesics are locally shortest we can lift these shortest paths and usethe fact that the lifts at interior points of the geodesic are unique.Note that there is an even stronger lifting property (Proposition 1.17) if X isa manifold. Several results in this work are based on examining thedifferential of a submetry. So let us explain what we mean by differentiability andthe differential of a map between Alexandrov spaces.
Remark.
The material presented in this section is mostly due to [Lyt02]. Butsince it is nonstandard material we include it here and present it in a way moresuitable for the needs of this thesis. .2. SUBMETRIES 9
In [BGP92, p.44] a Lipschitz function f : X → R on a finite dimensional Alexan-drov space is said to be differentiable if its restriction to any geodesic is differentiable(with respect to arc length) from the right.This is generalized in [Lyt02, Sect. 3] to Lipschitz maps f : X → Y betweenfinite dimensional Alexandrov spaces. Remark.
In the following we will be using ultralimits. We refer the readerto [KL97, Sect. 2.4] for a concise definition of ultralimits. In short this conceptallows us to coherently choose for any sequence ( x j ) in a compact space one of itslimit points. This limit point is called the ultralimit lim ω x j of ( x j ) and dependson the particular choice of the nonprincipal ultrafilter ω on the integers.Using this [KL97] considers sequences of pointed metric spaces ( X j , x j ) anddefines their ultralimits lim ω ( X j , x j ) as the set X ∞ consisting of all sequences ( y j )with y j ∈ X j such that d j ( y j , x j ) is uniformly bounded. Then x ∈ X ∞ is definedas ( x j ) and we get a pseudometric d (( y j ) , ( z j )) which is defined as the ultralimitlim ω d j ( y j , z j ). After identifying points y, z ∈ X ∞ for which d ( y, z ) = 0 this turns( X := X ∞ / ( d =0) , x ) into a pointed metric space. Remark.
If ( X j , x j ) is a sequence of proper spaces converging in the pointedGromov-Hausdorff topology towards the proper space ( X, x ) then for any ω theultralimit lim ω ( X j , x j ) is isometric to ( X, x ).The ultralimit approach has the advantage that we can extend this notionnaturally to maps between metric spaces: Let f j : ( X j , x j ) → ( Y j , y j ) be a sequenceof Lipschitz maps with uniform Lipschitz constant then the ultralimit f := lim ω f j is given by f (( z j )) = ( f j ( z j )).Now let us look in particular at the tangent cone of a finite dimensional Alexan-drov space X : The tangent space T x X at x is the pointed Gromov-Hausdorff limitof the scaled spaces ( r j X, x ) for any positive sequence ( r j ) tending to zero. By λX we mean the space X with the scaled metric λ · d . Remark.
The tangent space T x X defined in this way is isometric to the metriccone C Σ x over the space of directions at x (cf. [BBI01, Sect. 10.9]).Based on this [Lyt02] makes the following definition: Definition 1.9.
Let f : X → Y be a Lipschitz map between finite dimensionalAlexandrov spaces. We consider for any positive sequence ( r j ) tending to zero theultralimit lim ω f j of the sequence f j := f : ( r j X, x ) → ( r j Y, f ( x )).We say f is differentiable at x ∈ X if lim ω f j does not depend on the choiceof ( r j ) and call the resulting Lipschitz map f ∗ x : T x X → T f ( x ) Y the differential of f at x .In detail f ∗ x is given in the following way: Let p ∈ X be close to x and let γ be a shortest path connecting x to p with direction ξ at x . Then considering that( r j X, x ) converges to T x X we see that ( γ ( r j · | xp | )) converges towards | xp | · ξ andconsequently ( f ( γ ( r j · | xp | ))) tends to some η in T f ( x ) Y . If η is independent of ( r j )then f ∗ x ( | xp | · ξ ) = η .Note that by this property f ∗ x is homogeneous , i.e. f ∗ x ( tξ ) = tf ∗ x ( ξ ) for anynonnegative t . Application to Submetries. (1) By [Lyt02, Prop. 3.7] f : X → Y is differentiable at x ∈ X if and onlyif for any y ∈ Y with y = f ( x ) the function d y ◦ f is differentiable, thusreducing the question of differentiability to the case treated by [BGP92].(2) From [Lyt02, Lem. 4.3] we know that f : X → Y is a submetry if andonly if d f − ( y ) = d y ◦ f for any point y in Y . Since for any closed A ⊂ X the function d A is differentiable outside A (cf. [BGP92, p.44]) this impliesthat submetries are differentiable.(3) If f j : ( X j , x j ) → ( Y j , y j ) is a sequence of submetries then its ultralimit isa submetry as well. This is an immediate consequence of the definitionof ultralimits and shows that the differential of a submetry is itself asubmetry between the tangent spaces.Moreover, the fibres of f j converge to the fibres of f (cf. [Lyt02,Lem.a 4.6]).Thus the study of the differential of a submetry reduces to the study of homoge-neous submetries f : C Σ → C S between cones or simply to submetries f : Σ → S where Σ and S have curvature ≥ Proposition 1.10.
Let Σ and S be finite dimensional Alexandrov spaces ofcurvature ≥ and let f : C Σ → C S be a homogeneous submetry. Then the follow-ing assertions hold:(a) The preimage f − (0) of the apex is the cone over some totally convex set V ⊂ Σ .The directions in V are called vertical .(b) Let H be the polar set of V with respect to Σ . Then C H consists just of the horizontal vectors of f , i.e. those h ∈ C Σ such that | f ( h ) | = | h | .(c) For any x ∈ C Σ \ (C V ∪ C H ) there are unique v ∈ C V and h ∈ C H such that x = h + v , h h, v i = 0 and f ( x ) = f ( h ) .(d) The restriction f : C H → C S is a submetry. The proof for Proposition 1.10 can be found in [Lyt02, Prop. 6.4, Lem. 6.5,Cor. 6.10]. We give a detailed proof of part (c) since this result will be essentiallater on.
Proof.
First note that since H is polar to V there may be at most one shortestcurve in Σ connecting H and V and passing through ξ = x | x | . Otherwise we couldcombine two such geodesics in such a way as to produce a branch point. So thenotation h + v is well defined.Let y = f ( x ) and let c be the geodesic ray in C S emanating at 0 (i.e. c (0) = 0)and passing through y . There is a unique horizontal lift γ of c through x since y lies in the interior of c . Let ˜ γ be the ray parallel to γ and emanating at 0, i.e.˜ γ ( t ) + γ (0) = γ ( t ).We define v := γ (0), so v is contained in C V because f ( γ (0)) = c (0) = 0. Thus γ ( t ) = ˜ γ ( t ) + v and so f (˜ γ ( t ) + v ) = f ( γ ( t )) = tf ( γ (1)) . Now as f is 1-Lipschitz we get(1.1) (cid:12)(cid:12) f ( γ ( t )) f (˜ γ ( t )) (cid:12)(cid:12) ≤ (cid:12)(cid:12) γ ( t ) ˜ γ ( t ) (cid:12)(cid:12) = | v | .3. EQUIDISTANT FOLIATIONS 11 but on the other hand using that f is homogeneous and ˜ γ is a ray we get (cid:12)(cid:12) f ( γ ( t )) f (˜ γ ( t )) (cid:12)(cid:12) = (cid:12)(cid:12) ( tc (1)) f ( t ˜ γ (1)) (cid:12)(cid:12) = t (cid:12)(cid:12) c (1) f (˜ γ (1)) (cid:12)(cid:12) for arbitrarily large t . Using (1.1) this implies f ( γ ( t )) = f (˜ γ ( t )) for all t ≥ t such that γ ( t ) = x we define h := ˜ γ ( t ). Then h ∈ C H and f ( h ) = f ( x ).Finally, by construction γ is perpendicular to the geodesic ray { tv | t ≥ } andhence so is ˜ γ , i.e. h h, v i = 0. (cid:3) Remark.
Let f : X → Y be a submetry between Alexandrov spaces and con-sider f ∗ x : C Σ x → C Σ f ( x ) . The cone C V x is the tangent cone at x of the fibre of f containing x and C H x is the tangent cone at x of the set N x of points near to x (cf. [Lyt02, Chap. 5]). An equidistant foliation of R n is a partition F into complete,smooth, connected, properly embedded submanifolds of R n such that for any twoleaves F, G ∈ F and p ∈ F the distance d G ( p ) does not depend on the choiceof p ∈ F . Moreover, we demand the foliation to be smooth, i.e. any vector tangentto a leaf can be locally extended to a vector field that is everywhere tangent to theleaves of F .The space B = R n / F of the leaves of F bears the natural metric d B ( F, G ) =dist R n ( F, G ) and the canonical projection π : R n → B is a submetry. The leaves of F are then the fibres of π . Remark 1.12.
Note that this definition is a special case of that of a singularRiemannian foliation as given by [Mol88]: A partition L of a Riemannian manifoldinto connected immersed submanifolds such that(a) any vector tangent to a leaf can be locally extended to a vector field tangentto the leaves of L , and(b) the foliation is transnormal , i.e. every geodesic that is perpendicular at onepoint to a leaf remains perpendicular to every leaf it meets.Note that transnormality characterizes local equidistance of the leaves — and indeedglobal equidistance if the leaves are properly embedded.Concerning condition (a) observe that Lemma 1.20 already implies that we mayextend any vector tangent to a leaf to a local vector field everywhere tangent tothe leaves. However, this vector field need not — a priori — be smooth at singularleaves.It is, however, quite reasonable to stick to our more restrictive definition as theadditional structure we gain is very useful. For example the submetry π and thebase space B have some nice properties (cf. [Lyt02, Prop. 12.8–12.11]): Proposition 1.13. (a) Let p be any point in R n . Then the set N p of pointsnear to p is convex.(b) Let F be the leaf passing through p . Then any direction perpendicular to T p F ishorizontal, and there is a positive number ε such that for any direction ξ p ∈ ν p F there is a horizontal geodesic of length at least ε starting in the direction of ξ p .Consequently, at ¯ p := π ( p ) , for any ¯ ξ ∈ Σ ¯ p B there is a geodesic in B emanating at p of length at least ε with direction ¯ ξ . Moreover, we get from Chapter 13 of [Lyt02]:
Proposition 1.14.
The set of regular points in B is a smooth Riemannianmanifold over which π is a smooth Riemannian submersion. We call the fibres over regular points of B the regular leaves of F .We introduce some notation commonly used when dealing with Riemanniansubmersions:We denote the vertical space T p F at p ∈ F by V p and the horizontal space ν p F by H p . Note that V and H are (at least locally) spanned by smooth vector fields(see Lemma 1.20). We denote the set of vertical and horizontal vector fields by V and H respectively.Let ∇ be the standard Levi-Civita connection on R n , and v ∇ and h ∇ its projec-tions to V and H respectively.The shape operator S of F ∈ F is as usual the 1-form on H F with values inthe symmetric endomorphisms of V F that is dual to the second fundamental form α of F : S X V = − v ∇ V X, X ∈ H , V ∈ V . The integrability tensor or O’Neill tensor O is the skew symmetric 2-form on H with values in V , given by O X Y = 12 [ X, Y ] v = v ∇ X Y, X, Y ∈ H . A vector field ξ on the regular part of F which is everywhere horizontal andfor which π ∗ ξ is a well defined vector field on the regular part of B is called basichorizontal or Bott-parallel . We denote the set of Bott-parallel vector fields by B .Observe that on the regular part of F we have[ B , V ] ⊂ V and as a consequence h ∇ V ξ = h ∇ ξ V = −O ∗ ξ V, V ∈ V , ξ ∈ B where O ∗ ξ is the pointwise adjoint of O ξ . Remark 1.15.
As a consequence of O’Neill’s formula (using the constant cur-vature of R n ) the O’Neill vector fields O ξ η for ξ, η ∈ B have constant norm alongthe regular leaves of F . Lifting through singular leaves.
We are frequently in a situation where we wantto lift a curve that is the projection of a geodesic which at least starts horizontally.This means the start of the projected curve is a geodesic but the whole curve maynot be due to the fact that there may be points in the base, such as the boundary,beyond which a geodesic cannot be extended.Such projections of geodesics which start horizontally are quasigeodesics (seefor example [PP94] for a concise definition and further properties of quasigeodesics).We only mention a few key properties (cf. [Lyt02, Sect. 12.4]):
Proposition 1.16.
Let X be an Alexandrov space.(a) For any x ∈ X and ξ ∈ Σ x there is a quasigeodesic ¯ γ emanating from x withdirection ξ . .3. EQUIDISTANT FOLIATIONS 13 (b) If there is a shortest curve γ of length l starting at x with the same direction ξ then ¯ γ agrees with γ up to length l .(c) If X is the base space of a submetry f : M → X from a Riemannian manifoldthen any quasigeodesic in X defined on a bounded (not necessarily compact)interval consists of finitely many geodesic pieces.(d) Let γ be a geodesic in M starting horizontally, then f ◦ γ is a quasigeodesicin X . This enables us to prove:
Proposition 1.17.
Let f : M → X be a submetry between a Riemannian man-ifold and an Alexandrov space and let γ : [0 , l ] → M be a geodesic such that therestriction of γ to [0 , ε ] for some ε > is horizontal.Then for any p ′ in the same fibre as p := γ (0) it is possible to lift f ◦ γ as ageodesic to p ′ and this lift is unique if the lift of f ◦ γ | [0 ,ε ] is unique. First we show:
Lemma 1.18.
Let B be a connected Alexandrov space and f, g : S n → B a sub-metry with f ( p ) = g ( p ) for some point p ∈ S n . Then f ( − p ) = g ( − p ) . Proof.
We use induction over the dimension n of the sphere. For n = 0 thereis nothing to show since B has to be a single point.So suppose our claim holds for S k with k = 0 , . . . , n −
1. Let v, w be unit vectorsin T p S n , horizontal with respect to f and g respectively, such that f ∗ p ( v ) = g ∗ p ( w ).Denote by γ v and γ w the geodesics starting at p with direction v and w respectively.We show that f ◦ γ v = g ◦ γ w . Then f and g agree at γ v ( π ) = γ w ( π ) = − p .Note that up to some maximal time t the curves f ◦ γ v and g ◦ γ w are geodesicsin X starting at the same point in the same direction; hence they agree at thebeginning, up to the point ¯ q := f ◦ γ v ( t ) = g ◦ γ w ( t ). Denote by q and q thepoints γ v ( t ) and γ w ( t ) respectively and define˜ v := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 γ v ( t − t ) , ˜ w := ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 γ w ( t − t ) . We can then identify the space of directions S n − at q with that at q setting˜ v = ˜ w . Then f ∗ q , g ∗ q : S n − → Σ ¯ q B are submetries agreeing at a point and hence,by induction, at its antipode.Now remember that γ v and γ w are both quasigeodesics in X consisting offinitely many geodesic segments. Applying the above argument successively toeach of these segments finishes our prove.Note that the only problematic case, i.e. Σ ¯ q B not being connected, can ariseonly when n = 1 with Σ ¯ q B = S . But then f ◦ γ can be extended beyond ¯ q , so ¯ q isnot a hinge point of f ◦ γ . (cid:3) Proof of Proposition 1.17.
We only need to check what happens at thehinge points of the quasigeodesic f ◦ γ .Let t be the first time γ meets a singular fibre of f . Let γ ′ be the horizontallift to p ′ of f ◦ γ | [0 ,t ] .Identifying the spaces of horizontal directions at q := γ ( t ) and q ′ = γ ′ ( t ) weget that f ∗ q , f ∗ q ′ : S k → T f ( q ) X agree on the direction from which γ and γ ′ arriveand hence on their respective antipodes. This allows us to continue γ ′ smoothly by a lift of the next geodesic segmentin f ◦ γ . Repeating this for the remaining hinge points finishes the proof. (cid:3) Remark 1.19.
Define F + ξ to be { p + ξ p | p ∈ F } . If F is a regular leaf in F and ξ is Bott-parallel along F Proposition 1.17 implies that F + ξ is a leaf of F and the smooth map p p + ξ between these leaves is surjective. Note that it isbijective and hence a diffeomorphism, if F + ξ is regular.In particular the tangent space T p + ξ ( F + ξ ) is given by { v + ∇ v ξ | v ∈ T p F } .Even if F + ξ is singular the map p p + ξ is at least a submersion: Lemma 1.20.
Let F ∈ F be regular. Then the map P : F → G := F + ξ with P ( x ) = x + ξ x is a surjective submersion. Proof.
Observe that we can extend ξ to be a Bott-parallel normal field in aneighborhood of F such that F ′ + ξ = G for all leaves F ′ in that neighborhood.Using this we can also extend P : p p + ξ p to the same neighborhood renaming P | F : F → G to ˜ P .Note that for any point p the differential P ∗ p is just the orthogonal projectiononto V p + ξ p .Now assume there is a point p ∈ F such that ˜ P ∗ p : V p → V q , with q := p + ξ p ,is not surjective. We show that ˜ P ∗ is nowhere surjective along F .So take some v ∈ V q perpendicular to the image of ˜ P ∗ p . Then v , or rather itsparallel translate to p , is contained in ν p F since h v, x i = (cid:10) v, P ∗ p x (cid:11) for any vector x with base point p . Let η be the extension of v to a Bott-parallel normal fieldalong F .We get P ∗ η = η + ∇ η ξ = (cid:18) η + h ∇ η ξ (cid:19) + O η ξ and h ∇ η ξ is again a Bott-parallel normal field along F . By Remark 1.15 the normof P ∗ η is constant along F , which implies that P ∗ η = η for any point in F since P ∗ is an orthogonal projection at every point.Hence, the differential of ˜ P is nowhere surjective. But since ˜ P : F → G is asurjective map its singular values should be a set of measure zero in F by Sard’sTheorem. (cid:3) Using Proposition 1.17 we can prove the following rigidity result for the regularleaves of F (based on the idea of [HLO06, Lem. 6.1] for the case of Riemanniansubmersions). Proposition 1.21.
Let F be an equidistant foliation of R n and π : R n → B the corresponding submetry. Then for any regular leaf F the principal curvaturesin the direction of Bott-parallel ξ are constant along F . Proof.
Let λ = 0 be an eigenvalue of S ξ at p ∈ F and || ξ p || = 0. Let v ∈ T p F with || v || = 1 be a corresponding eigenvector.Consider the geodesic γ p ( t ) = tξ p and its horizontal variation α p ( s, t ) = tξ p − st ( O ∗ ξ v ) p .3. EQUIDISTANT FOLIATIONS 15 Figure 1.1.
Main curvatures are constant along regular leaves.yielding the Jacobi field J p ( t ) := ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 α p ( s, t ) = − t (cid:16) O ∗ ξ p v p (cid:17) γ ( t ) along γ p . Denote the leaf passing through γ ( t ) by F t and remember that T γ ( t ) F t isspanned by { v i + t ∇ v i ξ } if { v i } is a basis of T p F .In particular this implies for t = 1 /λ that J p (cid:18) λ (cid:19) = − λ O ∗ ξ p v p = v p + 1 λ ∇ v p ξ is vertical at γ ( λ ).Now ¯ α ( s, t ) = π ◦ α p ( s, t ) is a variation of ¯ γ = π ◦ γ p by quasigeodesics and wecan lift this variation to any point q ∈ F . Thus we get the α q ( s, t ) = q + tξ q − tsη where η is the Bott parallel continuation along F of O ∗ ξ p v p .Note that the image of O ∗ ξ is equal to the image of O ∗ ξ O ξ which is Bott parallelfor ξ ∈ B (cf. Remark 4.3). Hence there exists w q ∈ T q F such that η q = O ∗ ξ q w q .As a consequence of this lifting J q (cid:0) λ (cid:1) = − λ O ∗ ξ q w q is vertical at γ q ( λ ) whichmeans that there is a v q ∈ T q F such that J q (cid:18) λ (cid:19) = v q + ∇ v q ξ = (cid:18) I − λ S ξ q (cid:19) v q − λ O ξ q v q . In particular (cid:0) I − λ S ξ q (cid:1) v q vanishes, which proves our claim. Note that by continu-ity of the principal curvatures their multiplicities are constant along F as well. (cid:3) Remark.
A generalization of this result to singular Riemannian foliations hasrecently been proved by Alexandrino and T¨oben (cf. [AT07]).HAPTER 2
Existence of an Affine Leaf
Gromoll and Walschap show in [GW97] that a regular equidistant foliationalways has an affine leaf. To be more precise they show that the space of leaveshas a soul, which is a point, and that the leaf corresponding to the soul is an affinespace.In Section 2.1 we show that it is possible to perform the same soul constructionfor singular foliations as well and in Section 2.2 we prove that the soul in thesingular case also has to be a point. The approach used in the latter case iscompletely different to [GW97] since their argument uses the spectral sequence forthe homology of the fibration, which does not work at all in the singular setting.Thus we get:
Theorem 2.1.
Let F be an equidistant foliation of R n with π : R n → B thecorresponding submetry. Then B has a soul S which is a single point and the fibreover S is an affine subspace of R n .In short, F contains a leaf which is an affine subspace (possibly a single point)of R n . We will first use the Cheeger-Gromoll soul construction (cf. [CE75]) to arriveat a totally convex, compact subset of B without boundary.We will, however, concentrate on lifting this construction to R n since we aremore interested in π − ( S ) than in the soul S itself.Remember that a ray γ in a length space is a unit speed geodesic definedon [0 , ∞ ) such that any restriction γ | [0 ,T ] is a shortest path. By a ray in R n we willmean throughout this section a horizontal one (with respect to F ). The followinglemma ensures the existence of rays. Lemma 2.2.
For any point p in a locally compact, complete, noncompactlength space X there is a ray γ starting at p . Proof.
Since X is not compact it cannot be bounded (cf. the Hopf-Rinow-Cohn-Vossen Theorem [BBI01, Thm. 2.5.28]). So let ( p n ) be a sequence in X with | pp n | tending to infinity. Consider the sequence ( γ n ) of shortest paths, connecting p to p n and denote by γ Tn their restriction to [0 , T ]. By the compactness of B T ( p )an Arzela-Ascoli type argument (cf. [BBI01, Thm 2.5.14]) yields the uniform con-vergence of a subsequence of ( γ Tn ) towards some curve γ T . However, in a lengthspace, the limit of a sequence of shortest paths is itself a shortest path (cf. [BBI01,Prop. 2.5.17]).
178 2. EXISTENCE OF AN AFFINE LEAF
By increasing T and passing on to subsequences we arrive at a curve γ : R +0 → X starting at p and the restriction of γ to any [0 , T ] is a shortest path. (cid:3) Let γ be a ray starting at some point p of B . We define B γ to be the horosphere S t> B t ( γ ( t )) and C γ := B \ B γ . Finally let C be the intersection of all C γ where γ ranges over all rays starting in p . Remark.
It is easy to check that C is totally convex by simply using the sameproof as in the manifold case (cf. [CE75, pp. 135f]). The essential ingredient thereis Toponogov’s Theorem, which holds for Alexandrov spaces as well (cf. [BBI01,p. 360]). Remark.
Note that C is nonempty since it contains p and closed since the B γ are all open. Clearly C is also compact. If it were not, we could find a ray startingat p and lying in C by the argument used in the proof of Lemma 2.2 using thefact that C is closed. But by definition of C no point of this ray — apart from p — is contained in C .We will now pass on to the lift of this construction. For any lift ˜ γ of a ray γ starting in p we define B ˜ γ ⊂ R n in analogy to B γ ⊂ B . Note that the B ˜ γ are openhalfspaces of R n .Denote by ˜ B γ the union of the B ˜ γ where ˜ γ ranges over all lifts of γ along F ,and by ˜ C γ its complement. Finally let ˜ C be the intersection of the sets ˜ C γ .Obviously the latter are closed and convex being the intersection of closedhalfspaces and hence so is ˜ C . Proposition 2.3.
The set ˜ C is the preimage of C . Proof.
Let q be any point in R n \ ˜ C . That means q is contained in a ball q ∈ B t (˜ γ ( t )), where ˜ γ is a horizonal ray emanating from F . But since π is asubmetry this implies π ( q ) ∈ B t ( γ ( t )), with γ = π ◦ ˜ γ , so q cannot lie in π − ( C ).On the other hand, consider any q ∈ R n \ π − ( C ). Then π ( q ) must lie in some B t ( γ ( t )) for a ray γ starting at p .Now take a lift ˜ γ of γ such that ˜ γ ( t ) is near to q , i.e. | q ˜ γ ( t ) | = dist (cid:0) q, π − ( γ ( t )) (cid:1) = | π ( q ) γ ( t ) | . Thus q ∈ B t (˜ γ ( t )) which implies that q is not contained in ˜ C . (cid:3) Remark.
Thus ˜ C is foliated by F , i.e. any leaf of F intesecting ˜ C is containedin ˜ C . In particular ˜ C it is nonempty.In fact this is also true of its boundary but, since ˜ C may have empty interiorin R n we have to find the right notion of “boundary” first.Let V m be the unique affine subspace of minimal dimension m such that ˜ C iscontained in V . We will denote the interior and boundary of a set X ⊂ V withrespect to V by int V ( X ) and ∂ V X respectively.The convexity of ˜ C implies that int V ( ˜ C ) is nonempty: By definition of V wemay choose m + 1 points q , . . . , q m ∈ ˜ C such that the vectors q − q , . . . , q m − q are linearly independent. But then the convex hull of these points has nonemptyinterior in V and is contained in ˜ C . Remark.
Thus it makes perfect sense to call m the dimension of ˜ C . .1. A SOUL CONSTRUCTION 19 Lemma 2.4.
Let A be a closed, convex subset of R n foliated by F . Moreover,let V be the the minimal affine subspace of R n containing A . Then ∂ V A — if it isnonempty — is also foliated by F . Proof.
We have to make sure that fibres of π that contain a boundary pointof A are themselves completely contained in the boundary of A .So, suppose there is a leaf F ⊂ A and two points p, q ∈ F such that p ∈ int V ( A ) and q ∈ ∂ V A .By convexity of A and since F is smooth there is a geo-desic γ in V passing through q , perpendicular to F such that γ (0) = q and q := γ ( ε ) ∈ int V ( A ) and q := γ ( − ε ) / ∈ A for ε > q q ] is mapped by π onto a quasigeodesic in B , whichby Proposition 1.17 can be lifted to a geodesic γ ′ passingthrough p . Denote by p and p the points γ ′ ( ε ) and γ ′ ( − ε )respectively.Since [ qq ] is contained in A so is [ pp ] as A is foliatedby F . Hence γ ′ is a line segment in V and so for small ε thepoint p is contained in A , which is a contradiction because q and p lie in thesame leaf. (cid:3) Using this last result we can now continue the construction recursively until weend up with a compact set in B the preimage of which is an affine subspace of R n .To be more precise we set ˜ C (1) := ˜ C and construct ˜ C ( n + 1) from ˜ C ( n ) in thefollowing way:We will show inductively that ˜ C ( n ) is again closed, convex and foliated by F .Denote its dimension by m ( n ) and write ∂ ˜ C ( n ) for its boundary with respect to the m ( n )-dimensional affine subspace containing it. If this boundary is nonempty let˜ C ( n + 1) be the set of those points in ˜ C ( n ) whose distance from ∂ ˜ C ( n ) is maximal.More formally: For p in ˜ C ( n ) define ρ n ( p ) to be the distance function d ∂ ˜ C ( n ) ( p )relative to ∂ ˜ C ( n ) and let R ( n ) be the maximum of ρ n on ˜ C ( n ). Then ˜ C ( n + 1) isthe R ( n )-level set of ρ n . Remark.
Note the equality ρ n = d ∂ ˜ C ( n ) = d π ( ∂ ˜ C ( n )) ◦ π. Since ˜ C ( n ) is closed and foliated by F we get that π ( ˜ C ( n )) is closed and thuscompact being a subset of C . Hence, ρ n does indeed have a maximum, which ispositive since ˜ C ( n ) has nonempty interior. Proposition 2.5.
For any n ∈ N the set ˜ C ( n + 1) (if defined) is closed, convexand foliated by F . Moreover, if ∂ ˜ C ( n + 1) is nonempty, then it too is foliated by F .Finally, the dimension of ˜ C ( n + 1) is strictly less than that of ˜ C ( n ) . Proof.
Obviously, ˜ C ( n + 1) is closed as it is a level set of ρ n .To show its convexity assume p , p to lie in˜ C ( n + 1). By definition, B VR ( n ) ( p i ) is then con-tained in ˜ C ( n ), where V is the minimal affine sub-space containing ˜ C ( n ). By the latter’s convexity, the convex hull of the two balls is also contained in ˜ C ( n ) and hence also the balls B VR ( n ) ( q ) where q is any point on the line segment [ p p ]. Thus, [ p p ] is containedin ˜ C ( n + 1).We now show that ˜ C ( n +1) is foliated by F . We begin by showing this propertyfor the auxiliary setˆ C ( n + 1) := n p ∈ R n (cid:12)(cid:12)(cid:12) dist (cid:16) p, ∂ ˜ C ( n ) (cid:17) = R ( n ) o . Now dist (cid:16) p, ∂ ˜ C ( n ) (cid:17) = min F ( d F ( p )) , where the minimum is taken over all leaves F in ∂ ˜ C ( n ). As we have observedbefore, due to π being a submetry we get d F ( p ) = | π ( p ) π ( F ) | , which is constantalong the leaf through p . But this also holds for the minimum over all leaves F in ∂ ˜ C ( n ), so for p ∈ ˆ C ( n + 1) the whole leaf through p is contained in this set.Observe that ˜ C ( n + 1) = ˆ C ( n + 1) ∩ ˜ C ( n ) and the intersection of two setsfoliated by F is also foliated.Then ∂ ˜ C ( n + 1) being foliated by F is an immediate consequence of Lemma 2.4.Obviously m ( n + 1) ≤ m ( n ), so assume equality holds. Then ˜ C ( n + 1) hasinterior points with respect to the minimal affine subspace V containing ˜ C ( n ).Thus, ˜ C ( n + 1) contains some ball B Vε ( p ). But clearly there are points in this ballthat are closer to the boundary of ˜ C ( n ) than p , which is a contradiction. (cid:3) This implies that our recursive construction terminates at some n — at the verylatest, when ˜ C ( n ) is a point. The final ˜ C ( n ) then is a closed, convex subset of R n without boundary, i.e. an affine subspace, foliated by F . So, restricting ourselvesto this subspace, we have a submetry from a Euclidean space onto π ( ˜ C ( n )). Remark. In R n convexity and total convexity are the same. Hence, the set π ( ˜ C ( n )),i.e. the soul of B is a totally convex subset of B (since we can lift geodesics) and sois again an Alexandrov space of nonnegative curvature.Observe that indeed any nonnegatively curved finite dimensional Alexandrovspace X that is complete and unbouned has a soul, which can be obtained by thesame construction as above. The only ingredient still needed in that constructionis the fact that C ( n ) is convex, which follows from the fact that the distance to ∂C ( n −
1) is concave. This was proven (together with an Alexandrov space versionof the soul theorem) by Perelman in 1991. A proof of the above mentioned concavityresult can be found in [AB03, Thm. 1.1(3B)] (as far as we know Perelman’s resultstill exists only as a preprint).
Let F be an equidistant foliation of R n and assume the space of leaves B to becompact. We will show that B has to be a point.Assume, for now, that B is not a point. Since B is compact it has finite diameterdiam ( B ) >
0. So for any leaf F of F the closed diam ( B )-tube τ diam( B ) ( F ) := { p ∈ R n | d F ( p ) ≤ diam ( B ) } around F is R n .Consider a regular leaf F . Let ξ p be a unit normal vector in H p , p ∈ F , anddenote by ξ its Bott-parallel continuation along F . We denote by F t the leaf F + tξ .2. SUBMETRIES ONTO COMPACT ALEXANDROV SPACES 21 through p t := p + tξ p . Note that if F t is regular then ξ t with ξ t ( q + tξ q ) := ξ ( q ) isBott-parallel along F t . Remark.
Using Proposition 1.16 we can always make sure that F t is regular bypassing from t to t + ε , if necessary, for sufficiently small ε > S ξ t and O ∗ ξ t on F t in terms of S ξ and O ∗ ξ on F :Let γ be a smooth curve on F with γ (0) = p , ˙ γ (0) = v and denote by γ t itsBott-parallel translate γ t ( s ) := γ ( s ) + tξ ( γ ( s )). Recall from Remark 1.19 that γ t isa smooth curve on F t and we get γ t (0) = p t and ˙ γ t (0) = v + t ∇ v ξ =: v t .Using this we calculate S ξ t v t = − ( ∇ v t ξ t ) v = − (cid:18) ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 ξ t ( γ t ( s )) (cid:19) v = − ( ∇ v ξ ) v and O ∗ ξ t v t = − ( ∇ v t ξ t ) h = − (cid:18) ∂∂s (cid:12)(cid:12)(cid:12)(cid:12) s =0 ξ t ( γ t ( s )) (cid:19) h = − ( ∇ v ξ ) h with the vertical and horizontal parts taken with respect to F t . Since k v t k = (cid:13)(cid:13) v − tS ξ v − t O ∗ ξ v (cid:13)(cid:13) = k ( I − tS ξ ) v t k + t (cid:13)(cid:13) O ∗ ξ v t (cid:13)(cid:13) = k v k − t h v, S ξ v i + t k S ξ v k + t (cid:13)(cid:13) O ∗ ξ v (cid:13)(cid:13) (2.1)we see that both S ξ t v t k v t k and O ∗ ξ t v t k v t k tend to zero as t goes to infinity.Hence the leaves F t become more “flat” as t increases. To formalize this weintroduce the following notation:For any leaf G ∈ F let B GR ( p ) be the intrinsic metric ball in G around p of radius R . Furthermore, we will denote by E ( p, ξ ) the hyperplane through p with normal vector ξ and by E ε ( p, ξ ) the ε -tube around E ( p, ξ ), i.e. E ε ( p, ξ ) = { x ∈ R n | | h x − p, ξ i | < ε } . Proposition 2.6.
Let R and δ be positive, then for sufficiently large t theclosed intrinsic balls B F t R ( p t ) are contained in E δ ( p t , ξ t ) . Proof.
Let c t be a curve parameterized by arclength on F t starting in p t . Wedefine X t ( s ) := c t ( s ) − p t and ξ t ( s ) := ξ t ( c t ( s )). So, we only have to check that forsufficiently large t the estimate | h X t ( s ) , ξ t (0) i | < δ holds for all s ≤ R .Observe first that we can pull back this construction to F via the map q q + tξ .So, there is a curve ˜ c t on F such that c t ( s ) = ˜ c t ( s ) + ˜ ξ t ( s ) where we define ˜ ξ t ( s ) tobe ξ (˜ c t ( s )). Part 1.
The main step is to show that ˙ ξ t ( s ) = ˙˜ ξ t ( s ) = ∇ ˙˜ c t ( s ) ξ tends uniformlyto zero as t goes to infinity. We first show this convergence pointwise:For any fixed s ∈ [0 , R ] we apply Equation (2.1) to our situation:(2.2) 1 = k ˙ c t ( s ) k = k ( I − tS ξ ) w t k + t (cid:13)(cid:13) O ∗ ξ w t (cid:13)(cid:13) where we have used w t as a shorthand for ˙˜ c t ( s ). Obviously this implies that O ∗ ξ w t tends to zero as t goes to infinity. Figure 2.1.
In the proof of Proposition 2.6 we can pull back theconstruction on F t to the original leaf F .On the other hand the eigenvalues of ( I − t S ξ ) are 1 − t λ i where the λ i arethe eigenvalues of S ξ . So, for any λ i = 0 we get 1 − t λ i → ±∞ as t → ∞ and hencethe projection ( w t ) i of w t to the eigenspace of S ξ corresponding to λ i tends to zeroas t goes to infinity. Note that this argument only works because the eigenvaluesof S ξ are constant along F . Remark 2.7.
Observe that ∇ ξ is uniformly bounded on F , i.e. there is aconstant C such that k∇ v ξ k ≤ C k v k . This is obviously true pointwise. Considerthen k∇ v ξ k = k S ξ v k + (cid:13)(cid:13) O ∗ ξ v (cid:13)(cid:13) . The first term is bounded by (max {| λ i |} ) · k v k and the λ i are constant along F .For the second term consider any η ∈ B and observe that (cid:10) O ∗ ξ v, η (cid:11) = h v, O ξ η i ≤ kO ξ η k k v k and kO ξ η k is again constant along F .Now suppose λ = 0 then our conclusions from Equation (2.2) imply (cid:13)(cid:13)(cid:13) ˙˜ ξ t ( s ) (cid:13)(cid:13)(cid:13) = k∇ w t ξ k = (cid:13)(cid:13)(cid:13) ∇ ( P i =0 ( w t ) i ) ξ − O ∗ ξ ( w t ) (cid:13)(cid:13)(cid:13) ≤ X i =0 (cid:13)(cid:13) ∇ ( w t ) i ξ (cid:13)(cid:13) + (cid:13)(cid:13) O ∗ ξ ( w t ) (cid:13)(cid:13) and the last term tends to zero as t → ∞ as we have seen. The remaining termstend to zero as well because of Remark 2.7.But this also implies uniform convergence (cid:13)(cid:13)(cid:13) ˙˜ ξ t ( s ) (cid:13)(cid:13)(cid:13) → (cid:13)(cid:13)(cid:13) ˙˜ ξ t ( s ) (cid:13)(cid:13)(cid:13) is definedon the compact interval [0 , R ]. In particular we can choose t large enough such that (cid:13)(cid:13)(cid:13) ˙ ξ t ( s ) (cid:13)(cid:13)(cid:13) < εR uniformly in s . .2. SUBMETRIES ONTO COMPACT ALEXANDROV SPACES 23 Part 2.
We return to proving the assertion of the proposition:Writing ξ t ( s ) = R s ˙ ξ t ( σ ) dσ + ξ t (0) we get h ξ t ( s ) , ξ t (0) i = 1 + Z s D ˙ ξ t ( σ ) , ξ t (0) E dσ and the modulus of the integrand is bounded by εR . Hence h ξ t ( s ) , ξ t (0) i is containedin the interval (1 − ε, ε ).As a consequence we get the estimate k ξ t ( s ) − ξ t (0) k = k ξ t ( s ) k + k ξ t (0) k − h ξ t ( s ) , ξ t (0) i < ε since ξ t ( s ) is a unit vector for any s . Moreover we can write h ξ t ( s ) , X t ( s ) i = Z s (cid:18) ddσ h ξ t ( σ ) , X t ( σ ) i (cid:19) dσ since X t (0) = 0 and (cid:12)(cid:12)(cid:12)(cid:12) dds h ξ t ( s ) , X t ( s ) i (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)D ˙ ξ t ( s ) , X t ( s ) E(cid:12)(cid:12)(cid:12) < (cid:13)(cid:13)(cid:13) ˙ ξ t ( s ) (cid:13)(cid:13)(cid:13) · R < ε since ˙ X t ( s ) = ˙ γ t ( s ) ⊥ ξ t ( s ).So, | h ξ t ( s ) , X t ( s ) i | < ε · R . Hence, we can finally show | h X t ( s ) , ξ t (0) i | = | h X t ( s ) , ξ t ( s ) + ( ξ t (0) − ξ t ( s )) i | < εR + √ εR. Choosing ε sufficiently small proves our claim. (cid:3) Note that since ξ t is a Bott-parallel normal field along F t the assertion ofProposition 2.6 holds for every point of F t .Now, consider a sequence t n with t n → ∞ and denote by F n the leaf F t n .By compactness of the base B we may assume π ( F n ) to converge in B . We willcall the fibre over this limit ˜ F . The compactness of B also implies that the closedball B diam( B ) ( p ) meets all leaves. Choose now a sequence p n in B diam( B ) ( p ) with p n ∈ F n . Remember that ξ n := ξ t n ( p n ) is a unit vector for any n . By passing onto subsequences we may assume that p n converges towards some point ˜ p ∈ ˜ F and ξ n ( p n ) → ˜ ξ (˜ p ) for some unit vector ˜ ξ with base point ˜ p . We do not care if ˜ ξ iscontained in ν ˜ p ˜ F . Proposition 2.8.
The limit leaf ˜ F is contained in the hyperplane E (˜ p, ˜ ξ (˜ p )) . Proof.
Let ˜ γ : [0 , R ] → ˜ F be a simple curve parameterized by arclength start-ing at ˜ p . By Lemma 1.20 we may extend the velocity ˙˜ γ to a vertical vector field V in some neighborhood of the image of ˜ γ . Since the latter is compact we may choosethis neighborhood to be some compact tube around the image of ˜ γ .Choose some point ˜ q := ˜ γ ( t ) lying on ˜ γ and let γ n be the integral curve of V starting at p n . Using standard theory of ordinary differential equations we see thatchoosing p n sufficiently close to ˜ p implies that q n := γ n ( t ) is arbitrarily close to ˜ q and also the length of γ n | [0 ,t ] is arbitrarily close to t , in particular it is less than2 R , say.Let then 0 < ε < R and choose n to be sufficiently large such that the followinginequalities hold: B F n R ( p n ) ⊂ E ε ( p n , ξ n ) , k p n − ˜ p k < ε, (cid:13)(cid:13)(cid:13) ˜ ξ − ξ n (cid:13)(cid:13)(cid:13) < ε Figure 2.2.
The curve ˜ γ from the proof of Proposition 2.8 iscontained in the blown up hyperplane E ε ( p n , ξ n ).and increase n even further if necessary such that the aforementioned properties k q n − ˜ q k < ε, L (cid:0) γ n | [0 ,t ] (cid:1) < R also hold.This implies that ˜ q is contained in the blown up hyperplane E ε ( p n , ξ n ): | h ˜ q − p n , ξ n i | = | h (˜ q − q n ) + ( q n − p n ) , ξ n i | < ε, which in turn shows that ˜ q lies in the hyperplane E (˜ p, ˜ ξ ) because (cid:12)(cid:12)(cid:12)D ˜ q − ˜ p, ˜ ξ E(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)D (˜ q − p n ) + ( p n − ˜ p ) , ξ n + ( ˜ ξ − ξ n ) E(cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) h ˜ q − p n , ξ n i + D ˜ q − p n , ˜ ξ − ξ n E + h p n − ˜ p, ξ n i + D p n − ˜ p, ˜ ξ − ξ n E(cid:12)(cid:12)(cid:12) < ε + ( ε + R ) ε + ε + ε for arbitrarily small ε > q ∈ B ˜ FR (˜ p ) and indeed for any radius R it follows thatthe whole leaf ˜ F is contained in the hyperplane E (˜ p, ˜ ξ ). (cid:3) Now ˜ F being contained in a hyperplane means that it cannot be diam ( B )-closeto every point in R n . So B must be a point. Thus we have shown: Theorem 2.9.
Let F be an equidistant foliation of R n and suppose the spaceof leaves B to be compact. Then B is a single point.Remark. Observe that the leaves of F being connected, as we assumed in def-inition of an equidistant foliation, is essential for this assertion to hold. A simple .2. SUBMETRIES ONTO COMPACT ALEXANDROV SPACES 25 counterexample to the theorem, dropping connectedness, is given by the covering f : R → S , t e it . We denote the affine leaf of F by F and for the rest of this thesis we assumethat F = R k × { } ⊂ R k + n . Remark.
We end this chapter by observing that due to F being equidistant theaffine leaf F of course is the most singular leaf of F , i.e. the dimension of F issmallest.Note also that we may assume F to be unique. For assume there is anotheraffine leaf, F ′ say, then F and F ′ are parallel and the affine space A , spanned bythem is foliated by leaves of F parallel to F . Observe that A ∼ = R k + n is spannedby F and a line l = p + span [ v ] meeting F and F ′ perpendicularly.Consequently, for each point p ∈ l the or-thogonal complement p + span [ v ] ⊥ to the line l is invariant under F , i.e. each leaf meeting thatspace is contained in it. Since F is equidistant,the restrictions of F to any two such perpendic-ular complements B and B ′ of l differ only bya parallel translation along l . Hence, F is theproduct of the induced foliation of B and thediscrete foliation of the line l .HAPTER 3 The Induced Foliation in the Horizontal Layers
The existence of the affine leaf F leaves us in the special situation that F together with the horizontal distribution along F induces a further, refined foliation˜ F of R n + k by intersecting the leaves of F with the normal spaces of F .We first look at the homogeneous case and show that F is given by the orbits of G × R k with G compact and R k acting on R k + n by generalized screw motions aroundthe axis F . In particular we conclude that the induced foliation ˜ F is equidistant.In the remainder of this chapter we examine how much of this rather nicestructure of ˜ F can be recovered in the general case.For any p ∈ F we denote the affine space p + H p by L p and call it the horizontallayer through p . Definition 3.1.
For any p ∈ F we will denote by ˜ F p the foliation of L p induced by F , i.e. ˜ F p := { F ∩ L p | F ∈ F} . Consequently, the union ˜ F over all ˜ F p , where p is in F , is a foliation of R n + k . Wedenote the leaf F ∩ L p of ˜ F p by ˜ F p .Note that we have to make sure that the L p intersect the leaves of F as transver-sally as possible.Let us first introduce some tools and notation used throughout this chapter. Projections onto the affine leaf.
Let Ξ be the vector field on R k + n indicat-ing the position relative to F , i.e. for x = ( x , x ) ∈ R k + n we set Ξ x := (0 , − x ).Obviously, the shortest path from a point x to F is given by t x + t Ξ x , hencethe restriction of Ξ to the regular part of F is a Bott-parallel horizontal field. Definition 3.2.
Let P : R k + n → F be the orthogonal projection onto theaffine leaf F . We denote by P v and P h the restriction of P ∗ to the vertical andhorizontal distributions respectively.We can easily describe these projections using Ξ since P x = x + Ξ x . Conse-quently, its derivative is given by(3.1) P ∗ X = X + ∇ X Ξ , for any vector X . Lemma 3.3.
Let F be a regular leaf and ξ, η two Bott-parallel vector fieldson F . Then (cid:10) P h ξ, P h η (cid:11) is constant along F . Proof.
This follows immediately from the proof of Lemma 1.20 and Re-mark 4.3. (cid:3)
278 3. THE INDUCED FOLIATION IN THE HORIZONTAL LAYERS
By Lemma 1.20 the projection P v is surjective at any regular point of thefoliation F , which enables us to lift any tangent vector field on F to one on theregular leaves of F . Definition 3.4.
Let v be a vector in T p F and let x be a point in a regularleaf F ∈ F such that P x = p . We will call the unique vector L x ( v ) ∈ (ker P vx ) ⊥ ⊂ T x F such that P v L x ( v ) = v the vertical lift of v to x .After this digression we show that ˜ F is indeed a smooth foliation. Lemma 3.5.
For any p ∈ F the leaves of ˜ F p are complete smooth submani-folds of L p . Proof.
Let us first look at a regular leaf F of F . By Lemma 1.20 every p ∈ F is a regular value of the orthogonal projection P | F : F → F so the preimage ˜ F p of p is a smooth submanifold of F .To deal with the singular leaves of F note that we will show in Proposition 3.13that ˜ F p is equidistant. To be more precise, for any p ∈ F the restriction to L p of π ∗ p is a submetry and its fibres are the leaves of ˜ F p .But the regular fibres being smooth submanifolds already implies the sameproperty for the singular fibres (cf. [Lyt02, Prop. 13.5]). (cid:3) In order to understand the role of the induced foliation ˜ F better let us firstconsider the homogeneous case. So, in this section we assume the fibres of F tobe the orbits of a connected Lie group G ⊂ Isom( R k + n ) acting effectively on R k + n with F = R k × { } being its most singular orbit.Obviously, for any p ∈ F the foliation ˜ F p is then given by the orbits of theslice representation of G p . Hence, each ˜ F p is equidistant and since the isotropygroups along a fibre are conjugate any two ˜ F p and ˜ F q are isometric to each other.Moreover, we show: Theorem 3.6.
In the homogeneous case the induced foliation ˜ F is equidistant. To achieve this we must take a closer look on how G acts on R k and R n respectively. Since G leaves the affine space F invariant it must be a subgroup ofIsom( R k ) × SO( n ) = (cid:26)(cid:18)(cid:18) A B (cid:19) , (cid:18) a (cid:19)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12) A ∈ SO( k ) , B ∈ SO( n ) , a ∈ R k (cid:27) where any g ∈ G acts on ( x, y ) ∈ R k × R n via (cid:18)(cid:18) A B (cid:19) , (cid:18) a (cid:19)(cid:19) . ( x, y ) = ( Ax + a, By ) . Remark.
Consider the two natural projections P : G → Isom( R k ) and P : G → SO( n ) , both of which are continuous group homomorphisms. Note that P i ( G ) may not bea closed group. We will use the following notation: .1. THE HOMOGENEOUS CASE 29 We denote the kernel of P i by N i . For any subgroup H of G we will use ˆ H and˜ H for its image under the projections P and P respectively.We start by proving a reducibility result. Lemma 3.7.
Either N is trivial or F splits off a Euclidean factor. Proof.
Assume that N is not trivial. Observe that since N = (cid:26)(cid:18)(cid:18) A E (cid:19) , (cid:18) a (cid:19)(cid:19) (cid:12)(cid:12)(cid:12)(cid:12) ( A, a ) ∈ P ( G ) (cid:27) the projection P | N : N → ˆ N is an isomorphism.Consider the action of ˆ N on R k . By Theorem 2.1 one of the orbits of thisaction is an affine space A , which we may assume without loss of generality to passthrough the origin.Remember that ˆ G acts transitively on R k . Let x be an arbitrary point in R k and let g ∈ ˆ G be such that g. x . Since N is a normal subgroup of G we getˆ N ⊳ ˆ G . Thus the ˆ N -orbit passing through x , given byˆ N .x = ˆ N .g. g. ˆ N . g. A , is also an affine space, which we denote by A x . By the equidistance of the orbitsof ˆ N all these affine spaces A x must be parallel.Remember that N acts trivially on R n . So for any ( x, y ) ∈ R k + n the N -orbitthrough ( x, y ) is just the affine space ( x, y ) + A × { } . Hence, F splits off theEuclidean factor A × { } .Suppose that A = { } . Then N acts trivially on R k since ˆ N does. So N istrivial as we assumed the action of G to be effective. (cid:3) Remark 3.8.
In the following we will concentrate on the case of P being anisomorphism by passing on to the reduced foliation if necessary. Lemma 3.9.
The isotropy group G is equal to N and the projection ˆ G of G is abelian. Proof.
According to Remark 3.8 we have G ∼ = ˜ G = P ( G ) and since ˜ G iscontained in the compact Lie group SO( n ) we get the following decomposition forthe Lie algebra g of G :(3.2) g = z ⊕ g ′ , where z is its center and g ′ is semisimple. Remark.
Note that a priori we only get the decomposition g = rad g ⊕ h , where rad g is the solvable radical of g and h is semisimple (cf. [Var74, Thm. 3.8.1]).Let then R be the connected Lie group corresponding to rad g and consider its imageunder P .Obviously P ( R ) is solvable, i.e. there is a chain { } =: G ⊳ . . . ⊳ G n := P ( R ) ofnormal subgroups such that subsequent quotients G i +1 /G i are abelian. But clearlyby continuity of the group operations the property of being a normal subgroup ispreserved if we take closures and the subsequent quotients of the G i remain abelianas well. Hence, P ( R ) is solvable. But as a compact Lie group this can only be thecase if it is abelian. Now P ( G ) is contained in the normalizer of P ( R ) and actson P ( R ) by conjugation. But since P ( R ) is a torus its automorphism group isdiscrete. Also note that P ( G ) is connected, so P ( G ) is in fact contained in thecentralizer of P ( R ).In particular P ( R ) lies in the center of P ( G ) and hence R lies in the centerof G because P is a group isomorphism. The reverse inclusion follows from thedefinition of R .Let G ′ be the unique connected Lie subgroup of G corresponding to the Liealgebra g ′ . Note, that the decomposition (3.2) implies G ′ to be a normal subgroupof G .Observe that G ′ is a semisimple subgroup of Isom( R k + n ) = R k + n ⋉ SO( k + n )and consider the natural projection P : Isom( R k + n ) → SO( k + n ). Note that P isa Lie group homomorphism.Now the Lie algebra g ′ decomposes into a sum of simple Lie algebras g ′ i . Each g ′ i is either mapped to zero or to an isomorphic image of g ′ i . But P ∗ ( g ′ i ) = 0 meansthat the corresponding connected Lie group G ′ i consists only of translations of R k + n and hence is solvable, which contradicts G ′ being semisimple.So g ′ is isomorphic to a subalgebra of so ( n ) and hence G ′ is compact. Inparticular it has a fixed point ( x, y ) ∈ R k + n . Consequently, ˆ G ′ leaves x ∈ R k invariant. Remark 3.10.
Let G be any group acting transitively on some space X andsuppose H to be a normal subgroup of G which is contained in the isotropy group G x of some point x ∈ X . Then H acts trivially on X .To see this observe that the H -orbit passing through some y ∈ X is given by H.y = H.g.x = g.H.x = g.x = y, for some g ∈ G since G acts transitively.This implies that ˆ G ′ acts trivially on R k , i.e. G ′ is contained in N . So N hasLie algebra t ⊕ g ′ for some subalgebra t of z . Since ˆ G = P ( G ) ∼ = G/N and G/N has Lie algebra ( z ⊕ g ′ ) / ( t ⊕ g ′ ) it follows that ˆ G is abelian.Thus ˆ G is a normal subgroup of ˆ G and Remark 3.10 implies G ⊂ N , whichfinishes the proof. (cid:3) In particular this means that the isotropy group G p does not depend on thechoice of p ∈ F , hence, the induced foliations ˜ F p are equal up to parallel transportalong F , which proves Theorem 3.6.Also, this provides a convenient way to describe the action of G on R k + n : Proposition 3.11. If F is irreducible there exists a Lie group homomorphism Φ : R k → Centr( ˜ G ) into the centralizer of ˜ G relative to SO( n ) such that the orbitsof G are of the form (3.3) G. ( x, y ) = n ( x + v, Φ( v ) . ˜ G .y ) (cid:12)(cid:12)(cid:12) v ∈ R k o . Remember that G acts trivially on R k , thus ˜ G is just the trivial embeddingof G into Isom( R n ) and hence a Lie group. .1. THE HOMOGENEOUS CASE 31 Let us first show that ˆ G and thus G act on R k by translations. Since the actionof ˆ G on R k has trivial isotropy and ˆ G is abelian it suffices to prove: Lemma 3.12.
Let H be an abelian group acting simply transitively on R m byIsometries. Then H acts by translations. Proof.
Remember that H may be viewed as a subgroup ofIsom( R m ) = { ( A, a ) | A ∈ O( m ) , a ∈ R m } with the group multiplication given by ( A, a ) ◦ ( B, b ) = (
AB, a + Ab ).Since H acts simply transitively any h = ( A, a ) ∈ H is uniquely determinedby its translational part, i.e. there is a group homomorphism ϕ : R m → O( m ) suchthat any h ∈ H is of the form h = ( ϕ ( a ) , a ) for some a ∈ R m .Define V := ker ϕ and V := V ⊥ . Observe that since H is abelian the dimensionof its image under ϕ is at most the rank of O ( m ) which is strictly less than m so V has positive dimension.Assume V to be non-trivial. Let v ∈ V with v = 0 and w ∈ V . The group H being abelian then implies( ϕ ( v ) ϕ ( w ) , v + ϕ ( v ) w ) = ( ϕ ( w ) ϕ ( v ) , w + ϕ ( w ) v ) . In particular, using w ∈ ker ϕ , this means v + ϕ ( v ) w = w + v for all w ∈ V . So, ϕ ( H ) acts trivially on V and thus the image of ϕ is contained in O( V ).This yields the group homomorphism ϕ | V : V → O( V ) which by the aboverank argument must have a non-trivial kernel. But this contradicts ϕ | V beinginjective so V must be trivial. (cid:3) As a consequence of this and because G is a normal subgroup of G any elementof G is uniquely determined by a translation on R k up to multiplication with G .This yields a homomorphism φ : R k → SO( n ) such that the orbits of G are ofthe form described in (3.3). Remark.
Note that the image of φ has to be contained in Norm( ˜ G ) since G is a normal subgroup of G . But it need not, in general, be contained in Centr( ˜ G ).In fact, the map Φ we construct in the following may lead to a different groupaction, which, however, is orbit equivalent to that of G . Proof of Proposition 3.11.
Let us first take a look at φ at the level ofLie algebras: φ ∗ : R k → n maps the abelian Lie algebra R k into the normalizer n := norm (˜ g ) (relative to so ( n )) of the Lie algebra ˜ g of ˜ G . ✁✁✁✁✁✁✁✁✁ n := norm (˜ g ) ˜ g (˜ g ) ⊥ φ ∗ R k Consider the natural projection P : Norm( ˜ G ) → Norm( ˜ G ) / ˜ G and its derivative P ∗ e : n → n / ˜ g . Wemay assume that P ∗ e ◦ φ ∗ is injective for otherwise F would split off the kernel of P ∗ e ◦ φ ∗ as a Euclideanfactor (cf. also Section 4.2).Now n / ˜ g is canonically isomorphic to (˜ g ) ⊥ , withthe orthogonal complement taken with respect to theKilling form in n . And since ˜ g is an ideal of n sois (˜ g ) ⊥ (cf. [Hel78, Chap. 6]). Hence, (˜ g ) ⊥ is con-tained in the centralizer of ˜ g , in fact centr (˜ g ) = (˜ g ) ⊥ ⊕ z (˜ g ), where z (˜ g ) is the center of ˜ g . Thus there is a Lie algebra homomorphism ˜Φ : R k → centr (˜ g ) such that thefollowing diagram commutes: R k φ ∗ / / ˜Φ (cid:15) (cid:15) n P ∗ e (cid:15) (cid:15) centr (˜ g ) ⊃ (˜ g ) ⊥ n / ˜ g ∼ = o o And since R k is simply connected we can lift ˜Φ to a Lie group homomorphism Φfrom R k to the connected component of Centr( ˜ G ).By this construction we get for any v ∈ R k that Φ( v ) is equal to φ ( v ) upto multiplication by some element in ˜ G , which implies that the orbits of G mayindeed be written in the form (3.3). (cid:3) We have seen that in the homogeneous case each of the induced foliations ˜ F p is equidistant. This holds in general for equidistant foliations of R k + n : Proposition 3.13.
For any point p in the affine leaf F the induced foliation ˜ F p of the horizontal Layer L p is equidistant. Proof.
Let ¯ π be the restriction π ∗ p : H p → T π ( p ) B of the differential π ∗ p tothe horizontal space at p ∈ F . By Proposition 1.10 we know that ¯ π is a submetry,hence its fibres are equidistant.Now H p is isometric to L p via the normal exponential map at p . And fromSection 1.2.2 we know that ¯ π maps a vector h to ¯ h if and only if π maps the geodesicwith starting direction h to that with starting direction ¯ h . So the exponential mapcommutes for horizontal directions with the differential of the submetry. Hence,the fibration of H p by ¯ π is isometric to the induced foliation ˜ F p . (cid:3) We have seen that each individual induced foliation ˜ F p is equidistant. In thissection we will examine how these foliations change if we move along the affineleaf F . Remark 3.14.
Unless we say otherwise we will from now on assume each ofthe induced foliations ˜ F p to be homogeneous. In particular, for each p ∈ F theleaves of ˜ F p are the orbits of some connected closed subgroup G of SO( n ), where G acts on R n via the restriction of the standard representation of SO( n ).Note that for any fixed p this group G need not be unique. Therefore we willpass on to the maximal group that has the same orbits. Definition 3.15.
Let G ⊂ SO( n ) be a closed connected Lie group acting on R n via the restriction of the standard representation of SO( n ). Then G max := { g ∈ SO( n ) | g ( Gx ) = G. ( gx ) , ∀ x ∈ R n } is the maximal connected Lie subgroup of SO( n ) having the same orbits as G . .3. THE INDUCED FOLIATIONS IN DISTINCT HORIZONTAL LAYERS 33 By definition G max leaves the orbits of G invariant and acts transitively onthem, since G ⊂ G max . A straightforward calculation shows that G max is indeed aLie group.We will denote the maximal connected subgroup of SO( n ) whose orbits are theleaves of ˜ F p by G p . This notation already suggests that G p is just the isotropygroup of p in case F is homogeneous. Proposition 3.16.
For any p, q ∈ F the induced foliations ˜ F p and ˜ F q areisometric to each other. We first show that ˜ F p and ˜ F q are diffeomorphic to each other. Next we provethat ˜ F p → ˜ F q in a suitable way as p → q . We conclude then that G p and G q haveto be in the same conjugation class and G p → G q as p tends to q . Lemma 3.17.
Let p and q be any two points in F then ˜ F p and ˜ F q are diffeo-morphic to each other. Proof.
Consider a parallel vector field V on F such that p + V p = q and itsvertical lift L ( V ) to R k + n as introduced in Definition 3.4. By construction theflow φ t of L ( V ) maps horizontal layers onto each other preserving the leaves of F ,in particular p is mapped to q for t = 1. This yields the desired diffeomorphism. (cid:3) As we have seen in the previous section the induced foliations ˜ F p are equidis-tant, so it makes sense to contemplate the restriction of this foliation to the unitsphere in L p based at p even if we drop the homogeneity assumption made inRemark 3.14. We will denote this restriction by ˜ F p . Lemma 3.18.
Let ( p j ) be a sequence in F with p j → p ∈ F . Then ˜ F p j converges uniformly in Hausdorff distance towards ˜ F p .Remark. By ˜ F p j d H −→ ˜ F p we mean the following. Let us identify all horizontallayers L p by parallel translation along F . Thus we understand the ˜ F ⋆ to befoliations on the same euclidean space R n . Then ˜ F p j tends to ˜ F p in the Hausdorffdistance if and only if for any leaf F ∈ ˜ F p there is a sequence of leaves F j ∈ ˜ F p j such that F j d H −→ F and this convergence is uniform in the leaves F . Proof.
First we show that ˜ F p j converges towards ˜ F p leafwise, i.e. for any leaf F in F the leaves ˜ F p j tend towards ˜ F p in Hausdorff distance.For any j ∈ N consider the vertical lift γ j,x of the line segment pp j through x ∈ ˜ F p , which gives us the estimate d H (cid:16) ˜ F p , ˜ F p j (cid:17) ≤ max x ∈ ˜ F p L ( γ j,x ) . Using the lifting map L : R k + n × T F → T R k + n we can express the length of γ j,x via L ( γ j,x ) = Z (cid:13)(cid:13) L γ j,x ( t ) ( p j − p ) (cid:13)(cid:13) dt. By construction L is linear in its second argument. So L x ( p j − p ) tends to zero forfixed x as j tends to infinity. Since L is continuous this convergence is uniform in K × S n − , where K is any compact neighbourhood of p in F . Hence, ˜ F p j d H −→ ˜ F p and this convergence is uniform in the choice of F ∈ F . (cid:3) Remark.
Whereas the homogeneity assumption for the foliations ˜ F p was notnecessary for the previous two lemmas it is essential for the following arguments.Choose any biinvariant metric on G and consider the space S ( G ) of closedsubgroups of G equipped with the Hausdorff metric. Compactness of G impliesthat S ( G ) is compact as well. To see this consider the following: Remark 3.19.
For any compact metric space X the set M ( X ) of all closed sub-sets of X equipped with the Hausdorff distance is compact (cf. [BBI01, Thm. 7.3.8,p. 253]).Suppose A j → A in M ( X ) then A is the set of all limits of all sequences( a j ) ∈ X such that a j ∈ A j (cf. [BBI01, p. 253]).Now, suppose ( H j ) ∈ S ( G ) converges to H ∈ M ( G ). The previous remark thenclearly implies that the 1-element of G is in H . And since all of the H j are groupsso is H by continuity of the group operations. Thus, S ( G ) is a closed subset of M ( G ) and so is compact as well. Lemma 3.20.
Let G and G j , with j ∈ N , be closed Lie subgroups of SO( n ) and let F G and F G j be the foliations of S n − by the orbits of G and G j respectively.Assume the group actions to be the restrictions of the standard representation of SO( n ) and assume further that G = G max and G j = G max j for all j . Then, theuniform convergence of F G j towards F G in Hausdorff distance implies G j d H −→ G . Proof.
Since S (SO( n )) is compact we may assume for the moment withoutloss of generality that G j converges to some Lie subgroup H ⊂ SO( n ).The main part of this proof is to show that H is contained in G , which isto say that H leaves the orbits of G invariant. Assume the contrary, i.e. there isan h ∈ H and a point x ∈ S n − such that hx / ∈ Gx . By Remark 3.19 we get asequence g j ∈ G j tending to h . The uniform convergence of F G j towards F G thenimplies that the distance between g j x and Gx tends to zero, which contradicts ourassumption.Note that by an analogous argument H acts transitively on the leaves of F G ,which implies H max = G . But since all G j are maximal so is their limit, hence H = G .To finish the proof we drop the convergence assumption on ( G j ). Since anysubsequence of ( G j ) contains itself a convergent subsequence and the limit of theseis always G we get that G j d H −→ G . (cid:3) Hence the map F → S (SO( n )) , p G p is continuous and we finish the proof of Proposition 3.16 by showing: Lemma 3.21.
The conjugacy classes in G are the path connected componentsof S ( G ) . Proof.
Let us denote the conjugacy classes in G by ( K α ) α ∈ A . Obviously S ( G )is the disjoint union of these K α . .4. EQUIDISTANCE OF THE LEAVES IN DISTINCT HORIZONTAL LAYERS 35 First observe that G contains only countably many conjugacy classes: Thereare only finitely many semisimple Lie subgroups up to conjugation. And the toriare characterized by the slope of their embedding in a maximal torus. Since weonly consider closed subgroups of G this slope has to be rational. Hence, up toconjugation, there are only countably many closed abelian subgroups of G . Sinceany Lie subalgebra of g is the sum of an abelian and a semisimple Lie algebra thisproves our first claim.Now, consider the action of G on S ( G ) by conjugation: g.H = gHg − , for g ∈ G and H ∈ S ( G ). Obviously, this action is continuous. Moreover G acts byisometries since the Hausdorff metric on S ( G ) is based on a biinvariant metric on G .Hence, each G -orbit in S ( G ) is compact and path connected and the orbits forman equidistant decomposition of S ( G ).On the other hand, suppose K and K to be in the same path connectedcomponent of S ( G ) and γ a path connecting them. Clearly t dist ( γ ( t ) , K ) iscontinuous but takes only values in a countable set and, hence, is constant. So, thetwo conjugacy classes are identical, proving the lemma’s assertion. (cid:3) Remark 3.22.
As a consequence we may describe the leaves of F in analogyto Equation (3.3) from Proposition 3.11. That is to say, we can find for any x ∈ R k a smooth map Ψ x : R k → SO( n ) such that the leaf F passing through ( x, y ) ∈ R k + n is given by F = (cid:8) ( x + v, Ψ x ( v ) .G x .y ) (cid:12)(cid:12) v ∈ R k (cid:9) . We stress again that this map depends on x ∈ R k but not on y .We call Ψ x a screw motion map although Ψ x need not, a priori, be a grouphomomorphism. However, we can of course choose Ψ x such that Ψ x (0) = id holds. We have seen that in the homogeneous case the induced foliations ˜ F p are thesame for every point p up to parallel translation along F . We show that in generalthis property is characterized by the behavior of the projections of Bott-parallelfields.We first introduce some more notation. Definition 3.23.
We denote by ˜ P p : R k + n → L p the orthogonal projectiononto the horizontal layer L p and by ˜ P hp : H → R n the restriction of its differentialto the horizontal distribution H .We sometimes omit the index p and write just ˜ P h if it is not important whichspecific horizontal layer we are considering. Definition 3.24.
We call F horizontally full if at every regular point x of F the map P h : H x → T P x F is surjective.Let us now examine how the projections of Bott-parallel normal fields behave.Our first result states that F and ˜ F p are “compatible” via the projection ˜ P p . Lemma 3.25.
Let F be a regular leaf of F and ξ a Bott-parallel normal fieldalong F . For any p ∈ F consider the restriction of ξ to the induced leaf ˜ F p . Thenthe projection ˜ P hp ξ of ξ to the horizontal layer L p is Bott-parallel (with respect to ˜ F p )along ˜ F p . Proof.
We refer the reader to figure 3.1 for an illustration of the constructionused in this proof.
Figure 3.1.
The projection of a Bott-parallel normal field to ahorizontal layer is Bott-parallel with respect to the induced folia-tion in that layer. .4. EQUIDISTANCE OF THE LEAVES IN DISTINCT HORIZONTAL LAYERS 37
Choose an arbitrary point x ∈ ˜ F p . Denote by ˜ ξ x the projection ˜ P hp ξ x of ξ x andby ˜ ξ its Bott-parallel continuation (with respect to ˜ F p ). The leaf ˜ F p + ˜ ξ in ˜ F p willbe called ˜ G p .Consider the curve γ : [0 , → L p with γ ( t ) = x + t ˜ ξ x and denote its endpointby y . In the following we will examine the image of γ under both π and π ∗ p .Since the image of γ is a horizontal shortest path in ˜ F p it is mapped by π ∗ p toa shortest path in the tangent cone T π ( F ) B .Note that in general this might only yield a quasi-geodesic in T π ( F ) B but we geta proper geodesic if γ is sufficiently short. Since our argument works for arbitrarysmall | ξ x | > α t : s p + s ( γ ( t ) − p )are horizontal shortest paths with respect to F so π maps them to shortest pathsin B . Again we may have to assume the image of γ to be close to p , which we cando without loss of generality since the assertion we want to prove is left invariantby dilating radially from F .So the curve π ◦ γ is given by the endpoints of the shortest paths π ◦ α t : π ( γ ( t )) = π ( α t (1))and the starting direction of π ◦ γ is just π ∗ x ( ˜ ξ x ). By taking the horizontal partof ˜ ξ x with respect to F , i.e. ξ x , and using Proposition 1.10 we see that in fact thestarting direction of π ◦ γ is given by π ∗ x ( ξ x ).As an aside we observe that we need not bother to check whether π ◦ γ has awell defined starting direction, since for small | ξ x | the image of γ lies within theregular part of F and here π is given by a Riemannian submersion.Now if we choose another starting point on ˜ F p , x ′ say, and construct a curve γ ′ in analogy to γ using ˜ ξ x ′ we get π ∗ p ◦ γ ′ = π ∗ p ◦ γ in T π ( F ) B . Consequently, thevariation π ∗ p ◦ α ′ t does not depend on the choice of x ′ ∈ ˜ F p and so neither does π ◦ α ′ t since shortest paths in B are uniquely determined by their starting directionand their length.But this means that π ◦ γ ′ is independent of the choice of x ′ as well. Hence theabove argument implies that at any point x ′ ∈ ˜ F p it is exactly the F -Bott-parallelcontinuation of ξ x that projects onto the ˜ F p -Bott-parallel continuation of ˜ ξ x via ˜ P hp thus proving our claim. (cid:3) Proposition 3.26.
If the induced foliation ˜ F is equidistant then:( ∗ ) For any Bott-parallel vector field ξ and any p ∈ F the projection P h ξ of ξ to F is constant along any regular leaf ˜ F p of ˜ F p .Conversely, if ( ∗ ) holds and if F is horizontally full then ˜ F is equidistant. Proof.
Part 1:
We first assume ˜ F to be equidistant.Let p be a point in F and x ∈ F such that P x = p . Choose any ξ x ∈ ν x F anddefine q ∈ F by q := p + P h ξ x . We define ˜ ξ x := ˜ P h ξ x and denote by ˜ ξ its Bottparallel (with respect to ˜ F p ) continuation along ˜ F p .Then γ x : t x + tξ x , for t ∈ [0 , F and theleaf passing through x + ξ x , which we will denote by G . Note that we may haveto replace ξ x by εξ x for γ x to be not only locally shortest, but the assertion of thelemma is invariant under such a scaling of ξ . Moreover, choosing ε sufficiently smallguarantees the regularity of G . Figure 3.2.
The equidistance of ˜ F is equivalent to P h ξ beingconstant along ˜ F p .Now for any point y ∈ ˜ F p we define ξ y := P h ξ x + ˜ ξ y , where we have identifiedvectors differing only by parallel transport in R k + n .The equidistance of ˜ F implies that both x ′ := p + P h ξ x and y ′ := y + P h ξ x lie inthe same leaf of ˜ F q . In particular x ′′ := x ′ + ˜ ξ x = x + ξ x and y ′′ := y ′ + ˜ ξ y = y + ξ y both lie in ˜ G q since ˜ ξ is ˜ F -Bott parallel.On the other hand, by definition ξ has constant norm along ˜ F p , i.e. | xx ′′ | = | yy ′′ | = dist ( F, G ). So, ξ is the F -Bott parallel continuation of ξ x along ˜ F p and byconstruction ( ∗ ) holds. Part 2:
Let p and q be any two points in F and assume ( ∗ ) holds. We willshow that ˜ F p and ˜ F q are equidistant to each other.Let x ∈ F be a regular point of ˜ F p and let ξ be any Bott parallel normal fieldalong F . Other than this, we will use the same notation as in Part 1. Assertion ( ∗ )implies that y + ξ y lies in the same leaf ˜ G q of ˜ F q for any y ∈ ˜ F p , in fact ˜ F p + ξ = ˜ G q .On the other hand, assume G = F + ξ to be regular. Then ξ yields a Bottparallel normal field on G by defining ζ z + ξ z := − ξ z for z ∈ G .Using assertion (b), we conclude that ˜ ζ = ˜ P h ζ is ˜ F q -Bott parallel along ˜ G q , i.e.˜ G q + ˜ ζ is some leaf ˜ H q in ˜ F q . But by construction this is just the parallel translateby P h ξ x of ˜ F p . .5. ISOMETRIES OF THE INDUCED FOLIATION 39 Remark.
Note that F + tξ may be singular for certain values of t . However,this can only happen for finitely many values of t ∈ [0 ,
1] (cf. Proposition 1.16).So, the parallel translate of ˜ F p to L r is a leaf in ˜ F r for almost all points r lying onthe line pq . By continuity of ˜ F this holds indeed for all r in pq . (cid:3) In general condition (*) appears hard to verify. However, equidistance of ˜ F follows if we prescribe certain dimensional restrictions to the leaves of F . Corollary 3.27.
If the affine leaf F is 1-dimensional the induced foliation ˜ F is equidistant. Proof. If F is horizontally full this is an immediate consequence of Proposi-tion 3.26.Otherwise, H x is everywhere perpendicular to F , i.e. the leaves F of F arecylinders F × ˜ F ⋆ and hence the assertion holds. (cid:3) Remark.
Observe that if the regular leaves have codimension 2 horizonal full-ness implies F to be 1-dimensional and hence ˜ F is equidistant as we have seen.Of course ˜ F is equidistant if the regular leaves are hypersurfaces and hencespherical cylinders around F . Obviously, F cannot be horizontally full in thiscase. We close this chapter with some observations on the group of isometries of theinduced foliation in each horizontal layer. Though interesting in themselves theywill become particularly important in the following chapters.We are often interested in the objects related to the horizontal layer based ata generic point in F . Often these objects will be essentially independent of theparticular choice of base point and we will denote this generic point by ⋆ and theobjects based at this point by L ⋆ , ˜ F ⋆ , etc.The (effective) isometry group of ˜ F ⋆ is given by(3.4) Isom( ˜ F ⋆ ) = Norm( ˜ F ⋆ ) / Centr( ˜ F ⋆ ) , where the normalizer of ˜ F ⋆ consists of all g ∈ SO( n ) leaving ˜ F ⋆ invariant while thecentralizer of ˜ F ⋆ fixes each leaf of ˜ F ⋆ .If ˜ F ⋆ is homogeneous, i.e. given by the orbits of G ⋆ , then maximality of G ⋆ implies that Isom( ˜ F ⋆ ) is simply Norm( G ⋆ ) /G ⋆ . At least for irreducible ˜ F ⋆ we getsome a priori information about its isometry group. Lemma 3.28.
If the action of G ⋆ on L ⋆ is irreducible then the the connectedcomponent of Isom( ˜ F ⋆ ) is contained in either {± } , U(1) or Sp(1) depending onthe type of the G ⋆ -action. Remark 3.29.
Let N := Norm( G ⋆ ) and denote by G ⊥ ⋆ the Lie subgroupexp( g ⊥ ⋆ ) of N where g ⋆ is the Lie algebra of G ⋆ and the orthogonal complementis taken with respect to the Killing form on N (cf. the proof of Proposition 3.11).Then G ⊥ ⋆ is contained in the connected component of the centralizer of G ⋆ and itis isomorphic to Isom ( ˜ F ⋆ ) (cf. the proof of Proposition 3.11). Proof.
Obviously G ⊥ ⋆ acts on L ⋆ as a group of G ⋆ -invariant endomorphisms.Since the G ⋆ -action on L ⋆ is irreducible Schur’s Lemma implies that these endo-morphisms are either zero or invertible. Thus they form an associative divisionalgebra over R , namely R , C or H , depending on the type of the representation(cf. [BtD85, Chap.II], in particular Thm. (6.7)).As G ⊥ ⋆ acts by isometries it is contained in the respective group of units. Hence, G ⊥ ⋆ is either {± } , U(1) or Sp(1). (cid:3) Remark.
Let H denote {± } , U(1) or Sp(1) depending on the type of therepresentation. Note that the isometric H -action on ˜ F ⋆ need not be effective. Forexample consider the standard representation of U( n ) on C n , which is obviously ofcomplex type but the isometry group of the orbit foliation is trivial.So Isom ( ˜ F ⋆ ) may be much smaller than H . But at least the lemma providesan upper bound on Isom ( ˜ F ⋆ ).The main reason for our interest in the isometries of ˜ F ⋆ is the description ofthe leaves of F by the screw motion maps Ψ x as introduced in Remark 3.22. Fromthis description it is clear that the induced foliation ˜ F is equidistant if and onlyif the image of Ψ x is contained in the normalizer of G x , for one and thus for any x ∈ F .So, assuming ˜ F to be equidistant, Equation (3.4) implies even more, since itis not really Ψ x : R k → Norm( ˜ F ⋆ ) we are interested in but rather the inducedmap ¯Ψ x : R k → Isom( ˜ F ⋆ ). As a consequence we get a rather stronger result thanthat in Remark 3.22: Lemma 3.30.
Let ˜ F be equidistant and ˜ F ⋆ homogeneous. Then for any x ∈ R k there is a smooth map Ψ x : R k → G ⊥ ⋆ such that the leaf F passing through ( x, y ) ∈ R k + n is given by (3.5) F = (cid:8) ( x + v, Ψ x ( v ) .G ⋆ .y ) (cid:12)(cid:12) v ∈ R k (cid:9) . Proof.
As said above, Remark 3.22 yields a smooth map ψ x : R k → Norm( G ⋆ )satisfying (3.5). Also the image of ψ x is contained in the connected componentof Norm( G ⋆ ) as R k is connected.Let P be the canonical projection Norm( G ⋆ ) → Norm( G ⋆ ) /G ⋆ . According toRemark 3.29 there is a Lie group isomorphism ϕ : (Norm( G ⋆ ) /G ⋆ ) = Isom ( ˜ F ⋆ ) → G ⊥ ⋆ such that any h ∈ Norm ( G ⋆ ) differs from ϕ ( P ( h )) only by multiplication withsome element of G ⋆ . And G ⊥ ⋆ commutes with G ⋆ .In particular, setting Ψ x := ϕ ◦ P ◦ ψ x gives us the desired map since ψ x and Ψ x describe the same foliation F . (cid:3) Finally observe that F is homogeneous if and only if Ψ x : R k → G ⊥ ⋆ is a Liegroup homomorphism that is independent of the base point x .HAPTER 4 Reducibility of Equidistant Foliations
This chapter deals with two different notions of reducibility. The concept westart with, the existence of invariant subspaces, is well known from representationtheory and we show that fullness of regular leaves characterizes irreducibility evenin the inhomogeneous case. We then examine reducibility in the sense that the foli-ation splits as a product and examine how this is linked to the notion of horizontalfullness we introduced in the last chapter.
It is a well known fact that a homogeneous foliation of Euclidean space con-taining a non-full leaf is reducible. To be more precise, suppose G to be a Liegroup acting on R n by isometries. Let F be a G -orbit such that the minimal affinesubspace V containing F has dimension strictly less than n . Then V is invariantunder the action of G . This follows, using minimality of V , from the fact that theaction of G is affine.An analogous result holds for equidistant foliations: Proposition 4.1.
Let F be an equidistant foliation of R n and let F be a regularleaf. If F is not full the minimal affine space V containing F consists of leaves of F ,i.e. all leaves intersecting V are contained in V . To prove this proposition we show that there is a Bott-parallel subbundle of ν F such that at any point x ∈ F the affine space x + T x F + ν x F is equal to V . Weachieve this by studying the following tensor. Definition 4.2.
Let N : H × H → H be the tensor on the regular part of F given by(4.1) N X Y := O ∗ X O X Y , where O and O ∗ are the O’Neill-tensor of F and its pointwise adjoint. Remark 4.3.
Note that N is Bott-parallel, i.e. for ξ, η ∈ B the vectorfield N ξ η is Bott-parallel as well. To see this let ξ, η, ζ be Bott-parallel and observe that h N ξ η, ζ i = hO ξ η, O ξ ζ i , which is constant along the regular leaves of F (cf. [GG88, p. 145]).Observe also that the image of any linear map A is equal to the image of A ∗ A ,where A ∗ is the adjoint of A . In particular im (cid:16) O ∗ ξ (cid:17) = im ( N ξ ) is Bott-parallel if ξ is. Definition 4.4.
The k -th osculating space of F at p is the space O kp F spannedby the first k derivatives of curves γ : ( − ε, ε ) → F with γ (0) = p .
412 4. REDUCIBILITY OF EQUIDISTANT FOLIATIONS
The k -th normal space of F at p is the orthogonal complement ν kp F of O kp F in O k +1 p F .We will use the notation¯ ν kp F := k M i =1 ν kp F = O k +1 p F ∩ ν p F for the direct sum over the first k normal spaces and denote the sum over all ν kp F by ¯ ν p F . Remark 4.5.
Note that the dimension of these spaces may depend on thepoint p , hence, in general, they do not form bundles over F . However, if they dothen F is contained in the affine space p + T p F + ¯ ν p F = p + O ∞ p F for any p ∈ F and this space is minimal in that respect. (cf. [BCO03, Sect. 2.5 and p. 213]). Lemma 4.6.
Let F ∈ F be a regular leaf. Then for any k the space ¯ ν kp F formsa Bott-parallel bundle over F . Proof.
We will prove this lemma by induction over k .First we show that the first normal spaces ν p F are Bott-parallel, in particulartheir dimensions are constant along F . Let x ∈ ν p F be a vector in the orthogonalcomplement of ν p F , i.e. 0 = h x, α ( v, w ) i = h S x v, w i for all v, w ∈ T p F .Let X be the Bott-parallel continuation of x . Then S X =0, which is to say X is orthogonal to ν F , along F by Proposition 1.21. Hence, ( ν F ) ⊥ is Bott-paralleland consequently so is ν F .Suppose ¯ ν k F to be a Bott-parallel bundle over F and let ξ , . . . , ξ m ∈ B be anorthonormal frame of ¯ ν k F .Now ¯ ν k +1 p F can be viewed as the sum of ¯ ν kp F and the space spanned by thehorizontal part h ∇ v X of the covariant derivatives at p of vector fields X ∈ Γ (cid:0) ¯ ν k F (cid:1) in directions v ∈ T p F . In fact, writing such a vector field X as a C ∞ linearcombination P i f i ξ i of the ξ i , it is easily seen that ¯ ν k +1 F is spanned at each pointby the ξ i and the horizontal part of their covariant derivatives.Remember that for Bott-parallel normal fields ξ i the equality h ∇ v ξ i = −O ∗ ξ i v holds. Remark 4.3 then implies that ¯ ν k +1 F is a Bott-parallel bundle over F , whichproves our claim. (cid:3) Now, Proposition 4.1 is a simple corollary of Lemma 4.6: Let F be a non-fullregular leaf of F and V the minimal affine space containing it. By Remark 4.5, V = p + T p F + ¯ ν p F for any p ∈ F .Take any point q ∈ V \ F , let F ′ be the leaf passing through q and denote by p the point in F minimizing the distance to q . Consider the Bott-parallel continu-ation ξ of q − p ∈ ν p F along F . By Lemma 4.6, the horizontal geodesic t p + tξ p is contained in p + ¯ ν p F for any p , hence, F ′ = { p + ξ p | p ∈ F } is a subset of V . .2. THE NON-COMPACT CASE 43 The results of the previous section make no assumptions on the affine leaf beingcompact or not. In order to deal with the stronger reducibility concept of F beinga product we now concentrate on the non-compact case. Definition 4.7.
For any leaf F ∈ F and points x ∈ F and p = P x ∈ F let D Fp,x and E Fp,x be the subspaces of T p F defined by D Fp,x = P h ( ν x F ) , E Fp,x = ( D Fp,x ) ⊥ . We call F well projecting if for all p ∈ F the space D Fp,x (and hence E Fp,x ) onlydepends on p but not on x ∈ ˜ F p . The foliation F is called well projecting if allregular leaves are well projecting.If F is well projecting we omit the index x . By Lemma 3.3 the dimension of D Fp does not depend on p ∈ F so D F and E F are well defined distributions on F . Alsowe frequently omit the index F and write just D and E if it is clear from the contextwhich leaf the distributions are associated with. Remark.
Observe that Proposition 3.26 implies that F is well projecting if ˜ F is equidistant. In particular the regular leaves of a homogeneous foliation F arewell projecting. Finally, F is well projecting if it is horizontally full.We will show that there is a connection between F not being horizontally fulland F being reducible in the sense that it splits off a Euclidean factor. By the latterwe mean that there is an orthogonal vector space decomposition R k + n = V ⊕ W and an equidistant Foliation F ′ of V such that F = { F ′ × W | F ′ ∈ F ′ } .Let us first list some properties of the distribution E beginning with an auxiliarylemma: Lemma 4.8.
Let F be a leaf of F (not necessarily well projecting), x a pointin F and p = P x ∈ F . Identifying R k + n with its tangent space at any point avector v ∈ R k + n is contained in T x F and T p F if and only if v ∈ E Fp,x . Proof.
The vector v is contained in both T x F and T p F if and only if P v v = v (ignoring the base point). From elementary linear algebra we know that if P is anyorthogonal projection then h P v, P w i = h v, P w i = h P v, w i , ∀ v, w. The rest follows taking w ∈ ν x F . (cid:3) If F is well projecting this implies that E F lifts to F by parallel translation. Proposition 4.9.
Let F be a well projecting regular leaf of F . Then E F isintegrable. Moreover if M p is an integral manifold passing through p ∈ F and x ∈ ˜ F p then the parallel translate M p + ( x − p ) of M p to x is contained in F . Proof.
First note that by Lemma 4.8 we can lift E to F just by paralleltranslating it, i.e. the distribution ¯ E defined by ¯ E x := E P x is tangent to F .Hence, if X, Y are tangent vector fields on F with values in E their verticallifts ¯ X = L X and ¯ Y = L Y to F (see Definition 3.4) take values in ¯ E . Obviouslythe Lie brackets [ X, Y ] and (cid:2) ¯ X, ¯ Y (cid:3) are tangent to F and F respectively. Now X, ¯ X and Y, ¯ Y differ just by parallel translation, which yields an equality of Lie brackets: (cid:2) ¯ X, ¯ Y (cid:3) x = [ X, Y ] P x up to parallel transport. Lemma 4.8 then implies that [ X, Y ] can only have valuesin E , so the latter is integrable. The rest follows immediately. (cid:3) As mentioned above, we now examine the connections between horizontal full-ness and reducibility of F . Proposition 4.10.
Let F be horizontally full, then F does not split off aEuclidean factor. Proof.
Since the linear space W is contained in T x F for all x ∈ F and F ∈ F Lemma 4.8 implies that W is a subspace of E Fp for all p ∈ F . But since F ishorizontally full E F , and hence W , is trivial. (cid:3) Now, the natural question is whether the converse holds as well. At least forhomogeneous foliations we can show that F is reducible if it is not horizontally full. Let F be homogeneous, G the Lie groupacting on R k + n such that the leaves of F are the orbits of G . Remember fromProposition 3.11 that we can describe F by giving the isotropy group G ⋆ and aLie group homomorphism Φ : R k → Centr( G ⋆ ) from the affine leaf F ∼ = R k to thecentralizer of G ⋆ in SO( n ). We may assume that G ⋆ is the maximal connectedsubgroup of SO( n ) with the given orbits. Lemma 4.11.
The distribution ker Φ ∗ on F is parallel and F splits off theEuclidean factor ker Φ ∗ . In particular F splits if dim F > rk (cid:16) Isom( ˜ F ⋆ ) (cid:17) . Proof.
We start by proving that the distribution ker Φ ∗ is G -equivariant.Observe that the velocity field of a curve γ in F is everywhere tangent to ker Φ ∗ if and only if Φ( γ ( t )) .x = x for all x ∈ R n and all t . That is to say that γ can belifted into any leaf of F by parallel transport, which implies(4.2) ker Φ ∗ p = \ F ∈F E Fp , ∀ p ∈ F , where the inclusion of the right hand side in the left follows from Proposition 3.11.Now for any F ∈ F the distribution E Fp is G -equivariant ( P x = x + Ξ x and Ξis G -equivariant) and hence so is ker Φ ∗ .Consequently, ker Φ ∗ is parallel since G acts on F by translations. Thus F splits off the Euclidean factor ker Φ ∗ . (cid:3) Remark 4.12.
Assume ˜ F ⋆ to be irreducible, which is to say that the actionof G ⋆ is irreducible. By Lemma 3.28 the rank of Isom( ˜ F ⋆ ) is at most 1 and henceso is rk (cid:0) Φ( R k ) (cid:1) (cf. Lemma 3.30).So, if F does not split Lemma 4.11 asserts that the affine leaf F can be atmost 1-dimensional. Proposition 4.13. If F is homogeneous and not horizontally full then F splitsoff the Euclidean factor F or ˜ F ⋆ is reducible. .2. THE NON-COMPACT CASE 45 Remark.
N.b. the assertion does not hold if ˜ F is reducible. To illustrate thisconsider the homogeneous foliation of R given by F = n(cid:16) t, (cid:16) cos t sin t − sin t cos t (cid:17) x (cid:17) (cid:12)(cid:12)(cid:12) t ∈ R , x ∈ R o . Let F be the leaf passing through (0 , , ,
1) then D F is trivial while F does notsplit off a Euclidean factor. Proof of Proposition 4.13.
Assume ˜ F ⋆ to be irreducible. By Remark 4.12we may also assume F to be 1-dimensional so H := Φ( F ) is trivial or isomorphicto S . Let us assume the latter since in the former case we are already finished.Let F ∈ F be a regular leaf that is not horizontally full. This means that D F is trivial and F is a cylinder F = F × ˜ F ⋆ . Thus, ˜ F ⋆ is invariant under the actionof H .Observe first that we may assume H to act trivially on ˜ F ⋆ since by Proposi-tion 3.11 we can choose Φ such that its image is contained in Centr( G ⋆ ).Irreducibility of ˜ F ⋆ implies that any regular leaf, in particular ˜ F ⋆ , is full.Since H ∼ = S the horizontal layer L ⋆ splits into an orthogonal sum of 1- or 2-dimensional H -modules. We only have to consider the latter since the action onthe 1-dimensional modules is of course trivial. But ˜ F ⋆ being full means that forany H -module V we can find a point x ∈ ˜ F ⋆ such that the V -component of x isnonzero. Since H fixes ˜ F ⋆ pointwise the action of H on V must be trivial.Thus H acts trivially on L ⋆ , which means that all the leaves of F are cylinderssplitting off the Euclidean factor F . (cid:3) We show that a somewhat weaker analogue toProposition 4.13 holds even if we drop the homogeneity assumption for F . But letus first generalize some of the findings of the previous section.The key ingredient for the results in the previous section was describing F viathe Lie group homomorphism Φ : F → Norm ( G ⋆ ).Remember that by Remark 3.22 we can describe any equidistant foliation F of R k + n in a way similar to this as long as ˜ F ⋆ is homogeneous. This result is refinedby Lemma 3.30 for equidistant ˜ F .As noted before, the screw motion map Ψ a , a ∈ R k ∼ = F , need not be a Liegroup homomorphism. However, we can still use it as a tool to examine reducibilityof F .We first introduce a further distribution on F , which is motivated by Equa-tion (4.2). Definition 4.14.
Let E Ψ a be the distribution on F given by E Ψ a p := ker (cid:0) (Ψ a ) ∗ p (cid:1) , p ∈ F . The connection to (4.2) becomes clear in the next lemma:
Lemma 4.15.
Let a be an arbitrary point in R k . Then for any p ∈ F thespace E Ψ a p can be vertically lifted to any leaf in F by parallel translation to some x ∈ L p , i.e. we have the inclusion (4.3) E Ψ a p ⊂ \ F ∈F , x ∈ ˜ F p E Fp,x . Proof.
Let γ : ( − , → F be a smooth curve such that its derivative ˙ γ (0)is tangent to E Ψ a γ (0) , i.e. ddt (cid:12)(cid:12) t =0 Ψ a ( γ ( t )) = 0.Let F be an arbitrary leaf in F and x ∈ ˜ F γ (0) . Describing F in accordancewith Remark 3.22, choose b ∈ R n ≃ L a such that x = (cid:0) γ (0) , Ψ a ( γ (0) − a ) .b (cid:1) . Here we have identified γ ( t ) with just its first k coordinates (since the last n coor-dinates vanish anyway).Now, consider the lifted curve ¯ γ : ( − , → F given by¯ γ ( t ) = (cid:0) γ ( t ) , Ψ a ( γ ( t ) − a ) .b (cid:1) . Looking at its derivative, we obviously get ˙¯ γ (0) = (cid:0) ˙ γ (0) , (cid:1) which is just ˙ γ (0),abusing notation again. Hence, Lemma 4.8 implies that ˙ γ (0) is contained in E Fγ (0) ,x . (cid:3) Remark. For the remainder of this section we assume ˜ F to be equidistant and ˜ F ⋆ to be homogeneous. Then by Lemma 3.30 we can choose Ψ a such that its imageis contained in G ⊥ ⋆ and thus equality holds in (4.3).An immediate consequence is the following: Corollary 4.16. If Isom( ˜ F ⋆ ) is discrete F splits off the Euclidean factor F . Remember that an essential point in the proof of Proposition 4.13 was to assumethat F is at most 1-dimensional. We show that — provided ˜ F is equidistant and˜ F ⋆ is given by the orbits of an irreducible representation of complex type — F splitsif F has dimension larger than 1: Lemma 4.17.
Assume
Isom( ˜ F ⋆ ) to be 1-dimensional. Then either the affineleaf F of F is at most 1-dimensional or F splits off a Euclidean factor. Proof.
By Lemma 3.30 we may assume the image of Ψ a to be contained inthe 1-dimensional Lie group G ⊥ ⋆ . So for any p ∈ F the kernel of (Ψ a ) ∗ p is either ahyperplane or all of T p F .If the latter holds at any p ∈ F Lemma 4.15 clearly implies that E Fp = T p F for all F ∈ F . Since the dimension of E Fp is independent of p ∈ F it follows that F splits off the whole affine leaf F .So let us assume ker (cid:0) (Ψ a ) ∗ p (cid:1) to be a hyperplane at every point, which meansΨ a has only regular values. Consequently the level sets of Ψ a are regular hypersur-faces of F . We show that their connected components form the leaves of an equidis-tant foliation of F . We achieve this by showing that this foliation is transnormal,i.e. geodesics meeting any leaf perpendicularly meet all leaves perpendicularly (cf.Remark 1.12).Let p be any point in F and ξ ∈ T p F perpendicular to E Ψ a p . By Lemma 4.15there is some leaf F ∈ F such that ξ ∈ D Fp . Let then ¯ ξ ∈ ν x F be such that P h ¯ ξ = ξ with x ∈ ˜ F p . Then ¯ γ ( t ) := x + t ¯ ξ meets F perpendicularly and staysperpendicular to all leaves of F it meets. Hence, its projection γ : t p + tξ to F stays perpendicular to the distribution E Ψ a since˙ γ ( t ) = P h ( ˙¯ γ ( t )) ∈ D F t γ ( t ) , where F t is the leaf passing through ¯ γ ( t ). .2. THE NON-COMPACT CASE 47 Now the only equidistant foliation of Euclidean space by hypersurfaces is givenby parallel hyperplanes and lifting these to all leaves of F we see that F splits ifthe dimension of F is greater than 1. (cid:3) Assume ˜ F ⋆ to be given by the orbits of an irreducible representation. If therepresentation is of real type F splits off F since the isometry group of ˜ F ⋆ isdiscrete. If it is of complex type and F has dimension greater than 1 then F splits,as we have just shown. Remark.
Note, that we cannot use the proof of Lemma 4.17 if the representationis of quaternionic type:In the worst case Isom ( ˜ F ⋆ ) = Sp(1). Assume Ψ a to have only regular valuesand its fibres to be equidistant. Let G be the foliation of F given by the fibresof Ψ a and let ¯ G be the refinement of G given by the connected components of itsleaves. Then Ψ a factorizes in the following way F a / / Ψ a % % KKKKKKKKKKK F / ¯ G p (cid:15) (cid:15) F / G = Sp(1)where ¯Ψ a : F → F / ¯ G and p : F / ¯ G →
Sp(1) are the canonical projections.Both G and ¯ G are equidistant so Ψ a and ¯Ψ a are submetries if we take theinduced metrics on the respective quotients. Then p is a submetry as well (cf. Lem-ma 1.6).Observe that p has to be a covering map because the fibres of ¯Ψ a are allregular and p must be discrete (cf. [Lyt02, Thm. 10.1]). So F / ¯ G must be Sp(1)since Sp(1) ≃ S is simply connected. But on the other hand Theorem 2.9 impliesthat F / ¯ G cannot be compact. Hence our assumption was wrong.We close with the generalized version of Proposition 4.13: Proposition 4.18. If F is not horizontally full and ˜ F ⋆ is given by the orbitsof an irreducible representation of complex type then F splits off a Euclidean factor. Proof.
In analogy to the proof of Proposition 4.13 we choose a regular nothorizontally full leaf F ∈ F . Then G ⊥ ⋆ and hence the image of Ψ a leaves ˜ F ⋆ invariant, even pointwise by Lemma 3.30. The rest is exactly the same as in theproof of Proposition 4.13 replacing H with Isom ( ˜ F ⋆ ). (cid:3) HAPTER 5
Homogeneity Results
In this chapter we finally address homogeneity of F . First, we consider thequotient A = R k + n / ˜ F and show that — provided ˜ F is equidistant — the imageof F under the natural projection is an equidistant foliation of A . Moreover, thisnew foliation is described by the same screw motion map as the original one. Re-versing this construction we show how to construct new inhomogeneous equidistantfoliations of Euclidean space.We conclude with a homogeneity result for F if ˜ F ⋆ is homogeneous and ifIsom( ˜ F ⋆ ) fulfills certain conditions, e.g. if it is sufficiently small. Throughout this chapter we will assume ˜ F to be equidistant. In this section we will show that the submetry π factorizes into a composition π ◦ π such that both π i are submetries again. This yields a foliation A of theintermediate space A := π ( R k + n ) given by the fibres of π . We construct thefactorization of π in such a way that the leaves of A are exactly the images of theleaves of F under π .It turns out that A is more regular than F in the sense that the leaves of A are all of the same dimension. This regularity of A will be the key ingredient ofour study of F during the following sections. It is, however, bought at the expenseof A only being an Alexandrov space albeit of a rather nice type.In order to construct the map π consider the following: Let Σ denote thespace of directions of B at the point π ( F ), then C Σ is the tangent cone T π ( F ) B .Consider the map ¯ π : L → C Σ , where ¯ π is the restriction of π ∗ to the horizontallayer L , identifying L with H . As we have seen in Section 3.2, ¯ π is just thecanonical projection from L to L / ˜ F ∼ = C Σ . Definition 5.1.
We set π : R k + n ∼ = F × L → F × C Σ , π := id | F × ¯ π and A := F × C Σ . We define the map π : A → B by π (¯ x ) := π ◦ π − (¯ x ) . Remark.
Observe that π is a submetry since its components id | F and ¯ π are.Moreover, the fibres of π are the leaves of ˜ F because the latter is equidistant. So, π is just the canonical projection R k + n → R k + n / ˜ F .Since ˜ F is a subfoliation of F the map π is well defined and by Lemma 1.6 itis a submetry.
490 5. HOMOGENEITY RESULTS
So, the fibres of π define an equidistant foliation A of A , which by the remarkabove is given by the images of the leaves of F , i.e. A = (cid:8) π − ( x ) (cid:12)(cid:12) x ∈ B (cid:9) = { π ( F ) | F ∈ F} . We now study A and its foliation A in order to better understand F .As the essential information about A is contained in the structure of Σ under-standing Isom(Σ ) appears to be essential. In Section 3.5 we have already discussedthe isometry group of the induced foliation ˜ F ⋆ . Now, remember that Isom( ˜ F ⋆ ) actseffectively and by isometries on C Σ = L ⋆ / ˜ F ⋆ and hence on Σ as the action fixesthe apex of the cone. However, it is possible for the space of leaves to have moreisometries than the foliation. Remark.
The subgroup Isom( ˜ F ⋆ ) ⊂ Isom(Σ ) consists exactly of the isometriesof Σ that may be liftet to ˜ F ⋆ .For example consider an isoparametric hypersurface in a sphere and the foli-ation created by its parallel surfaces (cf. [PT88, Sect. 8.4] and [FKM81]). Sucha foliation always has two focal manifolds, hence the space of leaves is a compactinterval with the reflection at the midpoint being the only nontrivial isometry. Butthis reflection cannot always be lifted to an isometry of the foliation since the twofocal manifolds may have different dimension.It is not even clear whether the connected components of the two isometrygroups are the same. Nevertheless we will see that understanding the action ofIsom ( ˜ F ⋆ ) is quite sufficient in order to understand A .But first we mention a splitting result (cf. [Lyt02, Prop. 12.14]) for the submetry¯ π : R n → C Σ : Proposition 5.2. If diam (Σ ) > π then C Σ splits as C Σ = R l × C Σ ′ with diam (Σ ′ ) ≤ π . Moreover ¯ π : R l × R n − l → R l × C Σ ′ splits as ¯ π = id | R l × ¯ π ′ and ¯ π ′ is a submetry. In particular if Σ has diameter greater than π/
2, ˜ F ⋆ is reducible.Assuming ˜ F ⋆ to be homogeneous Section 3.5 shows that F is completely de-scribed by two data: the group G ⋆ acting on L ⋆ and a smooth map (or rather aset of maps) Ψ x : R k ∼ = F → G ⊥ ⋆ ∼ = Isom ( ˜ F ⋆ ). Thus the foliation A is completelydescribed by Ψ x interpreting it as a map into Isom ( ˜ F ⋆ ) ⊂ Isom (Σ ):(5.1) A = (cid:8)(cid:8) ( x + v, Ψ x ( v ) .a ) (cid:12)(cid:12) v ∈ R k (cid:9) (cid:12)(cid:12) ( x, a ) ∈ F × C Σ (cid:9) , and A is homogeneous if and only if Ψ x is a Lie group homomorphism independentof the base point x ∈ R k , i.e. if and only if F is homogeneous.Using the converse approach, we show how equidistant foliations F of R k + n may be constructed from the data mentioned above. In particular we give newexamples of inhomogeneous equidistant foliations of R k + n .So, let G be an equidistant foliation of S n , Σ := S n / G and G := Isom ( G ).Choose a smooth map Ψ : R k → G ⊂ Isom(C Σ ). Then, setting A := R k × C Σ this yields a foliation A of A with the leaf A passing through (0 , a ) given by A = (cid:8) ( v, Ψ ( v ) .a ) (cid:12)(cid:12) v ∈ R k (cid:9) . .2. NEW EXAMPLES FROM OLD 51 Viewing G as a subgroup of SO( n ) we can lift this construction to R k + n . Thus weget the foliation F with the leaf F ∈ F passing through (0 , x ) given by F = (cid:8) ( v, Ψ ( v ) .y ) (cid:12)(cid:12) v ∈ R k , y in the same G -leaf as x (cid:9) . This construction induces the two maps R k + n π −→ R k + n / ˜ F = A π −→ A / A =: B and F is given by the fibres of π ◦ π . Note that by construction ˜ F is automaticallyequidistant, hence π is a submetry. So, F is equidistant if and only if A is.In general, equidistance of A will be rather hard to check. However, it followsimmediately if A is homogeneous, i.e. if Ψ is a Lie group homomorphism. Remark.
Note that F inherits the remaining properties of an equidistant foli-ation from G since Ψ is smooth.Choosing Ψ to be a group homomorphism means that F is homogeneous ifand only if G is. Let us start then with G being inhomogeneous. As said beforethe only known examples are the ones generated by isoparametric hypersurfaces inspheres and the octonional Hopf fibration S ֒ → S → S . We already mentionedabove that in the former case the leaf space is a compact interval and hence G istrivial. So here our construction yields nothing new.Let us look at the Hopf fibration of S , which is given byS (cid:127) _ (cid:15) (cid:15) = Spin(8) / ^ Spin(7)S (cid:15) (cid:15) = Spin(9) / ^ Spin(7)S = Spin(9) / Spin(8)and ^ Spin(7) is the image of the standard embedding of Spin(7) in Spin(8) under a(non-trivial) triality automorphism of Spin(8).
Remark 5.3.
In general let G be a Lie group and K ⊂ H ⊂ G compactsubgroups. Thus we get the natural fibration p : G/K → G/H mapping gK to gH .Then a result by B´erard Bergery states that we can find suitable G -invariantmetrics on G/K and
G/H and an H -invariant metric on H/K such that p is aRiemannian submersion with totally geodesic fibres isometric to H/K (see [Bes87,p. 256f] for a detailed discussion).Since the fibre through gK is ( gH ) K = { ghK | h ∈ H } ∼ = H/K the submer-sion p is obviously G -equivariant.Note that in our case S and S bear just the standard metric and S is aEuclidean sphere of radius 1/2 (cf. [Bes87, 9.84]).We see that Spin(9) acting transitively on S leaves the Hopf fibration invari-ant. On the other hand let N ⊂ Spin(9) be the subgroup that maps fibres intothemselves, which hence has to be a normal subgroup. But SO(9) = Spin(9) / {± } is simple so N ⊂ {± } and − id obviously does map the fibres into themselves. This means that SO(9) acts transitively and effectively on the Hopf fibration.Since SO(9) is the isometry group of the space of fibres S it is already the fullisometry group of the Hopf fibration.Hence, we have proved: Proposition 5.4.
Taking any Lie group homomorphism Ψ : R k → SO(9) theabove construction yields an inhomogeneous non-compact equidistant foliation of R k + n with the induced foliation being given by the Hopf fibration S ֒ → S → S . Of course we can limit ourselves to k ≤ splits off as a Euclidean factor (cf. Lemma 4.11). We now present the main result of this chapter. The idea underlying it is thatwe do not have to know too much about Σ to understand A and thus F . Theimportant thing is rather how Isom ( ˜ F ⋆ ) acts on Σ . If this action is “similar” to arepresentation acting transitively on a sphere we can use Gromoll and Walschap’sresult to prove homogeneity of A and thus of F : Theorem 5.5.
Let F and ˜ F be equidistant and let ˜ F ⋆ be homogeneous. If theaction of H := Isom ( ˜ F ⋆ ) on C Σ has an orbit B isometric to a round sphere and H acts effectively on B then F is homogeneous. Proof.
Since H acts on C Σ by isometries, the partition B of A by the F -cylinders over these H -orbits is equidistant. Moreover, A is a refinement of B , soLemma 1.6 implies that the restriction A B of A to F × B is equidistant as well.Now, by assumption, F × B is isometric to a round cylinder R k × S lr ⊂ R k + l +1 for some l ≥
1. Let us call this isometry ϕ . Consequently, the image of A B under ϕ is equidistant and may be described via the maps ¯Ψ x with¯Ψ x : R k → SO( l ) , ¯Ψ x ( v ) .ϕ ( b ) := ϕ (Ψ x ( v ) .b ) , ∀ v ∈ F , b ∈ B such that the leaf ¯ A of ϕ ( A B ) passing through ( x, y ) is given by¯ A = (cid:8) ( x + v, ¯Ψ x ( v ) .y ) (cid:12)(cid:12) v ∈ R k (cid:9) . Now ϕ ( A B ) can be extended to an equidistant foliation of R k + l +1 and thisfoliation is regular. Thus, by [GW01] this foliation is homogeneous. In particular[GW97, Thm. 2.6] implies that the maps ¯Ψ x must be Lie group homomorphismsindependent of x . But then the same holds for the maps Ψ x and so F is homoge-neous. (cid:3) We immediately get the following important application for ˜ F ⋆ having smallisometry group: Corollary 5.6. If dim (cid:16) Isom( ˜ F ⋆ ) (cid:17) ≤ , in particular if(i) ˜ F ⋆ is given by the orbits of an irreducible representation of real or complextype or(ii) ˜ F ⋆ is given by an irreducible polar actionthen F is homogeneous. .3. HOMOGENEITY 53 Proof.
Assume H := Isom ( ˜ F ⋆ ) = U(1) then the H -orbits on C Σ are eithersingle points or diffeomorphic and hence isometric to S r . The latter holds if andonly if H acts effectively on that orbit. So if there is an effective H -orbit on C Σ Theorem 5.5 implies homogeneity of F . On the other hand, if there is no effective H -orbit the action of H is trivial and hence F splits off F .Now let us consider the special cases mentioned: If the representation is of realtype we have already seen that Isom ( ˜ F ⋆ ) is trivial and hence F splits off F . If itis of complex type H is a subgroup of U(1) and we are done by what we mentionedabove.If ˜ F ⋆ is given by a polar representation C Σ = L ⋆ / ˜ F ⋆ is the Weyl chamberof a principal orbit. In particular its isometry group is discrete, so F splits off F again. (cid:3) However, Isom ( ˜ F ⋆ ) being small is not necessary as the following result shows: Corollary 5.7. If ˜ F ⋆ is given by the complex or quaternionic Hopf fibrations S ֒ → S → S or S ֒ → S → S then F is homogeneous. Proof.
In both cases Σ is a sphere so to apply Theorem 5.5 we show thatIsom ( ˜ F ⋆ ) acts transitively and effectively on Σ . This can be done using Re-mark 5.3. However, a more direct approach is possible:Consider the U(1)-action on S ⊂ C by complex multiplication with unitcomplex numbers: λ. ( z , z ) = ( λz , λz ). The complex Hopf fibration is then thenatural projection to the orbit space C P ∼ = S . We show that Isom( ˜ F ⋆ ) = SO(3):Let G := (SU(2) × U(1)) / ∼ where we identify ( A, λ ) with ( − A, − λ ). Then G acts on S ⊂ C in the following way: ( A, λ ) . ( z , z ) := A ( z λ, z λ ) = λA ( z , z )and this action is effective.Obviously G leaves ˜ F ⋆ invariant as the G -action commutes with the U(1)-action. On the other hand, it is clear that the only elements of G leaving eachleaf of ˜ F ⋆ invariant are of the form (id , λ ). So G/ ( { id }× U(1)) ∼ = SU(2) / {± } actseffectively on the foliation and hence it is contained in Isom( ˜ F ⋆ ). ButSO(3) ∼ = G/ ( { id }× U(1)) ⊂ Isom ( ˜ F ⋆ ) ⊂ Isom (Σ ) = SO(3)and thus equality holds at every step.The quaternionic case is rather similar. Here we consider the action of Sp(1) onS ⊂ H by quaternionic multiplication from the right: h. ( q , q ) := ( q h − , q h − ).The orbits form the foliation ˜ F ⋆ . The remainder is analogous to the complex case:Let H := (Sp(2) × Sp(1)) / ∼ with ( A, h ) ∼ ( − A, − h ) and H acts effectively onS ⊂ H via ( A, h ) . ( q , q ) := A ( q h − , q h − ).Again it is clear that H leaves ˜ F ⋆ invariant and the only elements of H fixingeach leaf of ˜ F ⋆ are of the form (id , h ). The latter can easily be seen by letting ( A, h )act on ( a, b ) with a, b ∈ { , , i, j, k } . As before H/ ( { id }× Sp(1)) ∼ = Sp(2) / {± } actseffectively on ˜ F ⋆ andSO(5) ∼ = H/ ( { id }× Sp(1)) ⊂ Isom ( ˜ F ⋆ ) ⊂ Isom (Σ ) = SO(5)implies that Isom ( ˜ F ⋆ ) acts effectively and transitively on Σ = S . (cid:3) Open Questions.
Some problems that were addressed in this thesis still re-main open. In particular it has been essential for our homogeneity results to assumethe induced foliation ˜ F to be equidistant. Based on the findings of Chapter 3 it is my conjecture that indeed equidistance of F implies that of ˜ F . I even conjecturethat equidistance of F together with homogeneity of ˜ F ⋆ already implies F to behomogeneous.At the very least this should be true for ˜ F equidistant and ˜ F ⋆ homogeneousand irreducible. To see this one would have to show that the orbits of Isom ( ˜ F ⋆ )can only be S , S , S or one of the corresponding projective spaces. One couldthen try to modify the proof of Theorem 5.5 or indeed the approach used in [GW01]to work in the projective case as well.The first conjecture is obviously necessary for the second but is also interestingin itself. For example it implies that there are no further examples of noncom-pact inhomogeneous equidistant foliations of R n than those given in Section 5.2;in particular the [FKM81]-examples cannot appear as induced foliation of an irre-ducible F .On the other hand, proving this conjecture wrong would be most interestingas well since it would provide a whole new class of inhomogeneous equidistantfoliations. ibliography [AB03] Stephanie Alexander and Richard L. Bishop. F K -convex functions on metric spaces. Manuscripta Math. , 110(1):115–133, 2003.[AT07] Marcos M. Alexandrino and Dirk T¨oben. Equifocality of a singular riemannian foliation. preprint: arXiv:0704.3251v2 , 2007.[BBI01] Dmitri Burago, Yuri Burago, and Sergei Ivanov.
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