Totally Geodesic Riemannian Foliations on Compact Lie Groups
aa r X i v : . [ m a t h . DG ] J u l TOTALLY GEODESIC RIEMANNIAN FOLIATIONS ONCOMPACT LIE GROUPS
LLOHANN D. SPERANC¸ A
Abstract.
Let F be a Riemannian foliation with connected totally geodesicleaves on a connected compact Lie group with bi-invariant metric G . Weanswer a question of A. Ranjan proving that, up to isometry, the leaves of F coincide with the cosets of a fixed subgroup. Introduction
The present work is devoted to a simple question: how to fill a given a geometricspace with a geometric pattern? Or, conversely (following Thurston [33]): how toconstruct a manifold out of stripped fabric?For instance, starting with a Lie group G , we could use its algebraic structureto construct a pattern. A common example is given by cosets of a subgroup:any Lie subgroup H < G , induces a decomposition of G by both right cosets, F + H = { gH | h ∈ G } , and left cosets, F − H = { Hg | h ∈ G } . Such decompositionsare called as homogeneous foliations .In general, a foliation F on M is the decomposition of M into the integrablemaximal submanifolds of an involutive subbundle T F ⊂
T M . Such submanifoldsare called leaves . Existence, obstructions and classifications for foliations are deeptopological subjects (see e.g. Haefliger [14] and Thurston [31, 32, 33]). Foliationsacquire a very geometric flavor by imposing distance rigidity between leaves: afoliation is called
Riemannian if its leaves are locally equidistant (see e.g. Molino[26] or Ghys [9]).The decomposition into the fibers of a Riemannian submersion is a main exampleof Riemannian foliations: a submersion π : M → B is Riemannian if the restriction dπ p | (ker dπ p ) ⊥ is an isometry to T π ( p ) B for every p ∈ M (see e.g. O’Neill [25]or Gromoll–Walschap [12]). The classification of Riemannian submersions fromcompact Lie groups with bi-invariant metrics was asked by Grove [13, Problem5.4]. His question can be motivated in several ways. For instance, most examples ofmanifolds with positive sectional curvature are related to Riemannian submersionsfrom Lie groups (a more complete account can be found in Ziller [37]).On the other hand, given a compact Lie group G with bi-invariant metric, allknown Riemannian foliations whose leaves are totally geodesic are homogeneous. Mathematics Subject Classification.
MSC 53C35, MSC 53C20 and MSC 53C12.
Key words and phrases.
Riemannian foliations, holonomy, nonnegative sectional curvature,symmetric spaces, Lie groups. Only non-singular foliations are considered in this paper.
Therefore, it is natural to ask whether homogeneous foliations are the only Rie-mannian foliations with totally geodesic leaves or not (Ranjan [28]). The affirma-tive answer is supported by the following conjecture, commonly called “Grove’sConjecture” (see also Munteanu–Tapp [24]):
Conjecture 1.
Let G be a compact simple Lie group with a bi-invariant metric. ARiemannian submersion π : G → B with connected totally geodesic fibers is inducedeither by left or right cosets. Here Ranjan’s question together with Conjecture 1 are proved affirmatively.
Theorem 1.1.
Let F be a Riemannian foliation with totally geodesic connectedleaves on G , a compact connected Lie group with bi-invariant metric. Then F islocally isometric to a homogeneous foliation. One may ask whether Theorem 1.1 is local in nature or not. The proof we presentdoes use global objects, however the analyticity of the metric should allow a localanswer.We observe that the hypothesis on Theorem 1.1 can not be relaxed: Kerin–Shankar [18] presented infinite families of Riemannian submersions from compactLie groups with bi-invariant metrics that can not be realized as principal bundles(for instance, the composition h ◦ pr : SO (16) → S of the orthonormal framebundle pr : SO (16) → S with the Hopf map S → S ). Moreover, the simplegroup SO (8) admits a foliation, F SO (8) , by totally geodesic round 7-spheres (suchfoliation is produced by trivializing the orthonormal frame bundle SO (8) → S ).Kerin–Shankar examples does not have totally geodesic fibers and F SO (8) is notRiemannian. It is a curious fact that an intermediate step in this work does con-sider foliations by round 7-spheres. One may wonder if such foliations are theonly possible non-homogeneous (non-Riemannian) foliations with totally geodesicconnected leaves on simple compact Lie groups.The general classification of Riemannian foliations is wide open. For instance,classifications neither for totally geodesic Riemannian foliations on symmetric spaces,nor for generic Riemannian foliations on Lie groups are known (we refer to Lytchak[21], Lytchak–Wilking [22] and Wilking [35] for important developments in othercases). The author believes many resources provided here can be applied to thesymmetric space case.The main issue to prove Theorem 1.1 is to restrict the leaf type. Since we assumethat leaves are totally geodesic, each leaf is a symmetric space. Once proved that theleaf through the identity is a subgroup, we argue along the lines of Gromoll–Grove[10, Lemma 3.3] and Jimenez [17, Theorem 23] to prove homogeneity. Throughout,the strategy is to control infinitesimal holonomy transformations (see section 1.1 fora definition – by holonomy we mean the holonomy defined by displacement of leavesalong horizontal directions as in Gromoll–Walschap [12], which is fundamentallydifferent from the holonomy defined in Molino [26]).After developing the general theory and applications of infinitesimal holonomytransformations (sections 2-6), we proceed to the case of Riemannian foliationswith totally geodesic leaves on bi-invariant metrics. The first step in this case(section 7) is to relate the root system of G with a new, foliation-based, rootsystem introduced in section 6. Such algebraic step provides control over Grey–O’Neill’s integrability tensor (Proposition 1.9) and reduces the proof to irreduciblefoliations (Theorem 7.17) – in analogy to principal bundles, we call a Riemannian IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 3 foliation irreducible if every two points can be joined by a curve orthogonal to theleaves Gray–O’Neill’s integrability tensor measures how the horizontal connection(the distribution orthogonal to the leaves) is not integrable, analogous to the Levitensor in e.g. Piccione–Tausk [27]. In the case of Riemannian foliations with totallygeodesic fibers, the integrability tensor gives rise to (local) Killing fields along theleaves.Proposition 1.9 together with Theorem 1.2, a suitable version of Ambrose-Singertheorem ([2, Theorem 2]), shows that leaves have the local Killing property : aroundeach point, their tangent space has an orthonormal frame of Killing fields (we referto section 6 in Berestovskii–Nikonorov [4] for a detailed account of this property).The universal cover of a Riemannian manifold with local Killing property is isomet-ric to a product of an Euclidean space, constant curvature 7-spheres and compactsimple Lie groups with bi-invariant metrics ([4], Theorem 11).As a second step, we prove that compact Lie-groups with bi-invariant metricsdoes not admit Riemannian foliations of totally geodesic 7-spheres. Together withTheorem 1.5, it implies that only the Euclidean space and compact Lie groupsappear as (local) factors of the leaves. In particular, leaves are isometric to Liegroups with bi-invariant metric.The next step is to construct a special family of Killing fields that spans thetangent space of the leaves (as in Theorem 23 in Jimenez [17]). They are constructedin section 4.2 under assumptions in the holonomy group, which are verified in section9.3.The results in section 2-6 extrapolate the hypothesis in Theorem 1.1 and mightbe of independent interest. Below, we give a more detailed account of the paper.1.1.
Main Results.
Given a Riemannian foliation F on M , we might think of F locally as an stripped fabric (or a Riemannian submersion) with leaves verticallyplaced. At each point x ∈ M , we decompose T x M as the tangent to the leaf V x andits orthogonal complement H x = ( V x ) ⊥ . We call V x as the vertical space and H x as the horizontal space at x . A (local) vector field X is said to be basic horizontal ,if it is H -valued and, for every vertical field V , [ X, V ] is vertical. The flow of abasic horizontal vector field X induces local diffeomorphisms between leaves (asa standard computation shows – see e.g. Hirsch [16, Proposition 17.6]). These(local) diffeomorphisms are called (local) holonomy transofrmations . It is knownthat holonomy transformations are (local) isometries if and only if leaves are totallygeodesic (see e.g. Gromoll–Walschap [12, Lemma 1.4.3]).Given a horizontal curve c : [0 , → M , there is (locally) a natural holonomytransformation associated to c : any local basic horizontal extension of ˙ c induces alocal diffeomorphism between a neighborhood of c (0) in L c (0) to a neighborhood of c (1) in L c (1) . Given two local basic horizontal extensions of ˙ c , the induced diffeo-morphisms coincides on L c (0) (an equivalent construction using local submersionscan be found in Gromoll–Walschap [12, Examples and Remarks 1.3.1]). In par-ticular, for ξ ∈ V c (0) and Φ Xt , the flow of a basic horizontal extension X of ˙ c , thedifferential ( d Φ Xt ) c (0) ( ξ ) is well-defined for all t . The vector field ξ ( t ) = ( d Φ Xt ) c (0) ( ξ ) In modern terminology, F is irreducible if it has a single dual leaf (see Wilking [36]). Angulo–Guijarro–Walschap [3] calls such a foliation as twisted . We prefer the classical terminology alludingto the principal bundle case, from where our basic ideas comes from. L. D. SPERANC¸ A is called the holonomy field along c with initial value ξ (0) = ξ . We call the lin-ear map ˆ c ( t ) : V c (0) → V c ( t ) , ˆ c ( t ) ξ = ( d Φ Xt ) c (0) ( ξ ) as the infinitesimal holonomytransformation defined by c .Given a Riemannian foliation F , its Gray–O’Neill integrability tensor, A : H ×H → V , is defined as A X Y = 12 [ ¯ X, ¯ Y ] v , where ¯ X, ¯ Y are horizontal extensions of X, Y and v (respectively, h ) stands fororthogonal projection onto V (respectively, onto H ).The dual leaf passing through p ∈ M , L p , is the subset of points in M that canbe joined to p by horizontal curves (compare Wilking [36] or Gromoll–Walschap[12, section 1.8]). When the leaves of F are the fibers of a principal G -bundle π : P → B , the integrability tensor, infinitesimal holonomy fields and dual leavesreplace classical objects: the curvature 2-form satisfies Ω( X, Y ) = − A X Y ; given p ∈ M , for any horizontal curve c , ω c (1) ˆ c (1)( ξ ) = ω p ( ξ ), where ω : T P → g isthe connection one-form. Moreover, an action field ξ satisfies ˆ c (1) ξ ( c (0)) = ξ ( c (1)); P ( p ), the reduction of P through p , is the set of points that are reached by horizontalcurves starting at p ([19, section II]), i.e., P ( p ) = L p . The Ambrose-Singer theorem[2, Theorem 2] identifies the Lie algebra of the holonomy group of π with ω ( T p P ( p )).Our first result is an analogous characterization for Riemannian foliation: Theorem 1.2.
Let F be a Riemannian foliation with complete connection on apath connected space M . Then T L p ∩ V = span { ˆ c (1) − ( A X Y ) | X, Y ∈ H c (1) , c horizontal } . The proof of Theorem 1.2 follows [5, section 3.4], taking due attention to holo-nomy transformations.In section 3 we follow [29, section 2] and consider the bundle of infinitesimalholonomy transformations τ p : E p → M . If the leaves of F are the fibers of aRemannian submersion with totally geodesic fibers π : M → B , π ◦ τ p is isomorphicto the holonomy bundle of π ( ). For each p ∈ M , τ p : E p → M is the bundledefined by the set of infinitesimal holonomy transformations induced by horizontalcurves from p . It has a natural principal group H p ( F ), the set of infinitesimalholonomy transformations induced by horizontal loops at p , and a natural foliation˜ F = { τ − p ( L ) | L ∈ F} . In section 3 E p , τ p and ˜ F are proved to be smooth and˜ F to be Riemannian (for a suitable metric). The local holonomy transformationsinduced by a horizontal curve c define (not in a natural way) a diffeomorphismbetween the universal covers of the leaves, φ c : ˜ L c (0) → ˜ L c (1) (Lemma 2.1). Wecall the set of holonomy transformations induced by horizontal loops at p as theholonomy group at p , Hol p ( F ). H p ( F ) is a Lie subgroup of GL ( V p ) that recoversthe isotropy representation of Hol p ( F ) at the point p .In section 4 we specialize to foliations where H p ( F ) is a bounded subgroup of GL ( V p ). Such is the case when leaves are totally geodesic or coincide with theorbits of a locally free action. We call a foliation given by the orbits of a locallyfree action as principal . In Jimenez [17, Theorem 23], a foliation is guaranteed tobe principal given the existence of a subalgebra of vector fields satisfying special Let P q be the set of all holonomy transformations between π − ( p ) and π − ( q ). The holonomybundle of π is ˜ π : P → B , where P = ∪ P q and ˜ π ( f ) = q – see e.g. Gromoll–Walschap [12, Theorem2.7.2] IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 5 properties. Such subalgebra is constructed in section 4.2, provided H p ( F ) = { id } (Corollary 1.4). More generally, we prove: Theorem 1.3.
Let F be an irreducible Riemannian foliation with bounded H p ( F ) .Then Hol p ( F ) is a finite dimensional Lie group acting locally free on E p . Moreover,the orbits of Hol p ( F ) coincide with the leaves of ˜ F . By observing that τ p is a diffeomorphism if H p ( F ) = { id } , we get: Corollary 1.4.
Let F be as in Theorem 1.3. If H p ( F ) = { id } , then M admits alocally free Hol p ( F ) -action whose orbits coincide with the leaves of F . Furthermore,if the restriction of the Hol p ( F ) -action to each leaf is by isometries, then the actionon M is by isometries. From section 4, we specialize to foliations with totally geodesic leaves. We startby extending the (local) de Rham decomposition of a leaf to the entire foliation.
Theorem 1.5.
Let F be as in Theorem 1.3 and M simply connected. Let T L p = L i ˜∆ i be the de Rham decomposition of T L p . Then there are smooth integrabledistributions ∆ i on M such that: (1) ∆ i is vertical for every i and V = L ∆ i (2) ∆ i ⊥ ∆ j for i = j (3) F i , the foliation defined by ∆ i , is Riemannian and has totally geodesic leaves Theorem 1.5 is used to rule out 7-sphere factors in Theorem 1.1.In sections 5 and 6, the main general results are proven. Their roles are comple-mentary in the proof of Theorem 1.1. Theorem 1.6 , proved in section 5, greatlyrefines Theorem 1.2. Theorems 1.7 and 1.8 provide a root system (in the repre-sentation theory sense) based on the integrability tensor. The comparison betweenthe new root system and the Lie algebraic root system of G discloses special prop-erties of the integrability tensor (Propositions 1.9 and 8.6). Theorem 1.6 allows theLie-algebraic results based on Theorems 1.7 and 1.8 to be extended leafwise. Theorem 1.6.
Let F be a totally geodesic Riemannian foliation on a manifold M of non-negative sectional curvature. Then T L p ∩ V p = span { A X Y | X, Y ∈ H p } . Let ξ ∈ V and denote A ξ : H → H as the negative dual of A : (cid:10) A ξ X, Y (cid:11) = − h A X Y, ξ i . Theorem 1.7.
Let F be a totally geodesic Riemannian foliation, γ a vertical geo-desic and X a basic horizontal vector field along γ . Then A ˙ γ X is a basic horizontalfield along γ . Theorem 1.8.
Let F be a totally geodesic Riemannian foliation and suppose t v ⊂V p exponentiates to a totally geodesic flat in L p . If A has bounded norm along L p ,then A ξ A η = A η A ξ for all ξ, η ∈ t v . In sections 7-9 we specialize to totally geodesic foliations on bi-invariant met-rics. Section 7 is built upon Theorem 1.8 and Munteanu–Tapp [24, Theorem 1.5],providing the Lie algebraic relation between the two root systems. In section 8, wetake advantage of such algebraic to prove: The author was informed that M. Radeschi proved Theorem 1.6 assuming M a compact Liegroup with bi-invariant metric L. D. SPERANC¸ A
Proposition 1.9.
Let F be as in Theorem 1.1. If X, Y, Z, W are basic horizontalfields, then h A X Y, A Z W i is basic, i.e., it is locally constant along leaves. In particular, if F is irreducible, Theorem 1.6 guarantees that leaves have thelocal Killing property. Section 9 rules out the 7-sphere factors of the leaves andcompletes the proof of Theorem 1.1.1.2. Further remarks.
The classification of Riemannian foliations on symmetricspaces is greatly explored in literature (see [7, 8, 10, 21, 22, 23, 24, 28]). Ourarguments resemble ideas in [23, 24, 28].Ranjan [28] used the relation ( A ξ ) = ( ad ξ ) (induced by O’Neill’s formulas)to prove Conjecture 1, assuming there is a maximal torus contained in a leaf. Suchtorus provides a decomposition of the basic horizontal fields into candidates to leftand right invariant fields. Ranjan then uses the simplicity of the group to provethat the set of either left or right invariant basic horizontal fields is trivial.In contrast to the algebraic approach of Ranjan, Munteanu–Tapp [24] intro-duces the geometric concept of good triples : a triple { X, V,
A} ⊂ T p M is good ifexp p ( tV ( s )) = exp p ( sX ( t )) for all s, t ∈ R , where V ( s ) , X ( t ) denote the Jacobifields along exp( sV ) and exp( tX ), respectively, that satisfy V (0) = V , X (0) = X and V ′ (0) = A = X ′ (0). Such conditions are achieved in totally geodesic Rie-mannian foliations by a horizontal X and a vertical V (or vice-versa). In this case A = A V X . Theorem 1.5 in [24] provides a key identity that is used throughoutsection 7.Fixed a Riemannian manifold M , a Riemannian foliation with totally geodesicleaves on M is completely determined by its vertical space and its integrabilitytensor at a single point. In particular, basic horizontal fields along a fiber gathercomplete information about the foliation (see equation (2)). Part of our approach isto investigate horizontal fields, using them to determine the geometry of the leavesand the holonomy transformations, but not directly determining the integrabilitytensor.We observe that, given a homogeneous foliation induced by the right/left cosetsof H < G , the restriction to L id of the right/left invariant fields corresponding to h ⊥ = H id are basic horizontal: a holonomy field ξ along a horizontal geodesicexp( tX ) is the restriction of a left/right invariant field, therefore it satisfies ∇ X ξ = ∓ [ X, ξ ] (seeing X as a left/right invariant field); since holonomy fields commutes(as fields) with basic horizontal fields (see section 2), a basic horizontal field X alongexp( tξ ) must satisfy ∇ ξ X = ± [ ξ, X ] (seeing ξ as a right/left invariant field).A slightly more complicated situation happens in the following examples: givensubgroups H i < G i , let F be the foliation in G × G with L ( g ,g ) = { ( h g , g h ) | h i ∈ H i } ; given H < G , consider F ∆ as the foliation in G × G whose leaves are determinedby L ( g ,g ) = { ( hg , g h ) | h ∈ H } . In the first example, the set of basic horizontalfields along L id is the sum of a set of left invariant fields in G with a set of rightinvariant fields in G , corresponding to h ⊥ i ⊂ g i . In the second example, althougheach basic horizontal field is decomposed into a sum of a right and a left invariantfield, the space of basic horizontal fields does not split as a space of right invariantfields and a space of left invariant fields. This complication naturally appears insection 7 when dealing with the H ± ( F )-decomposition of horizontal vectors. IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 7
Section 7 provides a suitable splitting of the horizontal space which turns torepresent the splitting into left and right invariant components of basic horizon-tal fields. Section 8 explores the concept of good triple to prove Proposition 1.9.Sections 7-9 is the most involved part of this work.The author would like to thank C. Dur´an, K. Shankar and K. Tapp for sugges-tions and insightful conversations. Specially K. Shankar for pointing out [4]. Theauthor also would like to thank the Federal University of Paran´a for hosting theauthor for most part of this work.Part of the length of the paper lies in the author’s attempt to make it self-contained. The author apologizes for any missing reference.1.3.
Notation.
For convenience, we gather some notation and definitions in thissection. We mostly use the notation of Gromoll–Walschap [12]. We follow the usualnomenclature in Riemannian foliations, calling vectors tangent to leaves verticals and vectors orthogonal to leaves horizontals . They define the vector bundles V and H , respectively. Horizontal vectors will be denoted by capital Arabic letters: X, Y, Z, W ; vertical vectors by Greek lower case letter: ξ, η .Gray-O’Neill’s tensors will be denoted as in Gromoll–Grove [10] or Gromoll–Walschap [12]: A X Y = ∇ ¯ X ¯ Y = 12 [ ¯ X, ¯ Y ] , S X ξ = −∇ ξ ¯ X Where ¯ X, ¯ Y are horizontal extensions of X, Y . We observe that a foliation is totallygeodesic if and only if S ≡ c ( t ) are equivalently defined as thevertical solutions of(1) ∇ ˙ c ξ = A ξ ˙ c − S ˙ c ξ. Analogously, basic horizontal fields along a vertical curve γ can be defined ashorizontal solutions of(2) ∇ ˙ γ X = A ˙ γ X − S X ˙ γ. Holonomy fields along horizontal geodesics (respectively basic horizontal fieldsalong vertical geodesics, in the totally geodesic case) are the Jacobi fields inducedby local horizontal lifting (respectively, by holonomy transformations).Given F , the dual leaf through p ∈ M is the set L p = { q ∈ M | q can be joined to p via a horizontal curve } .L p is an immersed submanifold and F = { L p | p ∈ M } , a foliation (see [12, 36]).We call a foliation F irreducible if it has only one dual leaf.Both (4 ,
0) and (3 ,
1) Riemannian curvature tensors are denoted by R . In sections7 and 8, we work with the complexification of the Lie algebra, among other spaces,throughout the paper. The complexification of a space or an operator will bedenoted by a supindex C . We recall that the (4 ,
0) Riemannian curvature in abi-invariant metric is given by(3) R ( X, Y, Z, W ) = − h [ X, Y ] , [ Z, W ] i . L. D. SPERANC¸ A Holonomy transformations in Riemannian Foliations
Let π : M → B be a Riemannian submersion and ¯ c : [0 , → B a curve. Thehorizontal connection on M is called complete if for every point p ∈ π − (¯ c (0)),there is a horizontal curve c p : [0 , → M , c p (0) = p , such that π ◦ c p = ¯ c .If H is complete, by lifting horizontally ¯ c , one gets a holonomy transformation φ ¯ c : π − (¯ c (0)) → π − (¯ c (1)). For a foliation, neither ¯ c nor its lifts are naturallydefined. In this section we introduce a notion of completeness for horizontalconnections on foliations, suitable for this work, and observe how monodromyarguments produce diffeomorphisms between the universal covers of the leaves φ c : ˜ L c (0) → ˜ L c (1) by patching local ‘lifts’.We recall that a Riemannian foliation F is locally given by a Riemannian submer-sion (Gromoll–Walschap [12, Examples and Remarks 1.2.1, item (ii)]): each point p ∈ M has an open neighborhood U where the leaves of the restricted foliation F| U = { connected componentes of L ∩ U | L ∈ F} are the fibers of a Riemannian submersion π U : U → V (the metric on V is uniquelydetermined by π U [12, Theorem 1.2.1]). One can further choose U diffeomorphicto π − U ( π U ( p )) × V (take V a small geodesic ball around π U ( p ) and use holonomytransformations along radial geodesics). We call such U a submersive neighborhood.We call a horizontal connection H locally complete if, for every horizontal curve c : [0 , → M , there is a submersive neighborhood U of c [0 , H is locally complete if M is complete.Given a leaf L ∈ F , basic horizontal fields define a frame for H| L which isparallel with respect to the Bott connection ([12, Examples and Remarks 1.3.1(i)]). In particular, given a horizontal vector X ∈ H p , and a point q ∈ π − ( p ), π : ˜ L → L the universal cover, there is a unique field X on the pull-back π ∗ ( H| L ),where dπ q ( X ) = X and locally dπ ( X ) is basic horizontal. Therefore, given ahorizontal curve c : [0 , → M and a vertical curve γ : [0 , → M , c (0) = γ (0), upto technical assumptions on F (e.g. Lemma 2.1 below), there are unique horizontalcurves t c γ ( s ) ( t ) such that • c γ ( s ) (0) = γ ( s ); c γ (0) ( t ) = c ( t ); • for every t , s ˙ c γ ( s ) ( t ) is a basic horizontal field along s c γ ( s ) ( t ).Once fixed base points q ∈ ˜ L c (0) and q ′ ∈ ˜ L c (1) , monodromy arguments pro-duce a diffeomorphism ˜ φ c : ˜ L c (0) → ˜ L c (1) such that π ( φ c (˜ γ ( s )) = c γ ( s ) (1), where π : ˜ L c (1) → L c (1) is the universal cover and ˜ γ ( s ) is the unique lift of γ ( s ) with˜ γ (0) = q and ˜ φ c ( q ) = q ′ .We call a horizontal connection complete whenever, for any horizontal curve c ,the holonomy transformation ˜ φ c : ˜ L c ( o ) → ˜ L c (1) is well-defined. Lemma 2.1.
Let F be a Riemannian foliation with bounded A - and S -tensors andlocally complete H . Then H is complete.Proof. Let c be a horizontal curve and γ : [0 , → L c (0) be a curve connecting p = c (0) to some q ∈ L p . We prove the existence of a map F : [0 , × [0 , → M such that,(1) ∂F∂t ∈ H (2) for each submersive neighborhood U , π U ( F ( t, s )) does not depend on s IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 9 (3) F ( t,
0) = c ( t ), F (0 , s ) = γ ( s )(1) and (2) imply the curves s F ( t , s ), s F ( t , s ) are obtained from eachother by local holonomy translations (on each interval s ∈ [ s − ǫ, s + ǫ ]). In fact, Claim 2.2. If F satisfies (1) and (2), then ξ = ∂F∂s is a holonomy field along c s ( t ) = F ( t, s ) and X = ∂F∂t is basic horizontal.Proof. It is sufficient to compute (1) and (2) for X and ξ . But ∇ hX ξ = − S X ξ and ∇ vξ X = A ξ X are tensorial and [ X, ξ ] = ∇ X ξ − ∇ ξ X = 0. (cid:3) Uniqueness of integral manifolds in Frobenius theorem guarantees that everyintegral curve of
X π U -projects to a single curve, characterizing a holonomy trans-formation.We prove the existence of F by an extension argument. The local completenessof H guarantees there is such an F defined on a small square [0 , × [0 , ǫ ]. Let O ⊂ [0 ,
1] be the maximal interval containing 0 such that, if s ∈ O , then F isdefined on [0 , × { s } satisfying (1)-(3). From locally completeness, O is relativelyopenTo prove that O is also closed, suppose (by contradiction) that O = [0 , β ).Claim 2.2, together with standard ordinary differential equations arguments, uni-formly bounds X, ξ on F ([0 , × [0 , β )). In particular F | [0 , × [0 ,β ) , together withits derivatives, are uniformly bounded and can be extended to [0 , × [0 , β ], con-tradicting the maximality of O . (cid:3) We define the
Holonomy group of F at p , Hol p ( F ), as the indentity componentof the set of all holonomy transformations defined by horizontal curves that startsat p and ends in L p .In contrast with the holonomy transformation ˜ φ c , the infinitesimal holonomytransformation induced by c can be naturally defined using holonomy fields. Let ξ ∈ V c (0) , we set ˆ c ( t ) ξ = ξ ( t ), where ξ ( t ) is the holonomy field defined by ξ along c . Assuming that F is irreducible, we prove in section 3 that the set ofinfinitesimal holonomy transformations induced by curves starting on p forms asmooth manifold.2.1. Proof of Theorem 1.2.
For convenience, write¯ a p = span { ˆ c (1) − ( a c (1) ) | c horizontal } and let ¯ a = ∪ q ∈ M ¯ a q ⊂ V . By construction, ¯ a is closed under infinitesimal holonomytransformations, that is, ˆ c (¯ a c (0) ) = ¯ a c (1) . In particular, it has constant rank alongdual leaves. Let us fix p ∈ M . Once proved that ¯ a p ⊂ T p L (Claim 2.3), weassume M = L p by restricting the foliation to L p . In Claim 2.4 ¯ a | L is shown tobe smooth. The proof of Theorem 1.2 is concluded in Claim 2.5, by showing that¯ a ⊕ H ⊂ T L p is an involutive smooth distribution. In particular, every horizontalcurve starting from p must lie on the integral manifold of ¯ a ⊕ H that passes through p , concluding that this integral manifold must contain L p . Claim 2.3. ¯ a p ⊆ T L p .Proof. The claim follows by an usual construction of the A -tensor. Consider aneighborhood U of p ∈ M such that F| U is induced by the submersion π : U → V ,where V is some open set of an Euclidean space. Given X , Y ∈ H p , let X, Y bebasic horizontal extensions of X , Y such that [ dπ ( X ) , dπ ( Y )] = dπ [ X, Y ] h = 0. Denote by Φ Zt the flow of a vector-field Z . As in the proof of Lemma 2.1,the flow lines of X, Y are the horizontal lifts of flow lines of dπ ( X ) , dπ ( Y ). Since dπ ( X ) , dπ ( Y ) commutes, we conclude that, for t ≥ γ ( t ) = Φ X √ t Φ Y √ t Φ X −√ t Φ Y −√ t ( p )is a curve in L p with γ (0) = p and γ ′ (0) = 2 A X Y .In particular, if c is a horizontal curve starting at p and X , Y ∈ H c (1) , byconsidering a submersive neighborhood U , we can again consider extensions of X , Y and define the curve(4) γ ( t ) = φ − c Φ X √ t Φ Y √ t Φ X −√ t Φ Y −√ t ( p ) .γ again lies in L p with γ (0) = p and γ ′ (0) = 2ˆ c − ( A X Y ). In particular, ˆ c − A X Y is tangent to L p for every c, X , Y , as desired. (cid:3) From now on, we assume M = L p . Claim 2.4. ¯ a | L p is smooth.Proof. Since every two points in L are joined by horizontal curves and L isclosed under local holonomy transformations, it is sufficient to show that ¯ a has asmooth frame in a neighborhood of p .Let c i be a collection of horizontal curves and X i , Y i ∈ H c i (1) be horizontalvectors such that { ˆ c − i ( A X i Y i ) } ≤ i ≤ k forms a basis for ¯ a p . Being careful enoughwith neighborhoods, by extending X i , Y i as basic horizontal vectors, we can assumethat { dφ − c i ( A X i Y i ) } ≤ i ≤ k is still linearly independent in a small neighborhood of p in L p (here φ c i denotes the local holonomy transformation defined by c i ). Since ¯ a has constant rank, { dφ − c i ( A X i Y i ) } ≤ i ≤ k is a smooth basis of ¯ a | L p in a neighborhoodof p . Infinitesimal horizontal translation along horizontal curves defines a smoothbasis of ¯ a in a open neighborhood of p in M . (cid:3) Claim 2.5. ¯ a ⊕ H is integrable.Proof. Let
X, Y, Z, W be horizontal fields and ξ, η sections of ¯ a . We divide theproof into:( i ) [ X, Y ] is a section of
H ⊕ ¯ a , j( ii ) [ X, ξ ] is a section of
H ⊕ ¯ a ,( iii ) [ ξ, η ] is a section of H ⊕ ¯ a .Since [ X, Y ] v = 2 A X Y , [ X, Y ] ∈ H ⊕ ¯ a . For item ( ii ), it is sufficient to assume X basic horizontal and ξ holonomy along the integral curves of X , but then [ X, ξ ] = 0as in the proof of 2.1.As observed in the proofs of Claim 2.4, for some neighborhood U , ¯ a | U is generatedby a finite collection of vectors of the form dφ − c ( A X Y ), X, Y basic horizontal,satisfying [
X, Y ] h = 0. On the other hand, given dφ c ( A X Y ) , dφ c ′ ( A Z W ), we have[ dφ c ( A X Y ) , dφ c ′ ( A Z W )] = dφ c [ A X Y, ( dφ c ) − dφ c ′ ( A Z W )]= dφ c [ A X Y, dφ − cc ′ ( A Z W )] . Therefore, since ¯ a is closed under holonomy transformations, it is sufficient toprove that [ A X Y, dφ − c ( A Z W )] ∈ ¯ a for X, Y, Z, W basic horizontal and c a hori-zontal curve. Writing ξ = dφ − c ( A Z W ), we have2[ A X Y, ξ ] = [[
X, Y ] , ξ ] − [[ X, Y ] h , ξ ] = [[ X, ξ ] , Y ] + [ X, [ Y, ξ ]] − [[ X, Y ] h , ξ ] , which lies in ¯ a ⊕H since, putting items ( i ) and ( ii ) together, we have that the bracketof a horizontal fields with any section in ¯ a ⊕ H is again a section in ¯ a ⊕ H . (cid:3) IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 11 The Infinitesimal Holonomy Bundle
Given p, q ∈ M , consider the set of linear isomorphisms Iso( V p , V q ). The unionAut( V ) = ∪ p,q ∈ M Iso( V p , V q ) defines the Lie groupoid σ, τ : Aut( V ) → M where σ, τ : Aut( V ) → M , the source and the target map, are defined as σ ( h ) = p and τ ( h ) = q for h : V p → V q .Aut( V ) carries the partial multiplication ( h, h ′ ) h ◦ h ′ (composition of lin-ear maps), defined whenever σ ( h ) = τ ( h ′ ), and the inversion h h − , given bythe inversion of a linear map. Its core at p ∈ M , σ − ( p ) ∩ τ − ( p ), is the gen-eral linear group GL ( V p ). In particular, for every p , τ | σ − ( p ) : σ − ( p ) → M is a GL ( V p )-principal bundle whose principal action is right composition with elementsof GL ( V p ). From now one, we denote σ − ( p ) = Aut p ( V ) and τ = τ | σ − ( p ) (wheneverit is not ambiguous).Infinitesimal holonomy transformations are elements of Aut( V ), since they arelinear isomorphism between vertical spaces. They form a subset of Aut( V ) closedunder composition and inversion (the infinitesimal holonomy transformation in-duced by the concatenation c c is the composition ˆ c (1)ˆ c (1); setting c ˜( t ) = c (1 − t ),ˆ˜ c (1) = c (1) − . Compare [29, section 2]), although possibly not smooth. Our firstaim is to prove that the intersection of the set of infinitesimal holonomy transfor-mations with Aut p ( V ) (or equivalently, the infinitesimal holonomy transformationsinduced by horizontal curves starting at p ) is a smooth principal bundle over L p .This result is used in an essential way in sections 4 and 9.2.Given p ∈ M , we define the infinitesimal holonomy bundle at p , τ p : E p → L p ,as the restriction of τ to the set E p = { ˆ c (1) ∈ Aut( V ) | c horizontal, c (0) = p } . Theorem 3.1. E p ⊂ Aut( V ) is an immersed submanifold. Furthermore, H p ( F ) = E p ∩ GL ( V p ) is a Lie subgroup of GL ( V p ) and τ p : E p → L p is a smooth H p ( F ) -principal bundle. To prove 3.1, E p is realized as a dual leaf of the following foliation in Aut p ( V ):˜ F = {L q = τ − ( L q ) | q ∈ M } . To make sense of dual leaves, we must give a Riemannian structure to Aut p ( V )for which ˜ F is Riemannian (although a horizontal distribution should suffice forProposition 2.1 in [36], we will further explore the geometry of Aut p ( V ) in section9.2.) We start by defining the geometry along the leaves L q , then we pull-back partof the metric from M . Lemma 3.2. L q = τ − ( L q ) is a smooth embedded submanifold of Aut p ( V ) . Fur-thermore, the restriction τ | L q : L q → L q is isomorphic to the frame bundle of L q .Proof. The first assertion follows from standard Implicit Function Theorems (e.g.,[34, Theorem 1.39]). For the second, recall the frame bundle is the collection of lin-ear isomorphisms F ( L ) = ∪ s ∈ L Iso( R k , T s L ). Since T s L = V s , a linear isomorphism T : R k → V p induces a bundle isomorphism L q → F ( L q ). (cid:3) The vertical space of ˜ F is ˜ V = dτ − ( V ). A distribution ˜ H defines a horizontalconnection for ˜ F if and only if dτ ( ˜ H ) defines a horizontal distribution to F . Here,we use the definition of infinitesimal holonomy transformation to lift H . Given a horizontal curve c : [0 , → M , we define its τ -horizontal lift at h ∈ τ − ( c (0)) asthe curve ˆ c h : [0 , → Aut( V ) given byˆ c h ( t ) ξ = ξ h ( t ) , where ξ h ( t ) is the holonomy field along c with initial condition ξ h (0) = hξ ∈ V c (0) .Although a horizontal connection is completely determined by its horizontal lifts,not every set of candidates for horizontal lifts determine a linear distribution (e.g.´Alvarez–Dur´an [1, Page 2410] where Finsler horizontal lifts forms a cone, not alinear space). We now verify the linearity of the space ˜ H defined by the velocitiesof the curves ˆ c h . Lemma 3.3.
For each h ∈ Aut p ( V ) , let ˜ H h be the set of velocities of τ -horizontallifts at h . Then ˜ H = ∪ h ∈ Aut p ( V ) ˜ H h ⊂ T Aut p ( V ) is a smooth subbundle comple-mentary to ˜ V .Proof. As in the proof of Lemma 3.2, a linear isomorphism T : R k → V p identifiesAut p ( V ) with the frame bundle of V . Therefore, a linear connection ∇ V on V induces, for every h ∈ Aut p ( V ), a linear injection (a horizontal lift) ζ h : T τ ( h ) M → T h Aut p ( V ): a curve γ on Aut p ( V ) is tangent to ζ ( T M ) if and only if the vectorfield ξ ( t ) = γ ( t ) ξ is ∇ V -parallel along τ ◦ γ , for every ξ ∈ τ ( γ (0)). Let ∇ V bedefined as ∇ V X ξ = ( ∇ X ξ ) v + S X ξ. Then, if τ ◦ γ is a horizontal curve, γ ( t ) ξ is ∇ V -parallel if and only if it is aholonomy field. Thus, ˜ H h = ζ h ( H τ ( h ) ). (cid:3) We now define a metric h , i τ on Aut p ( V ) where ˜ F is Riemannian. Declare h ζ ( T M ) , ker dτ i τ = 0 and that dτ | ζ ( T M ) is an isometry onto its image. To de-fine h , i τ on vectors in ker τ , we recall that τ : Aut p ( V ) → M is a GL ( V p )-principalbundle and consider the connection one form ω : Aut p ( V ) → gl ( V p ) defined by ζ ( T M ). We fix an inner product Q on gl ( V p ) and set h V, U i τ = Q ( ω ( V ) , ω ( U )) for V, U ∈ ker dτ . Proposition 3.4. ˜ F is a Riemannian foliation with respect to h , i τ . Furthermore,a curve α : [0 , → Aut( V ) is ˜ F -horizontal if and only if α = ˆ c h for some horizontalcurve c : [0 , → M . In particular if H is complete, so it is ˜ H . ˜ F is Riemannian since h ˜ H , ˜ Vi τ = 0 and dτ | ˜ H is an isometry onto its image. Thecharacterization of ˜ F -horizontal curves given in Proposition 3.4 is a consequence ofthe definition of ˜ H since a curve α in Aut p ( V ) is tangent to ˜ H if and only if τ ◦ α is horizontal (since dτ ( ˜ H ) = H ) thus α ( t ) ξ is a holonomy field for every ξ ∈ V p (recall the proof of Lemma 3.3). In particular, E p is readily identified as the dualleaf associated to ˜ F passing through id V p ∈ Aut p ( V ). We conclude E p is a smoothimmersed submanifold of Aut p ( V ).Given g ∈ GL ( V p ), denote by µ g : Aut p ( V p ) → Aut p ( V p ) the right compositionby g . Since dµ g ( ˜ H h ) = ˜ H hg , µ g (ˆ c h ( t )) = ˆ c hg ( t ) for every horizontal curve c . Inparticular, H p ( F ) = E p ∩ GL ( V p ) is a subgroup of GL ( V p ) and τ p : E p → L p is H p ( F )-principal. We now show that τ p is a smooth submersion. Lemma 3.5.
For every p ∈ M , τ p : E p → L p is a submersion. In particular, H p ( F ) is a Lie subgroup of GL ( V p ) . IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 13
Proof. τ p is smooth since it is the restriction of the smooth map τ to an immersedsubmanifold. H is in the image of dτ p by construction. We use Theorem 1.2 todeal with T L p ∩ V . Theorem 1.2 implies that the rank of T L p ∩ V is constant andspanned by the velocities of curves of the form (4). The curve γ in equation (4) isdefined by composing the flows of four F -horizontal vector-fields. By lifting thesevector-fields to ˜ H h , we get a ˜ F -horizontal curve ˜ γ in E p with ˜ γ (0) = h . It satisfies τ p ◦ ˜ γ = γ , in particular, dτ p ( ˙˜ γ ) = ˙ γ , showing that T L p ∩ V is in the image of dτ p . (cid:3) Observe that ζ ( T L p ) defines a principal connection for the H p ( F )-principalbundle τ p : E p → L p , i.e., it is complementary to ker dτ p and H p ( F )-invariant.Moreover, since every point in E p can be joined to p by a curve tangent to ˜ H ⊂ ζ ( T L p ), τ p is irreducible as a principal bundle with connection ζ ( T L p ). We get: Corollary 3.6. If L p is simply connected, H p ( F ) is connected. A further remark on the geometry of ˜ F . Given q ∈ M , consider L q asa Riemannian manifold with the induced metric L q ⊂ M . The restriction of theconnection 1-form ω to T L q has the following geometric interpretation: a curve α in L q is ω -horizontal (i.e., ω ( ˙ α ) = 0) if and only if α ( t ) ξ ∈ T τ ( α ( t )) L q is a parallelfield along τ ◦ α , for every ξ ∈ V q . In this section we compute some quantities usedin section 9. Lemma 3.7.
Let α : [0 , → L p be a curve with α (0) = id . Then ω ( ˙ α (0)) : V p → V p is the morphism defined by ξ
7→ ∇ ˙ c (0) ( α ( t ) ξ ) .Proof. Denote γ ( t ) = τ ( α ( t )) and its ω -horizontal lift at id as ˜ γ ( t ) (i.e., ω ( ˙˜ γ ) =0). For each t , there is a unique g ( t ) ∈ GL ( V p ) such that α ( t ) = ˜ γ ( t ) g ( t ). Inparticular, ω ( ˙ α (0)) = g ′ (0) . On the other hand, for every vector-field ξ along γ , ddt | t =0 (˜ γ ( t ) − ξ ) = ∇ γ (0) ξ (observe that ˜ γ ( t ) − ξ is a curve in V p ). Therefore g ′ (0) ξ = ∇ γ (0) ( α ( t ) ξ ). (cid:3) Lemma 3.8.
Denote by A the integrability tensor of ˜ F . Let X, Y ∈ ˜ H id . Then ω ( A X Y ) : ξ
7→ ∇ ξ ( A dτX dτ Y ) .Proof. As in the proof of Proposition 2.3, consider U, ˜ U neighborhoods of p and idwhere F , ˜ F are given by Riemannian submersions π : U → V , ˜ π : ˜ U → V (where τ ( ˜ U ) ⊂ U ). Given X , Y ∈ H id , let X, Y be a basic horizontal extension of X , Y such that dπ ( X ) , dπ ( Y ) are commuting vector fields on V . We can describe theflow of ¯ X, ¯ Y , the horizontal lift of dπ ( X ) , dπ ( Y ) to ˜ U , through the flows of X, Y :denote by Φ Zt the flow of a vector-field Z . Given ξ ∈ V p , we get(5) Φ ¯ Xt ( ξ ) = ( d Φ Xt ) p ( ξ ) = dds (cid:12)(cid:12)(cid:12) s =0 Φ Xt ( γ ( s )) , where γ ( s ) is a curve on L p with γ (0) = p and γ ′ (0) = ξ . On the other hand, asobserved in Proposition 2.3,(6) 2 A X Y = ddt (cid:12)(cid:12)(cid:12) t =0 Φ X √ t Φ Y √ t Φ X −√ t Φ Y −√ t ( p ) . Therefore, by (5), chain rule and the analogous of (6),2 ω ( A ¯ X ¯ Y ) ξ = ∇ ∂∂t ( d Φ X √ t ◦ d Φ Y √ t ◦ d Φ X −√ t ◦ d Φ Y −√ t ( ξ ))= ∇ ∂∂t (cid:18) dds (cid:12)(cid:12)(cid:12) s =0 Φ X √ t Φ Y √ t Φ X −√ t Φ Y −√ t ( γ ( s )) (cid:19) = ∇ ∂∂s (cid:18) ddt (cid:12)(cid:12)(cid:12) t =0 Φ X √ t Φ Y √ t Φ X −√ t Φ Y −√ t ( γ ( s )) (cid:19) = 2 ∇ ξ ( A X Y ) . (cid:3) Foliations with bounded holonomy
Let π : E → M be a Riemannian submersion whose holonomy group is a finitedimensional compact Lie group. π enjoys special properties, mainly with respect tothe growth of the S -tensor along horizontal geodesics (see Tapp [30] and referencestherein).In [29], the author attempts to emulate a compact holonomy group on Riemann-ian foliations through a condition on holonomy fields. Here we explore a slightlyweaker condition (which we call by the same name as in [29]): we say that afoliation F has bounded holonomy if H p ( F ) is a relatively compact subgroup of GL ( V p ). As in [29], bounded holonomy is readily verified whenever the leaves aretotally geodesic ( H p ( F ) consists of isometries) or when the foliation is principal( H p ( F ) = { id } , see [29, Lemma 3.5]).Although Corollary 1.4 is a direct consequence of Theorem 1.3, we first proveCorollary 1.4 and then derive Theorem 1.3 from it. Theorem 1.5 is proved in section4.4.In view of Lemma 2.1, whenever M is not compact, we assume that A and S arebounded. Throughout the section, F is assumed irreducible (the author believesthat a similar result holds for the non-irreducible case: E p should be diffeomorphicto E p × Hol p ( F ) ˜ L p . )4.1. Bounded holonomy and totally geodesic leaves.
Proposition 4.1 gives aclose relation between foliations with bounded holonomy and foliations with totallygeodesic leaves. Proposition 4.1 plays a key role in the proof of Theorem 1.3.Let F be a Riemannian foliation on the Riemannian manifold ( M, g ) with verticaland horizontal sapces V , H . A metric g ′ is called a vertical variation of g if g ′ ( H , V ) = 0 and g and g ′ coincides on horizontal vectors. That is, for every X ∈ H and ξ ∈ V , g ′ ( X + ξ, X + ξ ) = g ( X, X ) + g ′ ( ξ, ξ ) . Proposition 4.1. [29, Theorem 6.5]
If the holonomy of F is bounded, there is avertical variation g ′ of g where the leaves of F are totally geodesic.Proof. Suppose that F has bounded holonomy. Since the closure of H p ( F ) on GL ( V p ) is compact, V p can be endowed with a H p ( F )-invariant inner product h , i .Observe that the inner product defined by h ξ, η i q = (cid:10) h − ξ, h − η (cid:11) in V q does notdepend on the choice of h ∈ τ − p ( q ): if h, k ∈ τ − p ( q ), (cid:10) h − ξ, h − η (cid:11) = (cid:10) h − kk − ξ, h − kk − η (cid:11) = (cid:10) k − ξ, k − η (cid:11) , since h − k ∈ H p ( F ). This metric is smooth since it descends from a smooth innerproduct on τ ∗ p V . IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 15
On the other hand, a Riemannian foliation is totally geodesic if and only ifholonomy fields have constant length (as equation (1) indicates). Given a holonomyfield ξ along a horizontal curve c , ξ ( t ) = ˆ c ( t ) ξ for ξ = ξ (0). Therefore, h ξ ( t ) , ξ ( t ) i c ( t ) = (cid:10) ˆ c ( t ) − ˆ c ( t ) ξ , ˆ c ( t ) − ˆ c ( t ) ξ (cid:11) = h ξ , ξ i . (cid:3) Given a foliation whose leaves are totally geodesic, every holonomy transforma-tion φ c , c (0) = p is an isometry. In particular, Hol p ( F ) is a subgroup of the groupof isometries of ˜ L p , Iso( ˜ L p ). We realize the embedding Hol p ( F ) ⊂ Iso( ˜ L p ) as thenext step in the proof of Theorem 1.3. In particular, Hol p ( F ) inherits a differentialstructure as a subgroup of Iso( ˜ L p ).Denote by ˜ π : ˜ L p → ˜ L p the frame bundle of ˜ L p . We realize ˜ L p as the coveringmap π ∗ : ˜ L p → L p , where π ∗ ( h ) = dπ ◦ h and π : ˜ L p → L p .From now on, we assume that F satisfies all hypothesis in Theorem 1.3. Thefollowing claim follows from Proposition 4.1 and the fact isometries are completelydefined by its value and its differential at a single point. Claim 4.2.
Let z ∈ π − ∗ (id V p ) and denote by µ : Hol p ( F ) × ˜ L p → ˜ L p the actiondefined by the differential of elements in Hol p ( F ) . Then µ is free and transitive. Theorem 1.2 applied to the foliation ˜ F characterizes the Lie algebra of Hol p ( F ).Corollary 4.3 is used in section 9. Corollary 4.3.
Let F be as in Theorem 1.3 and denote by hol p ( F ) the set of rightinvariant vector fields on Hol p ( F ) . Then hol p ( F ) = span { ( dφ c dπ ) − ( A X Y ) | X, Y ∈ H c (1) , c (0) = p, c horizontal } , where φ c is the holonomy transformation induced by c . Proof of Corollary 1.4.
Let G be a Lie group. We call a foliation F as G -principal if the leaves of F coincide with the orbits of a locally free G -action. Herewe show that a foliation F satisfying the hypothesis of Theorem 1.3 is principal ifand only if H p ( F ) = { id } .Assuming H p ( F ) = { id } , F has bounded holonomy and Hol p ( F ) is a finitedimensional Lie group. We proceed with few additional observations. Lemma 4.4.
Let F be a Riemannian foliation satisfying the hypothesis in Theorem1.3. If H p ( F ) = { id } , then (1) the action of Hol p ( F ) on π − ∗ ( L p ∩ E p ) is free and transitive; (2) τ p : E p → M is a diffeomorphism; (3) V → M is a trivial vector bundle. Item (1) follows from Claim 4.2, (2) from Lemma 3.5. For (3), we observe themore general fact that τ ∗ p V → E p is trivial: the trivialization map ¯ χ : E p × V p → τ ∗ p V is defined as ¯ χ ( h, ξ ) = hξ . Since τ p is a diffeomorphism throughout this section, we consider the trivializa-tion χ : M × V p → V , χ ( q, ξ ) = τ − p ( q ) ξ . We fix ξ ∈ V p and explore the vectorfield ξ ( q ) = χ ( q, ξ ). Lemma 4.5.
Let c be a horizontal curve, then ˆ c (1) ξ ( c (0)) = ξ ( c (1)) . Proof.
Let α : I → M be a horizontal curve joining p to c (0) and denote by cα theconcatenated curve. On one hand, c cα (1) = ˆ c (1) α (1). On the other hand, since τ p is a diffeomorphism, c cα (1) = τ − p ( cα (1)) = τ − p ( c (1)). Thus,ˆ c (1) ξ ( c (0)) = ˆ c (1) τ − p ( c (0)) ξ = ˆ c (1)ˆ α (1) ξ = c cα (1) ξ = τ − p ( c (1)) ξ = ξ ( c (1)) . (cid:3) According to Lemma 4.5, the action µ in Claim 4.2 induces an embedding µ z ( h ) = τ p ( µ ( h, z )) of Hol p ( F ) on ˜ L . This embedding is µ -equivariant, i.e., µ z ( gh ) = gµ z ( h ). In particular, µ z sends right-invariant fields to µ -action fields.Given ξ ( q ) = χ ( q, ξ ), consider ˜ ξ as the only vector field on Hol p ( F ) which is π ◦ µ z -related to ξ | L p . Claim 4.6. ˜ ξ is left invariant.Proof. Since Hol p ( F ) is a finite-dimensional Lie group, it is sufficient to show that˜ ξ commutes with a set of generators in hol p ( F ). According to Corollary 4.3, a setof generators is given by { ( dφ c dπ ) − ( A X Y ) | X, Y basic horizontal } . Since both( dφ c dπ ) − ( A X Y ) and ˜ ξ are π -related and dφ c ( ξ ) = ˆ c (1) ξ = ξ , it is sufficient tocompute [ A X Y, ξ ]. Taking
X, Y such that [
X, Y ] h = 0 as in Claim 2.3, we have2[ A X Y, ξ ] = [[
X, Y ] , ξ ] = [[ X, ξ ] , Y ] + [ X, [ Y, ξ ]] . Both terms in the right-hand-side is zero since the restriction of ξ to any horizontalcurve is a holonomy field (Lemma 4.5). (cid:3) We are ready to prove the main result of this section:
Proposition 4.7.
Let X be the collection of vertical fields of the form ξ ( q ) = χ ( q, ξ ) . Then X is a subalgebra of vector-fields isomorphic to hol p ( F ) .Proof. Let ξ, η ∈ X , ξ ( p ) = ξ , η ( p ) = η . Using the notation in Claim 4.6, it issufficient to show that [ ξ, η ]( q ) = χ ( q, [ dπ ˜ ξ, dπ ˜ η ]) = χ ( q, dπ [ ˜ ξ, ˜ η ]).Observe that [ ξ, η ] is vertical, since ξ, η are vertical, and [ ξ, η ] | L q = [ ξ | L q , η | L q ]for every q ∈ M . On the other hand, according to Claim 4.6 the differential d ( µ z )induces an isomorphism between the algebra of left invariant fields on Hol p ( F ) tothe restriction X | L p = { ξ | L p | ξ ∈ X } . In particular, if ξ, η ∈ X , then [ ξ | L p , η | L p ] =[ ξ, η ] | L p ∈ X | L p .Furthermore, given a horizontal curve c connecting p to q , consider φ c , its localholonomy transformation between neighborhoods U ⊂ L p of p and V ⊂ L q of q .We have:[ ξ, η ] | V = dφ c [ dφ − c ( ξ | U ) , dφ − c ( η | U )] = dφ c [ ξ | U , η | U ] = dφ c [ ξ, η ] | U . Showing, in particular, that X is an algebra. Moreover, the restriction map X → X | L p is an injective epimorphism, since the holonomy invariance and irreducibility of F guarantees that a field ξ ∈ X whose restriction ξ | L p vanishes, vanishes identicallyon M . (cid:3) From standard action theory, X integrates to a smooth locally free G -action,where G is the universal cover of Hol p ( F ). By construction, the action is transitiveon leaves, proving Corollary 1.4.Generically the new Hol p ( F )-action is not by isometries. Nevertheless, it alwayspreserves basic horizontal fields: from holonomy invariance, [ ξ, X ] = 0 whenever ξ ∈ X and X is basic horizontal. In particular: IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 17
Proposition 4.8.
The principal G -action is by isometries if and only if its restric-tion to each leaf is by isometries. When F has totally geodesic leaves, it is sufficient to verify the hypothesis ofProposition 4.8 in a single leaf, since q χ ( q, ξ ) commutes with holonomy tans-formations, which are isometries.4.3. Proof of Theorem 1.3.
Using Corollary 1.4, Theorem 1.3 follows once ob-served that H id ( ˜ F ) = { id } (Proposition 4.9). In this case, Hol p ( F ) acts freely on L p ∩ E p since Hol id ( ˜ F ) is the kernel of the isotropy representation of Hol p ( F ). Proposition 4.9.
Let F be as in Theorem 1.3. Then H id ( ˜ F ) = { id } .Proof. First we describe H id ( ˜ F ) explicitly. Denote H ′ = ζ ( T M ) | E p (where ζ is themap in the proof of Lemma 3.3). We decompose ˜ V in two factors: T E p ∩ ker τ p ,which we identify with h p ( F ), the Lie algebra of H p ( F ), via the restriction of theconnection 1-form ω : T E p → h p ( F ) of section 3.1; and the H ′ -factor, V ′ = H ′ ∩ ˜ V .Given a horizontal loop α at id ∈ E p , its infinitesimal holonomy is decomposed asˆ α = (cid:18) ˆ α ˆ α ˆ α ˆ α (cid:19) : V ′ id ⊕ h p ( F ) → V ′ id ⊕ h p ( F ) . That is, ˆ α ij : V i → V j is a linear map where V = V ′ id and V = h p ( F ). We showthat ˆ α = 0 and both ˆ α , ˆ α are identity maps.The elements ˆ α , ˆ α are easy to understand: if ζ is an H p ( F )-action field, therestriction ζ | α is a holonomy field in ˜ F , thereforeˆ α (1)( ζ (id)) = ζ ( α (1)) = ζ (id) . In particular, ˆ α = 0 and ˆ α = id h p ( F ) .The remaining terms ˆ α , ˆ α are related to lifts of variations of horizontal curvesin M , analogous to the construction of projectable Jacobi fields in Riemanniansubmersions (see [12]). Let ζ ( t ) be a holonomy field with ζ (0) = ζ ∈ V ′ id . Let γ ( s )be a curve in L p which is tangent to H ′ and satisfies ˙ γ (0) = ζ (0). By the definitionof holonomy fields, there is a variation α s of α through ˜ F -horizontal curves realizing ζ , such that γ ( s ) = α s (0). Since, for every s , ˙ α s ∈ ˜ H , the curves c s = τ p ◦ α s definea variation of c = τ p ◦ α through F -horizontal curves. Furthermore, ξ ( t ) = dτ ( ζ ( t ))is a holonomy field along τ ◦ α . From uniqueness of ˜ H -horizontal lifts, we concludethat α = ˆ c . Therefore, ξ (1) = ˆ c (1) ξ (0) = α (1) ξ (0) = ξ (0) (recall that α is a loop atid.) In particular, since ˆ α ( ζ (0)) is the V ′ -component of ζ (1), ˆ α ( ζ (0)) = ζ (0).To conclude that ˆ α = 0, we observe that ˜ F has totally geodesic leaves in themetric constructed in Proposition 3.4, therefore H p ( ˜ F ) must be bounded. On theother hand, if ˆ α = 0, ˆ α k = (cid:18) id 0ˆ α id (cid:19) k = (cid:18) id 0 k ˆ α id (cid:19) . defining an unbounded subgroup of H p ( ˜ F ), a contradiction. (cid:3) Splitting of totally geodesic foliations.
Here we prove Theorem 1.5. Let F be as in the hypothesis of Theorem 1.5 and fix p ∈ M . Let T L p = L i ˜∆ i be thede Rham decomposition of T L p . Claim 4.10. If H p ( F ) is connected, then Hol p ( F ) preserves ˜∆ i . Proof.
Since F has totally geodesic leaves, ˆ c (1) is the differential of a local isometry.Moreover, since F is irreducible, H q ( F ) is isomorphic to H p ( F ) for every q ∈ M . Since H q ( F ) is isomorphic to the isotropy representation of Hol q ( F ) at any z ∈ π − ( q ), every holonomy transformation of a loop at p preserves the de Rhamdecomposition T L p = L ˜∆ i (recall that an isometry subgroup does not preserve thedecomposition only if it interchanges factors – as it follows from uniqueness of thede Rham composition – in particular, it has non-connected isotropy representationat some point.) (cid:3) We extend the distribution ˜∆ i via holonomy transportation: given q , let c be ahorizontal curve joining q to p and define ∆ i ( q ) = ˆ c (1)( ˜∆ i ( p )). ∆ i is well-definedsince, if c , c are horizontal curves joining p to q , ˆ c (1) − ˆ c (1) ∈ H p ( F ), thus:ˆ c (1)( ˜∆ i ) = ˆ c (1)(ˆ c (1) − ˆ c (1) ˜∆ i ) = ˆ c (1)( ˜∆ i )It is sufficient to analyze integrability of ∆ i leafwise since ∆ i ⊂ V . But, ∆ i | L q =˜∆ i is integrable and ∆ i | L q is an isometric translation of a de Rham factor of T L p ,therefore, ∆ i | L q is both integrable and integrates a Riemannian totally geodesicfoliation on L q . Since L q is totally geodesic on M , the integral submanifolds of ∆ i are totally geodesic on M . Claim 4.11. F i , the foliation defined by ∆ i , is Riemannian.Proof. To show that F i is Riemannian, we show that L U g ∆ ⊥ i = 0 for every U ∈ ∆ i (see [12, Theorem 1.2.1]). But ∆ ⊥ i = H ⊕ ( ⊕ i = j ∆ j ) and: L U g ( H , V ) = 0 and L U g ( H , H ) = 0 since F is Riemannian; L U g (∆ j , ∆ k ) = 0, j, k = i , since the re-striction of ∆ i to each leaf is Riemannian. (cid:3)(cid:3) An Ambrose-Singer theorem for totally geodesic foliations onnon-negatively curved manifolds
Here we use Theorem 2 to prove Theorem 1.6. The core of Theorem 1.6 lies inthe following inequality.
Lemma 5.1.
Let F and M be as in Theorem 1.6. Then, for every x ∈ M , thereis a neighborhood of x and a τ > such that τ || X |||| Z |||| A ξ X || ≥ | (cid:10) ( ∇ X A ξ ) X, Z (cid:11) | (7) for all horizontal X, Z and vertical ξ .Proof. Given
X, Z ∈ H and ξ ∈ V , O’Neill’s equations ([12, page 44]) states thatthe unreduced sectional curvature K ( X, ξ + tZ ) = R ( X, ξ + tZ, ξ + tZ, X ) satisfies(8) K ( X, ξ + tZ ) = t K ( X, Z ) + 2 t h ( ∇ X A ) X Z, ξ i + || A ξ X || . Since K ( X, ξ + tZ ) ≥ t ) must be non-negative:0 ≤ K ( X, Z ) || A ξ X || − h ( ∇ X A ) X Z, ξ i . On small neighborhoods, continuity of K guarantees there exists some τ > K ( X, Z ) ≤ τ || X || || Z || . On the other hand, computing for a holonomy field ξ along a horizontal curve c such that ˙ c = X , we conclude for all Y, Z horizontal: h ( ∇ X A ) Z Y, ξ i = − (cid:10) ( ∇ X A ξ ) Z, Y (cid:11) . (cid:3) IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 19
Proposition 5.2.
Let F be as in Theorem 1.6. Let X ∈ H p , ξ ∈ V p be such that A ξ X = 0 . Then ξ ( t ) , the holonomy field along c ( t ) = exp( tX ) with ξ (0) = ξ ,satisfies A ξ ( t ) ˙ c ( t ) = 0 for all t .Proof. Taking || X || = 1 and replacing Z = A ξ ˙ c in (7), we get: τ || A ξ ˙ c || ≥ (cid:10) ( ∇ ˙ c A ξ ) ˙ c, A ξ ˙ c (cid:11) = (cid:10) ∇ ˙ c ( A ξ ˙ c ) , A ξ ˙ c (cid:11) = 12 ddt || A ξ ˙ c || . (9)Inequality (9) is Gronwall’s inequality for u ( t ) = || A ξ ( t ) ˙ c ( t ) || and implies || A ξ ( t ) ˙ c ( t ) || ≤ || A ξ (0) ˙ c (0) || e τt for all t >
0. In particular, if A ξ (0) X (0) = 0, A ξ ( t ) ˙ c ( t ) = 0 for all t >
0. The sameargument works for t <
0, by replacing X by − X . (cid:3) Fixed a holonomy field ξ ( t ), our next task is to understand the distribution D ( t ) = ker( A ξ : H c ( t ) → H c ( t ) ). The main result of this section is the constancy ofits rank (Proposition 5.5). We prove two technical lemmas for this aim. Lemma 5.3.
Let
X, Y ∈ H be orthonormal with A ξ X = 0 . Then, τ || A ξ Y || ≥ (cid:10) ( ∇ X A ξ ) Y + ( ∇ Y A ξ ) X, A ξ Y (cid:11) . (10) Proof.
We use (7) to get:2 τ || A ξ ( Y + X ) || ≥ (cid:10) ( ∇ X A ∗ ) X ξ + ( ∇ X A ξ ) Y + ( ∇ Y A ξ ) X + ( ∇ Y A ξ ) Y, Z (cid:11) τ || A ξ ( Y − X ) || ≥ − (cid:10) ( ∇ X A ξ ) X − ( ∇ X A ξ ) Y − ( ∇ Y A ξ ) X + ( ∇ Y A ξ ) Y, Z (cid:11)
Now one just sums up both inequalities and observes that A ξ ( X + Y ) = A ξ ( Y − X ) = A ξ Y . (cid:3) Consider the non-negative symmetric operator D = − A ξ A ξ . We recall thatker A ξ = ker D . Furthermore, if DY = λ Y for λ >
0, we can set ¯ Y = λ − A ξ Y ,so that || ¯ Y || = || Y || and A ξ ¯ Y = − λY . In particular, if || Y || = 1, || DY || = λ and || A ξ Y || = || A ξ ¯ Y || = λ . Lemma 5.4.
Let
X, Y be unitary horizontals satisfying A ξ X = 0 and DY = λ Y =0 . Then, (cid:10) ( ∇ Y A ξ ) X, A ξ Y (cid:11) + (cid:10) ( ∇ ¯ Y A ξ ) X, A ξ ¯ Y (cid:11) = (cid:10) ( ∇ X A ξ ) ¯ Y , A ξ ¯ Y (cid:11) Proof.
Lemma 1.5.1 in [12, page 26] gives, (cid:10) ( ∇ Y A ξ ) X, ¯ Y (cid:11) = − (cid:10) ( ∇ X A ξ ) ¯ Y , Y (cid:11) − (cid:10) ( ∇ ¯ Y A ξ ) Y, X (cid:11) . Observing that A ξ Y = λ ¯ Y and A ξ ¯ Y = − λY , we have (cid:10) ( ∇ Y A ξ ) X, A ξ Y (cid:11) = λ (cid:10) ( ∇ Y A ξ ) X, ¯ Y (cid:11) = − λ [ (cid:10) ( ∇ X A ξ ) ¯ Y , Y (cid:11) + (cid:10) ( ∇ ¯ Y A ξ ) Y, X (cid:11) ]= (cid:10) ( ∇ X A ξ ) ¯ Y , A ξ ¯ Y (cid:11) − (cid:10) ( ∇ ¯ Y A ξ ) X, A ξ ¯ Y (cid:11) . (cid:3) Proposition 5.5.
Let X , ξ satisfy A ξ X = 0 . If λ is a continuous eigenvalueof D along c ( t ) = exp( tX ) , then either λ vanishes identically, or λ never vanishes.Proof. We argue by contradiction, assuming that λ vanishes at t = 0 but there is l ′ > λ ( t ) > t ∈ (0 , l ′ ). From the semi-continuity of the rankof symmetric operators (in particular, of the multiplicity of its eigenvalues), there exists an l ′ > l > D has a smooth frame of eigenvectors along c ((0 , l )), c ( t ) = exp( tX ). We now prove that(11) 83 τ λ ≥ ddt λ . In particular, λ ( t ) ≤ λ ( ǫ ) e τt for all ǫ ∈ (0 , l ), t ∈ ( ǫ, l ). Thus, λ must vanish on(0 , l ), a contradiction.Inequality (11) follows from Lemmas 5.3, 5.4. Let Y be a smooth unitary vectorfield satisfying DY = λ Y . Since || A ξ Y || = λ , Lemma 5.3 gives2 τ λ ≥ (cid:10) ( ∇ X A ξ ) Y + ( ∇ Y A ξ ) X, A ξ Y (cid:11) . (12)Taking ¯ Y = λ − A ξ Y , we have D ¯ Y = λ ¯ Y and || A ξ ¯ Y || = λ . Therefore, replacing Y by ¯ Y in Lemma 5.3 gives(13) 2 τ λ ≥ (cid:10) ( ∇ X A ξ ) ¯ Y , A ξ ¯ Y (cid:11) + (cid:10) ( ∇ ¯ Y A ξ ) X, A ξ ¯ Y (cid:11) . Summing up equations (12) and (13):4 τ λ ≥ (cid:10) ( ∇ X A ξ ) Y, A ξ Y (cid:11) + (cid:10) ( ∇ X A ξ ) ¯ Y , A ξ ¯ Y (cid:11) + (cid:10) ( ∇ Y A ξ ) X, A ξ Y (cid:11) + (cid:10) ( ∇ ¯ Y A ξ ) X, A ξ ¯ Y (cid:11) . According to Lemma 5.4, the two last terms in the right-hand-side satisfy (cid:10) ( ∇ Y A ξ ) X, A ξ Y (cid:11) + (cid:10) ( ∇ ¯ Y A ξ ) X, A ξ ¯ Y (cid:11) = (cid:10) ( ∇ X A ξ ) ¯ Y , A ξ ¯ Y (cid:11) . On the other hand, (cid:10) ( ∇ X A ξ ) ¯ Y , A ξ ¯ Y (cid:11) = (cid:10) ∇ X ( A ξ ¯ Y ) , A ξ ¯ Y (cid:11) − (cid:10) A ξ ( ∇ X ¯ Y ) , A ξ ¯ Y (cid:11) = 12 ddt λ − λ (cid:10) ∇ X ¯ Y , ¯ Y (cid:11) = 12 ddt λ . Equivalently, (cid:10) ( ∇ X A ξ ) Y, A ξ Y (cid:11) = ddt λ , concluding the proof. (cid:3) Theorem 1.6 follows from Proposition 5.5 and Theorem 1.2.
Proof of Theorem 1.6.
Let p ∈ M . Observe that a ⊥ p = { ξ ∈ V p | A ξ = 0 } . Claim 5.6.
Under the hypothesis of Theorem 1.6, for any horizontal curve c , ˆ c (1)( a ⊥ p ) = a ⊥ c (1) .Proof. It is sufficient to prove the claim for horizontal geodesics, since c can besmoothly approximated by piece-wise horizontal geodesics. If c is a horizontalgeodesic, c (0) = p , and ξ ( t ) be a holonomy field with ξ (0) ∈ a ⊥ p , then ker A ξ (0) = H p and dim ker A ξ ( t ) is constant with respect to t (Proposition 5.5). Thus A ξ ( t ) = H c ( t ) for all t . (cid:3) Since ˆ c (1) is an isometry, Claim5.6 implies ˆ c (1)( a p ) = a c (1) . Theorem 1.2 com-pletes the proof. (cid:3) A root decomposition for basic horizontal fields
The usual setting for a root system consists of an abelian Lie algebra (over R ) t acting on a linear space V through a Lie algebra morphism ρ : t → End( V ) (wherethe Lie brackets on End( V ) is precisely the matrix commutator). For instance, onemay endow V with an inner product and suppose that ρ ( t ) is a subspace of com-muting skew-adjoint linear endomorphisms of V . In this case, the complexification IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 21 of A ∈ t (which we also denote by A ) acts in the complexification of V , V C , viaoperators with pure imaginary eigenvalues. The root decomposition induced by ρ is V C = X α ∈ Π V α , where, V α is the weight space of the linear function α : t → i R , V α = { X ∈ V C | ρ ( A ) X = α ( A ) X, ∀ A ∈ t } , and Π, the root system of t , is the set of linear maps α : t → i R such that V α = { } .Let F be a Riemannian foliation and ı : t ⊂ ˜ L p be a totally geodesic immersedflat with ı (0) = z . Here we prove Theorems 1.7 and 1.8. As a result, we producean action of t on the set of basic horizontal fields on ı ∗ H .6.1. Proof of Theorem 1.7.
Given ξ ∈ V p , let γ ( s ) = exp( sξ ) be a verticalgeodesic. Recall that a vector field X along γ is basic if and only if ∇ ξ X = A ξ X (see equation (2) or O’Neill [25]). We verify that ∇ ξ ( A ξ X ) = A ξ ( A ξ X ).Since fibers are totally geodesic, H and V are parallel along vertical geodesics.Therefore, given X and Y basic horizontal fields, ∇ ξ ( A ξ X ) is horizontal and (cid:10) ∇ ξ ( A ξ X ) , Y (cid:11) = ξ (cid:10) A ξ X, Y (cid:11) − (cid:10) A ξ X, ∇ ξ Y (cid:11) = − h∇ ξ ( A X Y ) , ξ i − (cid:10) A ξ X, A ξ Y (cid:11) = (cid:10) A ξ A ξ X, Y (cid:11) . Where the last equality follows since A X Y is Killing. (cid:3) Proof of Theorem 1.8.
Consider
X, Y , basic horizontal fields on ı ∗ H , andcommuting parallel vector fields ξ, η ∈ T t . Since ı ( t ) is an immersed flat, ξ, η satisfy R ( ξ, η ) = ∇ ξ ∇ η − ∇ η ∇ ξ = 0. In particular ∇ η ( A ξ X ) = ∇ η ∇ ξ X = ∇ ξ ∇ η X = ∇ ξ ( A η X ) . On the other hand, (cid:10) ∇ η ( A ξ X ) , Y (cid:11) = η (cid:10) A ξ X, Y (cid:11) + (cid:10) A η A ξ X, Y (cid:11) = − h∇ η ( A X Y ) , ξ i + (cid:10) A η A ξ X, Y (cid:11) . Analogously, h∇ ξ ( A η X ) , Y i = − h∇ ξ ( A X Y ) , η i + (cid:10) A ξ A η X, Y (cid:11) . Therefore,0 = (cid:10) ∇ η ( A ξ X ) − ∇ ξ ( A η X ) , Y (cid:11) = (cid:10) ( A η A ξ − A ξ A η ) X, Y (cid:11) − h∇ η ( A X Y ) , ξ i + h∇ ξ ( A X Y ) , η i . To prove that h∇ ξ ( A X Y ) , η i = h∇ η ( A X Y ) , ξ i = 0, we first observe that(14) ξξ h A X Y, η i = h∇ ξ ∇ ξ ( A X Y ) , η i = − h R ( A X Y, ξ ) ξ, η i = 0 , where the second equality follows since A X Y is Killing and the last since R ( η, ξ ) ξ =0. Let γ ( t ) = exp p ( tξ ) and ϕ ( t ) = (cid:10) A X ( γ ( t )) Y ( γ ( t )) , η ( γ ( t )) (cid:11) . Equation (14) impliesthat ϕ ( t ) is linear; the boundedness of A, | X | , | Y | , | η | and | ξ | implies ϕ bounded.Therefore ϕ ( t ) is constant and h∇ ξ ( A X Y ) , η i = ξ h A X Y, η i vanishes. (cid:3) Good triples and Totally geodesic foliations on Lie groups
We specialize to the case of a Riemannian foliation, F , with totally geodesicconnected leaves on a compact Lie group with bi-invariant metric, G . This sectionhas a technical aim: to split the elements in H id into commuting subspaces H ± ( F )which behave as spaces of left and right invariant horizontal fields (Theorem 7.7 –see section 1.2 for a geometric motivation). Such splitting is fundamental in theproof of Proposition 1.9 (section 8).Theorem 1.5 of [24] (Theorem 7.1 below) lays the ground for Proposition 7.4, themain algebraic identity used in Theorem 7.7. Theorem 7.1 provides a fundamentalbracket relation between vertical and horizontal vectors, which is explored usingthe root system in section 6. Theorem 7.1 (Theorem 1.5, [24]) . Let G be a compact Lie group with a bi-invariantmetric and denote its Lie algebra by g . The triple { J, V, A } ⊂ g is good if and onlyif, for all integers n, m ≥ , [ad nJ B, ad mV ¯ B ] = 0 , where B = ad V J − A and ¯ B = − ad V J − A . Given a Riemannian manifold M , we recall that a triple { X, V,
A} ⊂ T p M is goodif exp p ( tV ( s )) = exp p ( sX ( t )) for all s, t ∈ R , where V ( s ) , X ( t ) denote the Jacobifields along exp( sV ) and exp( tX ), respectively, satisfying V (0) = V , X (0) = X and V ′ (0) = A = X ′ (0). If F is endowed with a totally geodesic Riemannian foliation, { X, ξ, A ξ X } is a good triple for every X ∈ H , ξ ∈ V .7.1. Decomposition of the Horizontal space at the identity.
Consider amaximal vertical abelian subalgebra t v ⊂ V id completed to a maximal abeliansubalgebra t = t v ⊕ t ′ . t (and, in particular, t v ) acts on g through ρ ad ( ξ ) = ad ξ . t v has an additional action on H given by ρ A ( ξ ) = A ξ (Theorem 1.8).Given a linear map α : t v → i R , we call X ∈ g C (respectively X ∈ H C id ) a vertical α -weight (respectively an α - A -weight ) if, for all ξ ∈ t v , ad ξ ( X ) = α ( ξ ) X (respec-tively, A ξ ( X ) = α ( ξ ) X ). If there is a non-trivial vertical α -weight (respectively, α - A -weight), α is called a vertical root (respectively, an A -root ), denoting the set ofvertical roots as Π v ( t v ) (and the set of A -roots as Π V ( t v )). We complete a verticalroot α to a root ( α, β ) : t → i R using a linear function β : t ′ → i R (observe that( t ⊕ t ′ ) ∗ = t ∗ ⊕ ( t ′ ) ∗ . In particular, if ( α, β ) is a root, α is a vertical root.) Denotethe set of roots of the form ( α, β ) as Π( t ). We consider the weight spaces g ( α,β ) ( t ) = { X ∈ g C | ad ξ + ξ ′ X = ( α ( ξ ) + β ( ξ ′ )) X, for all ξ ∈ t v , ξ ′ ∈ t ′ } , g α ( t v ) = { X ∈ g C | ad ξ X = α ( ξ ) X, for all ξ ∈ t v } , H α ( t v ) = { X ∈ H C id | ad ξ X = α ( ξ ) X, for all ξ ∈ t v } , taking advantage of a two-level decomposition: g = t v + X α g α ( t v ) = t + X ( α,β ) g ( α,β ) ( t )Standard theory also guarantees that H C id = L H α ( t v ). The aim of this subsectionis to prove Proposition 7.4, which refines Theorem 7.1. The next Lemma settlesthe connection between vertical weights and A -weights. IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 23
Lemma 7.2.
Let t v be a maximal vertical abelian subalgebra. Then, Π v ( t v ) =Π V ( t v ) . Moreover, if X ∈ H C id is an α - A -weight, then X = X α + X − α . That is X ∈ g α ( t v ) + g − α ( t v ) .Proof. Since we are dealing with a totally geodesic Riemannian submersion in a bi-invariant metric, O’Neill’s equations gives for every ξ ∈ V id the string of identities(compare Ranjan [28], equation (1.3)):(15) − A ξ A ξ X = R ( X, ξ ) ξ = −
14 ad ξ X In particular, if X is an α - A -weight,( A ξ ) X = α ( ξ ) X = 14 ad ξ X for all ξ ∈ t . On the other hand, if α, α ′ : V id → R are linear functionals on V id suchthat α ( ξ ) = α ′ ( ξ ) for every ξ , then α = ± α ′ : if we suppose there exist ξ , ξ suchthat α ( ξ ) = α ′ ( ξ ) = 0 and α ( ξ ) = − α ′ ( ξ ) = 0 then the identity α ( ξ ) = α ′ ( ξ ) gives α ( ξ + tξ ) = α ( ξ ) + 2 tα ( ξ ) α ( ξ ) + t α ( ξ ) = α ′ ( ξ + tξ ) = α ( ξ ) − tα ( ξ ) α ( ξ ) + t α ( ξ ) , for all t , contradicting α ( ξ ) α ( ξ ) = 0. The Lemma follows since ∩ ξ ∈ t v ker( ad ξ − α ( ξ ) ) = g α + g − α . (cid:3) Given a maximal vertical abelian subalgebra t v , Lemma 7.2 provides projections π ± : H C id → g C defined by sending X ∈ H α ( t v ) to its g ± α ( t v ) component. Let H ± ( t v ) = π ± ( H id ) , H ( t v ) = ker π + ∩ ker π − . Although, H ( t v ) ⊂ H C id , H ± ( t v ) are not necessarily horizontal as it can be seen from F ∆ in section 1.2. We further observe that H ( t v ) = ∩ ξ ∈ t v ker ad ξ = ∩ ξ ∈ t v ker A ξ since(16) ker ad ξ = ker ad ξ = ker( A ξ ) = ker A ξ . Moreover, since both A ξ and ad ξ are real linear maps (i.e., commute with complexconjugation), H ( t v ) , H + ( t v ) , H − ( t v ) are real subspaces, i.e., are complexification ofsubspaces H R ǫ ( t v ) ⊂ g , ǫ = 0 , + , − . Furthermore, H ( t v ) ∩ ( H + ( t v )+ H − ( t v )) = { } . Lemma 7.3.
Let t v be a maximal vertical subalgebra and t ⊃ t v a maximal torus.Then, t decomposes orthogonally as t = t v ⊕ t ′ , with t ′ ⊂ H ( t v ) .Proof. Let t ∈ t , l ∈ t v and decompose t in its vertical and horizontal components, t = t v + t h . On one hand, equation (3) gives R ( t, l ) = 0. On the other hand,O’Neill’s equation (or [12, page 44]) gives R ( H , l, ξ, η ) = 0 for all ξ, η ∈ V id . Thus0 = R ( t, l, ξ, η ) = R ( t h , l, ξ, η ) + R ( t v , l, ξ, η ) = R ( t v , l, ξ, η ) . In particular, h R ( t v , l ) l, t v i = || [ t v , l ] || = 0. Since l ∈ t v is arbitrary and t v maximal, t v ∈ t v and t h = t − t v ∈ t . Since [ t h , l ] = 0 for all l ∈ t v , t h ∈ H ( t v ). (cid:3) Conversely, if t v is a maximal vertical subalgebra, then any maximal abeliansubalgebra t ′ ⊂ H ( t v ) gives a maximal abelian t = t v + t ′ ⊂ g . Lemma 7.3 is usedin the proof of Lemma 7.6. Bracket identities.
In this section we prove:
Proposition 7.4.
Let t = t v + t ′ be a maximal torus and X ∈ H + ( t v ) , Y ∈ H − ( t v ) .Then, for every pair of roots ( α, β ) , ( α ′ , β ′ ) ∈ Π , [ X ( α,β ) , Y ( α ′ ,β ′ ) ] = 0 . From now on, we fix a maximal vertical abelian subalgebra t v and a complement t ′ ⊂ H ( t v ). We prove two auxiliary lemmas. The next Lemma is essentially arestatement of Theorem 7.1. Lemma 7.5.
Let ξ ∈ t v , X ∈ H C id and denote by X ǫ , ǫ = 0 , + , − , the H ǫ ( t v ) -component of X . Then, for all n, m ≥ , [ad mX ad ξ X − , ad n +1 ξ X + ] = 0 . Proof.
Let ξ ∈ t v and X ∈ ⊕ α =0 H α ( t v ) be a horizontal vector without componentsin H ( t v ). We denote the decomposition of X into A -weights as X = P X α , andthe projection of each α - A -weight into H ± ( t v ) as X α ± . Observe that ad ξ X α ± = ± α ( ξ ) X α ± and A ξ ( X α + + X α − ) = α ( ξ ) X α . Thus, for the good triple { X, ξ, A ξ X } , B = 12 ad ξ X − A ξ X = X α =0 (cid:18)
12 ad ξ ( X α + + X α − ) − A ξ ( X α + + X α − ) (cid:19) = X α =0 α ( ξ ) (cid:0) ( X α + − X α − ) − ( X α + + X α − ) (cid:1) = − X α =0 α ( ξ ) X α − = − ad ξ X − . Analogously, ¯ B = − ad ξ X + . (cid:3) Lemma 7.6.
Given X ∈ H C id , ξ ∈ t v , for all integers m, n ≥ , [ad ξ X − , ad mX ad n +1 ξ X + ] = 0 . Proof.
The proof is through induction on s for(17) [ad rX X ′− , ad sX X ′ + ] = 0 , where X ′− = ad ξ X − and X ′ + = ad n +1 ξ X + . Observe that (17) holds for s = 0 and r ≥ s ≤ k and r ≥
0. We compute [ad rX X ′− , ad k +1 X X ′ + ] backwards:0 = [ad r +1 X X ′− , ad kX X ′ + ] =[[ X , ad rX X ′− ] , ad kX X ′ + ] + [[ X ′− , ad rX X ′− ] , ad kX X ′ + ] + [[ X ′ + , ad rX X ′− ] , ad kX X ′ + ]= ad X [ad rX X ′− , ad kX X ′ + ] − [ad rX X ′− , ad k +1 X X ′ + ] + ad X ′− [ad rX X ′− , ad kX X ′ + ] − [ad rX X ′− , [ X ′− , ad kX X ′ + ]] + [[ X ′ + , ad rX X ′− ] , ad kX X ′ + ] = − [ad rX X ′− , ad k +1 X X ′ + ]Where the last equality holds since, by the induction hypothesis, [ad rX X ′− , ad kX X ′ + ] =[ad rX X ′− , ad kX X ′ + ] = [ X ′− , ad kX X ′ + ] = [ X ′ + , ad rX X ′− ] = 0. (cid:3) Proof of Proposition 7.4.
Let X ∈ H C id be a horizontal field without H ( t v )-componentand l ∈ t ′ ⊂ H ( t v ). Consider Z = l + X . Let X ± ( α,β ) be the g ( α,β ) ( t )-component of X ± . Replacing X by Z in Lemma 7.6, we have for all n ≥ m ≥ ξ ∈ t v , l ∈ t ′ :0 = [ad ξ X − , ad nl ad mξ X + ] = X ( γ,δ ) ∈ Π γ ( ξ ) m δ ( l ) n X ( α,β ) ∈ Π α ( ξ )[ X − ( α,β ) , X +( γ,δ ) ] . IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 25
Let { ξ i } , { l i } be bases for t v and t ′ where α ( ξ i ) = 0, β ( l i ) = 0 whenever both α, β are non-zero roots. Replacing ξ, l by ξ i , l i and taking values enough of m, n , weconclude that(18) X ( α,β ) ∈ Π α ( ξ )[ X − ( α,β ) , X +( γ,δ ) ] = 0for every ( γ, δ ) ∈ Π. On the other hand [ X − ( α,β ) , X +( γ,δ ) ] ∈ g ( α + γ,β + δ ) ( t ), thus eachterm in the sum (18) lies in a different weight space concluding that [ X − ( α,β ) , X +( γ,δ ) ] =0 for all ( α, β ) , ( γ, δ ) ∈ Π( t ).We observe that we can assume G simple. If not, we consider the projection ofeach element X ± ( α,β ) into simple components of G . Thus, the brackets [ , ] : g ( α,β ) ( t ) × g ( α ′ ,β ′ ) ( t ) → g ( α + α ′ ,β + β ′ ) ( t ) is either zero, when g ( α + α ′ ,β + β ′ ) ( t ) = { } , or inducesa non-degenerate bi-linear pairing from a pair of one-dimensional subspaces to aone-dimensional subspace.Let π ( α,β ) : H C id → g ( α,β ) ( t ) be the linear projection into g ( α,β ) ( t ) and let π ± ( α,β ) = π ( α,β ) ◦ π ± . Suppose that ( α + α ′ , β + β ′ ) is a root. Then the pairing [ , ] : g ( α,β ) ( t ) × g ( α ′ ,β ′ ) ( t ) → g ( α + α ′ ,β + β ′ ) ( t ) is non-degenerate. Since [ π +( α,β ) ( X ) , π − ( α ′ ,β ′ ) ( X )] = 0for every X ∈ H , H = ker π +( α,β ) ∪ ker π − ( α,β ) (recall that all root spaces have one-dimension), which is only possible if one of the kernels is H . In particular, for everypairs ( α, β ) , ( α, β ′ ), [ π +( α,β ) ( H + ( t v )) , π − ( α ′ ,β ′ ) ( H − ( t v ))] = { } . (cid:3) The left-right horizontal splitting.
In this section we refine the H ǫ ( t v )-splitting to a, possibly new, t v -independent splitting. Let H ֒ → G be a connectedimmersed subgroup of G whose adjoint representation leaves V id invariant and istransitive in the set of maximal vertical abelian subalgebras. That is, if h ∈ H thenAd h ( V id ) = V id , moreover, given t v , every maximal vertical abelian subalgebra isof the form Ad h t v . We recall a few points:(1) if L id is a symmetric space then there is an H <
Iso( L id ) ֒ → G satisfyingthe hypothesis above (see [6])(2) if L id is a subgroup, then H can be taken as L id (3) by denoting ( α ◦ Ad − h , β ◦ Ad − h ) = h ∗ ( α, β ),Π(Ad h t ) = { h ∗ ( α, β ) | ( α, β ) ∈ Π( t ) } (4) g h ∗ ( α,β ) (Ad h t ) = Ad h ( g ( α,β ) ( t ))Define the vector spaces H ± ( F ) = P h ∈ H H ± (Ad h t v ) and H ( F ) = H C id ∩ ( H + ( F ) + H − ( F )) ⊥ . It is clear that H ǫ ( F ) is independent of H . We state themain result of the section. Theorem 7.7.
For F as in Theorem 1.1, H + ( F ) ⊥H − ( F ) and [ H + ( F ) , H − ( F )] = 0 . Theorem 7.7 follows from Proposition 7.4 and Proposition 7.8 below. We fixarbitrary maximal abelian subalgebras t v ⊂ t for the proof. Proposition 7.8.
Let t v be a maximal vertical abelian subalgebra and h ∈ H . Then H ± (Ad h t v ) = Ad h H ± ( t v ) . The proof of Proposition 7.8 takes advantage of Proposition 7.4 to control theset of Ad h t -roots. We proceed with three Lemmas. Denote by Υ ± ( t ) the set of roots that appear as components of elements in H ± ( t v ). Since H C id is the complex-ificaion of H id , it posses a natural complex conjugation. H ± ( t v ) is closed undersuch conjugation. In particular, ( α, β ) ∈ Υ ± ( t ) if and only if ( − α, − β ) ∈ Υ ± ( t ). Lemma 7.9. Υ + ( t ) ∩ Υ − ( t ) = ∅ .Proof. Recall that, given a set of positive roots Σ + ( t ), t has a generating set { H ( α,β ) } ( α,β ) ∈ Σ + ( t ) , satisfying[ X ( α,β ) , Y ( − α, − β ) ] = (cid:10) X ( α,β ) , Y ( − α, − β ) (cid:11) H ( α,β ) . According to Proposition 7.4, [ X +( α,β ) , Y − ( − α, − β ) ] = 0 for all X + ∈ H + ( t v ) and Y − ∈ H − ( t v ). Since h , i : g ( α,β ) × g ( − α, − β ) → C is non-degenerate and the rootspaces are one-dimensional, either π +( α,β ) ( H id ) = { } or π − ( − α, − β ) ( H id ) = { } . (cid:3) Lemma 7.10.
For every g ∈ H , H (Ad g t v ) = Ad g H ( t v ) .Proof. H ( t v ) = H C id ∩ ξ ∈ t v ker ad ξ . Therefore,Ad g H ( t v ) = (Ad g H C id ) ∩ ξ ∈ t v (Ad g ker ad ξ ) = H C id ∩ ξ ∈ t v ker ad Ad g ξ = H C id ∩ ξ ∈ Ad g t v ker ad ξ . (cid:3) Let Υ( t ) be the set of roots that appears as components of elements in H id .Since Ad g fixes V id , Ad g H id = H id and g ∗ Υ( t ) = Υ(Ad g t ). Furthermore, since H (Ad g t v ) = Ad g H ( t v ), for any X ∈ ⊕ α =0 H α ( t v ), Ad g X ∈ ⊕ α =0 H g ∗ α (Ad g t v ).In particular, if ( α, β ) ∈ Υ + ( t ) ∪ Υ − ( t ), then g ∗ ( α, β ) ∈ Υ + (Ad g t ) ∪ Υ − (Ad g t ),i.e., Υ + (Ad g t ) ∪ Υ − (Ad g t ) = g ∗ (Υ + ( t ) ∪ Υ − ( t )). We refine this identity in thenext Lemma. Lemma 7.11.
For every g ∈ H , Υ ± (Ad g t ) = g ∗ (Υ ± ( t )) .Proof. Given ( α, β ) ∈ Υ + ( t ) ∪ Υ − ( t ), consider H ± ( α,β ) = { g ∈ H | g ∗ ( α, β ) ∈ Υ ± (Ad g t ) } . H +( α,β ) ∩ H − ( α,β ) = ∅ since Υ + (Ad g t ) ∩ Υ − (Ad g t ) = ∅ (Lemma 7.9).Moreover, H +( α,β ) ∪ H − ( α,β ) = H since g ∗ Υ( t ) ⊂ Υ(Ad g t ) for all g ∈ H . Since H is connected, the proof is completed by showing that H +( α,β ) and H − ( α,β ) are opensubsets. Claim 7.12.
The rank of H ± (Ad g t v ) does not depend on g .Proof. Observe that \ ξ ∈ t v ker(( A ξ ) − α ( ξ ) ) = H α ( t v ) ⊕ H − α ( t v ) . On the other hand, ( A Ad g ξ ) = ad g ξ = Ad g ad ξ = Ad g ( A ξ ) . Thus H g ∗ α (Ad g t v ) ⊕ H − g ∗ α (Ad g t v ) = Ad g ( H α ( t v ) ⊕ H − α ( t v )) . Assume α = 0. Denote H ± α ( g ) = H ± g ∗ α (Ad g t v ), ξ g = Ad g ξ and let π ± g denotethe H ± (Ad g t v )-projections. Then, for any ξ ∈ t v ,( g ∗ α ( ξ g ) + ad ξ g )( H α ( g ) ⊕ H − α ( g )) = π + g ( H α ( g )) ⊕ π − g ( H − α ( g )) , (19) ( g ∗ α ( ξ g ) − ad ξ g )( H α ( g ) ⊕ H − α ( g )) = π − g ( H α ( g )) ⊕ π + g ( H − α ( g )) . (20)On one hand, the ranks of the left hand sides in (19) and (20) are constant withrespect to g (since they are Ad g -equivariant). Therefore, π + g ( H α ( g ) ⊕ H − α ( g )) ⊕ π − g ( H α ( g ) ⊕ H − α ( g )), has constant rank. IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 27
To prove constancy of the rank of π ± g ( H α ( g ) ⊕ H − α ( g )), observe that( A ξ g + ad ξ g )( H α ( g ) ⊕ H − α ( g )) = π + g ( H α ( g ) ⊕ H − α ( g )) . (21) ( A ξ g − ad ξ g )( H α ( g ) ⊕ H − α ( g )) = π − g ( H α ( g ) ⊕ H − α ( g )) . (22)Thus k ± ( g ) = rank π ± g ( H α ( g ) ⊕ H − α ( g )) is lower upper-continuous, ( k ± satisfies:if k ± | A is constant, then k ± | ¯ A ≤ k ± | A for every A ⊂ H (see Lewis [20])). On theother hand, k + ( g ) + k − ( g ) = k is constant. Let A ⊂ H be the set where k + hasits minimum value. It is closed by upper semi-continuity and it is open since itcoincides with the set where k − admits its maximum value. Therefore A = H and k ± are constant functions. (cid:3) For any given root ( α, β ) ∈ Υ + ( t ) ∪ Υ − ( t ), we now prove that H ± ( α,β ) are opensets. Taking advantage of Claim 7.12, we define the smooth subbundles¯ π ± : H ± ( t v ) = { ( g, X ) ∈ H × g | X ∈ H ± (Ad g t v ) } → H, and consider the continuous maps¯ π ± ( α,β ) : H ± ( t v ) → g ( g, X ) X g ∗ ( α,β ) . Observe that g ∗ ( α, β ) ∈ Υ ± (Ad g t ) if and only if ¯ π ± ( α,β ) (( g, H ± (Ad g t ))) has a non-zero element. That is, H ± ( α,β ) = π ± ((¯ π ± ( α,β ) ) − ( g − { } )). The proof is concludedby observing that ¯ π ± ( α,β ) are continuous and ¯ π ± are open maps. (cid:3) For the proofs of Proposition 7.8 and Theorem 7.7, let h ± ( t ) be the subalgebragenerated by ⊕ ( α,β ) ∈ Υ ± ( t ) g ( α,β ) ( t ). Proposition 7.4, Lemma 7.9 and invariance bycomplex conjugation guarantees that [ h + ( t ) , h − ( t )] = 0, h + ( t ) ⊥ h − ( t ) and h + ( t ) ∩ h − ( t ) = { } . Moreover, Lemma 7.11 implies h ± (Ad g t ) = Ad g h ± ( t ) for g ∈ H . Proof of Proposition 7.8.
Let π ± ( t ) : H C id → h ± ( t ) be the projections defined by thedecomposition g = h + ( t ) + h − ( t ) + h ( t ), where h ( t ) = ( h + ( t ) + h − ( t )) ⊥ . Observethat, for every g ∈ H , π ± (Ad g t ) = Ad g ◦ π ± ( t ) ◦ Ad g − . Therefore, H ± (Ad g t v ) = π ± (Ad g t )( H C id ) = Ad g ( π ± ( t )(Ad g − H C id )) = Ad g ( π ± ( t )( H C id )) . (cid:3) We have reached a new characterization of H ± ( F ): given any t v , H ± ( F ) is thesmallest Ad H -invariant subset containing H ± ( t v ).7.4. Proof of Theorem 7.7.
We first observe that [ H + ( F ) , H − ( F )] = { } ifand only if there is t v such that [ad kθ H + ( t v ) , H − ( t v )] = { } for every θ ∈ h and k ≥
0, where h is the Lie algebra of H : given t v , Proposition 7.8 implies that[ H + ( F ) , H − ( F )] = { } if and only if [Ad H H + ( t v ) , H − ( t v )] = { } . Moreover, h isa compact Lie algebra, since it is a subalgebra of a compact Lie algebra, and H isconnected by definition. Thus every element in H can be written as e θ for some θ ∈ h .If L id is a subgroup (respectively, an irreducible symmetric space), h can betaken as V id (respectively, [ V id , V id ] – [4], Lemma 7). Since L id might be reducible,we write V id = L ∆ i , where exp(∆ i ) are locally irreducible symmetric spaces. Claim 7.13. If i = j , then [∆ i , ∆ j ] = 0 . In particular, h can be taken as h = L h i where h i = ∆ i whether ∆ i is a subalgebra or h i = [∆ i , ∆ i ] otherwise. Proof.
Since L id is totally geodesic and is locally isometric to a metric prod-uct exp(∆ ) × · · · × exp(∆ s ), the curvature tensor of G at the identity satisfies R (∆ i , ∆ j ) = 0. Therefore, h R ( ξ, η ) η, ξ i = k [ ξ, η ] k = 0 for all ξ ∈ ∆ i , η ∈ ∆ j .In particular, exp( h ) integrates a subgroup which is, up to covering, a product H = ˜ H × · · · × ˜ H s . To see that H is transitive in the set of maximal verti-cal abelian subalgebras, note that a maximal abelian subalgebra of V id splits as t v = L t v ∩ ∆ i (e.g., by using arguments as in Lemma 7.3). Thus, since each ˜ H i acts transitively on the set of abelian subalgebras on ∆ i , H acts transitively on theset of maximal abelian subalgebras of V id . (cid:3) Whenever ∆ i is not a subalgebra, [4, Lemma 7] guarantees that (∆ i ⊕ h i , h i )is a symmetric pair (note that [∆ i , h i ] ⊂ ∆ i , since ∆ i is a Lie triple system –see e.g. [15]). In particular, h i is orthogonal to V id (it is orthogonal to ∆ j since h [∆ i , ∆ i ] , ∆ j i = h ∆ i , [∆ j , ∆ i ] i = 0, and orthogonal to ∆ i since [∆ i , h i ] ⊂ ∆ i ). Thus h i is horizontal whenever ∆ i is not a subalgebra. We decompose h = h h ⊕ h v in itshorizontal and vertical component and denote ∆ v (respectively, ∆ h ) as the sum ofthe ∆ i -components which are (respectively, which are not) subalgebras.Observing that [ h h , h v ] = 0, we decompose θ ∈ h in its horizontal and verticalcomponents, Z and ξ , and prove that(23) [ad mξ ad nZ H + ( t v ) , H − ( t v )] = 0 . We use induction on m : first we show that (23) holds for m = 0 and n ∈ N ,then, assuming that (23) holds for m ≤ k and all n ≥
0, we show that it holds for m = k + 1. Claim 7.14. [ad nZ H + ( t v ) , H − ( t v )] = 0 for all n ≥ .Proof. Let Z ∈ h h and decompose Z = Z + Z + + Z − according to H ǫ ( t v ). Wechoose t ′ such that Z ∈ t ′ . Since [ h + ( t ) , H − ( t v )] = 0 and H + ( t v ) ⊂ h + ( t ), it issufficient to show that h + ( t ) is ad Z -invariant. But ad Z ( h + ( t )) ⊂ h + ( t ) since Z ∈ t ;ad Z + ( h + ( t )) ⊂ h + ( t ) since h + ( t ) is a subalgebra and Z + ∈ h + ( t ); ad Z − ( h + ( t )) = { } ⊂ h + ( t ) (Proposition 7.4). (cid:3) From now on, we assume h v = { } – Claim 7.14 proves Theorem 7.7 whenever h v = { } . Let X + ∈ H + ( t v ), Y − ∈ H − ( t v ) and assume that [ad mξ ad nZ X + , Y − ] = 0for all m ≤ k and n ≥
0. Denote X ′ = ad kξ ad nZ X + . Claim 7.15. ad mξ ad nZ X + ∈ H C id ∩ ( H ( t v ) + H + ( t v )) for all n ≥ , ≤ m ≤ k + 1 .Moreover, if X + ∈ H C id ∩ H + ( t v ) , then ad nZ X + ∈ H C id ∩ ( H ( t v ) + H + ( t v )) .Proof. By the definition of H , ad h preserves H id . Moreover, for any ξ ∈ ∆ v , ad ξ preserves H id , since ∆ v is a subalgebra, ad ξ is skew-symmetric and [∆ i , ∆ j ] = 0.Let X α be an α - A -weight. Then, X α ± ∈ H C id for every α - A -weight α whose A -root α satisfies α ( ξ ) = 0 for some ξ ∈ ∆ v ∩ t v (recall that X α ± = (2 α ( ξ )) − ( A ξ ± ad ξ ) X α ).Moreover, [( X α + ) v , ∆ v ] = 0: as observed, ( X α + ) v = 0 only if α ( ξ ) = 0 for every ξ ∈ ∆ v ∩ t v . Therefore, supposing ( X α + ) v = 0, there is ξ ∈ t v ∩ ∆ h such that X α + = (2 α ( ξ )) − ad ξ ( X α + ). Since [ ξ , ∆ v ] = 0, we conclude that h X α + , ∆ v i = 0. IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 29
On the other hand, [( X α + ) ∆ h , ∆ v ] = 0. In particular,(24) ad mξ ad nZ X + = ad m − ξ ad nZ ad ξ X + = ad mξ ad nZ X α ∈ Π v α (∆ v ∩ t v ) =0 X α + , where X α + ∈ H C id ∩ H + ( t v ) for each X α + in (24), therefore ad mξ ad nZ X + ∈ H C id . Onthe other hand, using the induction hypothesis,(25) (cid:10) ad mξ ad nZ X + , H − ( t v ) (cid:11) = h ξ, [ad m − ξ ad nZ X + , H − ( t v )] i = 0 , thus ad mξ ad nZ X + ∈ H ( t ) + H + ( t v ) for all 1 ≤ m ≤ k + 1. For the secondstatement, if X + ∈ H C id ∩ H + ( t v ), ad nZ X + ∈ H C id and Claim 7.14 guarantees thatad nZ X + ⊥H − ( t v ). (cid:3) In view of equation (24), we assume X + ∈ H C id ∩ H + ( t v ), thus X ′ ∈ H C id ∩ ( H + ( t v )+ H ( t v )). Since ad ξ X ′ has no H − ( t v )-component , [ad ξ X ′ , Y − ] = [(ad ξ X ′ ) , Y − ](recall that the H + ( t v )-component commutes with H − ( t v ) – Proposition 7.4). Onthe other hand, by taking care of the root decompositions we conclude that(ad ξ X ′ ) = (ad ξ X ′ ) = 0 , since t v is maximal and ad t v H ( t v ) = 0. Since X ′ has no H − ( t v )-component,(ad ξ X ′ ) = (ad ξ X ′ + ) . Replacing X + by X ′ + in equation 25 shows that ad ξ X ′ + ∈H ( t v ) + H + ( t v ). Therefore[ad ξ X ′ , Y − ] = [(ad ξ X ′ ) , Y − ] = [(ad ξ X ′ + ) , Y − ] = [ad ξ X ′ + , Y − ] . Taking t ′ such that (ad ξ X ′ + ) ∈ t ′ , we conclude that [ad ξ X ′ + , Y − ] = [(ad ξ X ′ + ) , Y − ] ∈ h − ( t ). However, for every W − ∈ h − ( t ), (cid:10) [ad ξ X ′ + , Y − ] , W − (cid:11) = (cid:10) [ad ξ Y − , X ′ + ] , W − (cid:11) = (cid:10) ad ξ Y − , [ W − , X ′ + ] (cid:11) = 0 . Since h , i is non-degenerate on h − ( t ), [ad ξ X ′ + , Y − ] = [ad ξ X ′ , Y − ] = 0. (cid:3) Splitting of the dual foliation.
We present the first applications of Theorem7.7.
Lemma 7.16. If F is as in Theorem 1.1, then A ξ X = ad ξ ( X + − X − ) . Proof.
Fix X ∈ H id and ξ ∈ V id . Let t v be a maximal abelian vertical sub-algebra such that ξ ∈ t v and consider the decomposition into A -weights X = X + P α ∈ Π v ( t v ) X α . Since ad ξ X = 0, it is sufficient to consider X as an α - A -weight. Momentarily denote the H ǫ ( t v ) decomposition of X as X = X ′ + X ′ + + X ′− ,so that A ξ X = α ( ξ ) X , ad ξ X ′± = ± α ( ξ ) X ′± . But then, A ξ X = α ( ξ )( X ′ + + X ′− ) = 12 ad ξ X ′ + −
12 ad ξ X ′− . The proof is concluded by observing that ad ξ X ′± = ad ξ X ± since H ± ( F ) differsfrom H ± ( t v ) only by elements in H ( t v ) (By Claims 7.14, 7.15, ad H H + ( t v ) ⊥H − ( t v )). (cid:3) Due to possibly ambiguity, we clarify that the subindex here refer to the H ǫ ( t v )-decomposition, not to the t -root decomposition. Lemma 7.16 is used throughout, either as in its original form, or in its strongerversion, Proposition 8.5.As a last step in this section, we reduce Theorem 1.1 to the irreducible case. Insection 5, we identify
T L p with A (Λ H p ). Here we show that the dual foliationsplits as a product whenever G is simply connected. More precisely, G is isometricto a product L × exp ( s ), where the dual leaves are the submanifolds L × { s } . Theorem 7.17.
Let F be as in Theorem 1.1. Then s = { ξ ∈ V id | A ξ = 0 } isan ideal. Moreover, if G is simply connected, the dual foliation is isometric to themetric product L × exp( s ) .Proof. We divide the proof in three steps: Claims 7.18 and 7.19, and the conclusion.
Claim 7.18. s is an ideal.Proof. Since g = H id + V id , we prove that bracketing with horizontals and verticalsstabilizes s . Let ξ ∈ s and X ∈ H id , then, by equation (16), [ ξ, X ] = 0. Inparticular, if η ∈ V id , then [ ξ, η ] ∈ V id , since h [ ξ, η ] , H id i = h η, ad ξ H id i = 0.Furthermore, A [ ξ,η ] X = 0 for all X ∈ H id : let X = X + X + + X − be the H ǫ ( F )-decomposition. For all Y ∈ H id ,2 h A [ ξ,η ] X, Y i = (cid:10) ad [ ξ,η ] ( X + − X − ) , Y (cid:11) = h ad ξ ad η − ad η ad ξ ( X + − X − ) , Y i = − h A η ( X ) , ad ξ Y i − (cid:10) ad η A ξ ( X ) , Y (cid:11) = 0 . Thus, s is an ideal. Now assume G simply connected and identify G = G × S ,where S = exp( s ) and G = exp( s ⊥ ). (cid:3) Claim 7.19. L = G × { id } .Proof. Section 5 shows that A (Λ H ) ⊥ is parallel along horizontal geodesics. On theother hand, T S is a parallel subbundle of
T G . Therefore A (Λ H ) ⊥ | L = T S | L .In particular, L is an open subset of G . Since ( T L ) ⊥ ⊂ V , when leaves aretotally geodesic the dual foliation is Riemannian in the sense of Molino [26], that is,a geodesic that starts horizontal, stays horizontal . In particular, there is no dualleaf of positive codimension that intersects the closure of L in G × { id } . (cid:3) Proposition 4.1 in [24] (see [7] also) shows that two Riemannian foliations withtotally geodesic leaves on a complete manifold M coincide, provided their verticalspaces and A -tensors coincide at a single point. Define F ′ = { pr − ( L ∩ L ) | L ∈ F} , where pr : G × S → G is the projection in the first coordinate. F ′ is a Riemannianfoliation with totally geodesic leaves, whose vertical space and A -tensor coincidewith the vertical space and A -tensor of F . Therefore, F = F ′ and L ,s ) = L × { s } . (cid:3) The author thanks M. Alexandrino for pointing it out.
IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 31 Totally geodesic foliations on Lie groups
The algebra of bounded Jacobi fields.
Let G be a Compact Lie groupwith bi-invariant metric and γ a geodesic with γ (0) = id. There are three (usuallyintersecting) families of bounded Jacobi fields along γ : the parallel fields, the re-striction of left invariant fields and restrictions of right invariant fields. In [23, 24] itis shown that every bounded Jacobi field along γ is uniquely expressed as the sum ofone element in each family. The aim of this section is to couple this decompositionon horizontal fields along a leaf with the H ǫ ( F )-decomposition (Theorem 8.1 andProposition 8.3). We start by recalling the construction in [24], then generalize itand extend the bracket identity in Theorem 7.7 to horizontal fields.Given a geodesic γ , γ (0) = id, one decomposes g as the sum of eigenspaces V i = ker( ℜ ˙ γ (0) − k i ), where ℜ ˙ γ (0) X = R ( X, ˙ γ (0)) ˙ γ (0) and 0 = k < k < ... < k s are the eigenvalues of ℜ ˙ γ (0) . Then every Jacobi field J can be expressed as(26) J ( t ) = E + tF + s X i =1 cos( t p k i ) E i + sin( t p k i ) F i , where E i , F i are parallel fields satisfying E i (0) , F i (0) ∈ V i . J is completely definedby its initial conditions J (0) = P si =0 E i , J ′ (0) = F + P si =1 √ k i F i .A Jacobi field as in (26) has bounded norm if and only if F = 0. In this case, J ′ (0) ⊥ V , thus ad − γ (0) ( J ′ (0)) is well defined. So does(27) J + = 12 (cid:16) J (0) − E + ad − γ (0) ( J ′ (0)) (cid:17) , J − = 12 (cid:16) J (0) − E − ad − γ (0) ( J ′ (0)) (cid:17) . Thus the decomposition of J in the three families is given by: J = E , the parallelfield; J L (respectively J R ), the left (respectively right) invariant field with J L (0) = J + (respectively J R (0) = J − ) – recall that a left (respectively, right) invariantJacobi field is characterized by the relation J ′ (0) = ad ˙ γ (0) J (0) (respectively, J ′ (0) = − ad ˙ γ (0) J (0)). It is straightforward to see that J (0) = E + J + + J − and J ′ (0) = J ′ L (0) + J ′ R (0).When we deal with totally geodesic foliations, basic horizontal fields restrict tobounded (actually, constant norm) Jacobi fields along vertical geodesics, so theycan be decomposed accordingly to [24]. Theorem 8.1.
Let F be as in Theorem 1.1. Let X be a basic horizontal field along ˜ L id and consider the decomposition X (id) = X + X + + X − , with X ǫ ∈ H ǫ ( F ) , ǫ = 0 , + , − . Then X = X B + X L + X R , where(1) X L , X R are the restrictions of a left invariant field with X L (id) = X + anda right invariant field with X R (id) = X − , respectively;(2) X B is the parallel translation of X and can be realized by the restrictionof a left invariant field as well as a right invariant field.Proof. Given X (id) = X + X + + X − , the conditions X B (id) = X , X L (id) = X + , X R (id) = X − completely determine fields ˜ X B , ˜ X L , ˜ X R , where ˜ X B is parallel, ˜ X L is left invariant and ˜ X R is right invariant. To show that X = ˜ X B + ˜ X L + ˜ X R , itis sufficient to show that X ( e tξ ) = ˜ X B ( e tξ ) + ˜ X L ( e tξ ) + ˜ X R ( e tξ ) for every ξ ∈ V id and t ∈ R . On one hand, the restriction of both X and ˜ X = ˜ X B + ˜ X L + ˜ X R along e tξ are Jacobi fields. On the other hand, Corollary 7.16 gives ∇ ξ X (id) = A ξ X (id) = 12 ad ξ X + −
12 ad ξ X − = ∇ ξ ˜ X (id) . (cid:3) In order to compute the A -tensor of F (taking advantage of Theorem 7.7) weconsider the left translation of the Lie bracket as a (2 , J , K , on G . Toavoid ambiguity, we denote the Lie bracket of g as [ , ] g and the usual Lie bracket ofvector-fields as [ , ] X .Let X, Y be vector fields on G . We define J X, Y K as the vector-field J X, Y K ( g ) = dl g [ dl − g X ( g ) , dl − g Y ( g )] g , where l g : G → G stands for left multiplication by g ∈ G . Proposition 8.2. J , K is the only (2 , tensor that satisfies (28) J X L , Y L K = [ X L , Y L ] X for any pair X L , Y L of left invariant fields. In particular J , K is parallel, satisfies theJacobi identity and ∇ X L Y L = J X L , Y L K . Moreover, for any pair X R , Y R of rightinvariant fields, J X R , Y R K = [ X R , Y R ] X and ∇ X R Y R = − J X R , Y R K .Proof. J , K is a tensor since it is defined as a fiber-wise bilinear map in a trivialization T M ∼ = M × g . Equation (28) follows from J X L , Y L K = dl g [ dl − g X L ( g ) , dl − g Y L ( g )] g = dl g [ X L (id) , Y L (id)] g . The Jacobi identity and the identity ∇ X L Y L = J X L , Y L K follows by computing J , K on left invariant fields. To observe that J X R , Y R K = [ X R , Y R ] X , first recall that avector field Z is right invariant if and only if Z ( g ) = dl g Ad − g Z (id). Therefore,[ X R , Y R ] X = dl g Ad − g [ X R (id) , Y R (id)] g = dl g [Ad − g X R (id) , Ad − g Y R (id)] g = dl g [ dl − g X R ( g ) , dl − g Y R ( g )] g = J X R , Y R K . (cid:3) We now extend the bracket identity from Theorem 7.7.
Proposition 8.3.
Let F be as in Theorem 1.1. Let X, Y be basic horizontal fieldswith decomposition X = X B + X L + X R , Y = Y B + Y L + Y R . Then:(1) J X L , Y B K and J X R , Y B K are restrictions of left and right invariant fields,respectively(2) J X R , Y L K = 0 Proof. (1) follows from Proposition 8.2 and Theorem 8.1 item (2). For (2), we usegeometric arguments to improve Theorem 7.7.
Lemma 8.4.
Let p − ∈ L id . Then [ H + ( F ) , Ad p H − ( F )] g = 0 .Proof. Given p ∈ F , we consider the translated foliation F p = { l p ( L ) | L ∈ F} .Since l p : G → G is an isometry, we conclude that F p is a Riemannian foliationwith totally geodesic leaves. Furthermore, its vertical and horizontal spaces at pq ∈ M are given by V ppq = dl p ( V q ), H ppq = dl p ( H q ). Therefore, if X is a basic F -horizontal field along ˜ L p − , dl p X is a basic F p -horizontal field along l p ( ˜ L p − ).Moreover, if X = X B + X L + X R , then dl p X = dl p ( X B ) + dl p ( X L ) + dl p ( X R ) isone, therefore the only, decomposition of dl p ( X ) as a parallel basic horizontal, a leftinvariant and a right invariant field. Since dl p ( X ǫ ( p − ))(0) = X ǫ (id) for ǫ = B, L and dl p ( X R ( p − ))(0) = Ad p X − , we conclude that H ( F p ) = H ( F ) , H + ( F p ) = H + ( F ) , H − ( F p ) = Ad p H − ( F ) . Since F p satisfies the hypothesis in Proposition 7.7, [ H + ( F p ) , H − ( F p )] = 0. (cid:3) IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 33
With Lemma 8.4, we compute J X R , Y L K : J X R , Y L K ( p − ) = dl p − [ dl p X R , dl p Y L ] g = dl p − [Ad p X R (id) , Y L (id)] g = dl p − [Ad p X − , Y + ] g = 0 / (cid:3) Before proving Propositon 1.9, we prove two more results. Proposition 8.5 is themain step into the proof of Proposition 1.9. Proposition 8.6 is used in section 9.2.
Proposition 8.5.
For F as in Theorem 1.1, let X, Y be basic horizontal fieldsalong ˜ L id . Then A X Y = ( J X L , Y L K − J X R , Y R K ) v .Proof. Let ξ ∈ V id and consider the restriction of X to the geodesic e ξt , so that wecan think of ξ as a left invariant field along e ξt . Since X B , Y B are parallel horizontal, h A X Y, ξ i = (cid:10) A ξ X, Y (cid:11) = h∇ ξ X, Y i = h∇ ξ ( X L + X R ) , Y i = − (cid:10) X L + X R , A ξ Y (cid:11) = (cid:10) A ξ ( X L + X R ) , Y L + Y R (cid:11) . Thus, according to Propositions 8.2 and 8.3, h A X Y, ξ i = 12 h J ξ, X L K − J ξ, X R K , Y L + Y R i = 12 h ξ, J X L , Y L K − J X R , Y R K i . (cid:3) We are ready to prove Proposition 1.9.
Proof of Proposition 1.9.
We first assume the leaves of F locally irreducible. Definethe auxiliary tensor ˜ A X Y = ( J X L , Y L K − J X R , Y R K ). On one hand, h ˜ A X Y, ˜ A Z W i is basic since the mixed terms in4 h ˜ A X Y, ˜ A Z W i = h J X L , Y L K − J X R , Y R K , J Z L , Z L K − J W R , W R K i vanishes by the Jacobi identity of J , K and Proposition 8.3. The remaining terms, h J X L , Y L KJ Z L , W L K i and h J X R , Y R K , J Z R , W R K i , are basic since they are the innerproduct of either right invariant or left invariant fields.Proposition 8.5 states that the vertical part of ˜ A coincides with A . We concludethe proof by observing that ˜ A differs from A by a basic horizontal field. Observethat a horizontal field is basic if and only if its inner product with basic horizontalfields is basic. Let Z be a basic horizontal field, then2 h ˜ A X Y, Z i = h J X L , Y L K − J X R , Y R K , Z i = h J X R , Y R K , Z B + Z R i − h J X L , Y L K , Z B + Z L i , where the second equality follows from Proposition 8.3, item (2). The terms h J X R , Y R K , Z B + Z R i and h J X L , Y L K , Z B + Z L i are constant since they can be real-ized as the inner products of right, respectively left, invariant fields. Therefore h A X Y, A Z W i = h ˜ A X Y, ˜ A Z W i − h ( ˜ A X Y ) h , ( ˜ A Z W ) h i is the difference of two basic functions, so it is basic.If the leaves of F are locally reducible, we consider the decomposition V = L ∆ i in Theorem 1.5 and observe that the integrability tensor A i of F i satisfies(29) A iX Y = ( A X Y ) ∆ i for every X, Y ∈ H . Given a leaf L , A X Y , seem as a Killing field in L , locallysplits as a sum of the Killing fields A X Y = P A iX Y . Given X, Y, Z, W basic F -horizontal fields, we conclude that h A iX Y, A iZ W i is constant along L . Therefore, h A X Y, A X W i = P i h A iX Y, A iZ W i is basic. (cid:3) Proposition 8.6. If F is as in Theorem 1.1, ξ ∈ V and X, Y ∈ H , then ∇ vξ A = 0 .Proof. For the computation, (possibly using a translated foliation as in Lemma 8.4)we assume that ξ, X, Y ∈ g and take t v a maximal vertical subalgebra that contains ξ . We compute ∇ ξ ( A X Y ) directly by taking X an α - A -weight and Y a β - A -weight.According to Propositions 8.5 and 8.2, we have (compare with Lemma 7.16) ∇ ξ ( A X Y ) = 12 ∇ ξ ( J X L , Y L K − J X R , Y R K ) v = 12 ( J ∇ ξ X L , Y L K + J X L , ∇ ξ Y L K − J ∇ ξ X R , Y R K − J X R , ∇ ξ Y R K ) v = ( α ( ξ ) + β ( ξ )) 12 ( J X L , Y L K − J X R , Y R K ) v . (cid:3) Proof of Theorem 1.1
Conjecture 1 can be divided in two problems:
Problem 1.
Prove that leaves are (locally isometric to) subgroups.Once settled Problem 1, it is still left to prove that the foliation is homogeneous.
Problem 2.
Suppose that the leaves of F are locally isometric toa subgroup. Prove that F is homogeneous.We observe that Problem 2 is not straightforward. It is settled in Jimenez [17,Corollary 24] assuming that leaves are subgroups. To this aim, Jimenez uses [17,Theorem 23] which requires the existence of a special algebra of Killing fields.Section 4.2 produces such an algebra assuming triviality of H p ( F ) among otherconditions (see Proposition 4.8). We observe that the triviality of H p ( F ) is notsufficient: the Gromoll–Meyer fibration Sp (2) → Σ , [11], is a principal bundlewhich is not isometric to a homogeneous foliation.This section combine the algebraic and geometric results so far to show that: (1) H id ( F ) = { id } ; (2) the field χ ( q, ξ ) in section 4.2 is a constant length Killing fieldfor every ξ ∈ V id . Item (1) shows that S can not be a factor in the decompositionof the leaf (answering question one) and proves that F is principal. Knowingthat F is principal, (2) guarantees that the group action is either by right or leftinvariant fields on each simple component of G , completing the proof (up to coveringarguments – section 9.1).9.1. The non-simply connected case.
We recall that, if G is connected, π ( G )is naturally a central subgroup of the universal cover ˜ G . Let F be as in Theorem1.1. We argue that, if the induced foliation on ˜ G is homogeneous, so it is F . Let π : ˜ G → G be the universal cover and let ˜ F = { π − ( L ) | L ∈ F} . Suppose ˜ F isgiven by the right cosets of H . Since π = π ( G ) is central, π \ H is naturally asubgroup of π \ ˜ G = G . Therefore, F is given by the cosets of π \ H .For the rest of the section, we assume G simply connected.9.2. Ruling out the 7-sphere.
The main difficult to prove Conjecture 1 is tocontrol the leaf type. We know that the leaves of F must be locally symmetricspaces, since they are (immersed) totally geodesic submanifolds of a symmetricspace. Here we prove that S can not appear as a factor in the leaf. For thisaim, we show that H id ( F ) = { id } . In particular, Hol id ( F ) is a transitive group ofisometries acting locally free on ˜ L id . IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 35
Theorem 9.1. L id is an immersed Lie subgroup of G . Suppose there is a foliation (under the hypothesis of Theorem 1.1) where ˜ L id is not an immersed subgroup. Since ˜ L id has the Killing property, it must be ametric product of a Lie gorup and constant curvature 7-spheres. In particular,Theorems 7.17 and 1.5 guarantees the existence of a foliation (as in Theorem 1.1)such that ˜ L id is isometric to a constant curvature 7-sphere. To prove Theorem 9.1,we argue by contradiction, supposing that F is an irreducible foliation whose leavesare isometric to a round 7-sphere. We proceed with the following argument:We compute A , the integrability tensor of the foliation ˜ F on Aut( V ) and provethat ( ∇ vX A ) X Y = 0 for all X, Y ∈ ˜ H . Following arguments similar to section 5,we show that the subset A (Λ ˜ H ) ⊂ ˜ V is invariant under infinitesimal holonomytransformations. Recalling that E τ ( h ) is the dual leaf of ˜ F through τ ( h ), Theorem1.2 gives T h E τ ( h ) = A (Λ ˜ H h ) for all h ∈ Aut( V ). This done, Theorem 1.3 gives hol p ( F ) = A (Λ ˜ H h ) ∼ = span { A X Y ∈ X ( ˜ L p ) | X, Y ∈ X ( ˜ L p ) , X, Y basic horizontals } . (30)The last isomorphism follows since Killing fields are completely determined by their1-jet extension. On the other hand, Proposition 1.9 implies that evaluation at h induces an isomorphism from (30) to A (Λ H p ). In particular Hol p ( F ) acts locallyfreely and transitively on ˜ L h . Proposition 9.2.
Let p ∈ G . Then T id E p = A (Λ ˜ H id ) . The proof depends on Lemmas 9.3-9.7. Lemma 9.3 does not assume hypothesison the ambient space. We consider Aut p ( V ) with the Riemannian metric definedin section 3: h X + ζ, Y + χ i τ = h dτ X, dτ Y i + X i h ω ( ζ ) ξ i , ω ( χ ) ξ i i where { ξ i } is an orthonormal basis for V . Given ζ ∈ ˜ V , we orthogonally decompose ζ = ζ M + ζ ω , with ω ( ζ M ) = 0. In particular ζ M is the inverse image of a vectorin T M by the isomorphism dτ p | ker ω . We use the principal structure of Aut p ( V ) toidentify the set spanned by the ζ ω -components, ˜ V ω , as the subset˜ V ωh = h End( V p ) = { hh ′ ∈ End( V p , V τ ( h ) ) | h ′ ∈ End( V p ) } . V ω is the set spanned by the action fields of the principal GL ( V p )-action on Aut p ( V ). Lemma 9.3.
Suppose F has totally geodesic leaves and that ζ is a ˜ F -holonomyfield along a ˜ F -horizontal curve ˜ c . Then both ζ M and ζ ω are holonomy fields.Furthermore: ( i ) dτ ( ζ M ) is a holonomy field along c = τ ◦ ˜ c ( ii ) ζ ω is the restriction of an action field to ˜ c . In particular, for every ξ ∈ V p , ζ ω ( t ) ξ is a holonomy field along c .Proof. Let ˜ V M be the space spanned by the ζ M components of ˜ V . Since ˜ V M =ker ω ∩ ˜ V and ˜ V ω = ker dτ , we have h ˜ V ω , ˜ V M i τ = 0. Given ˜ c , we split the spaceof holonomy fields along ˜ c into two subspaces: the restriction of the GL ( V p )-actionfields and the fields with initial data in ˜ V M . We use the totally geodesic conditionto show that a field with initial data in ˜ V M stays in ˜ V M . For simplicity, by restricting F to a tube along c , we assume that holonomytransformations are well defined between leaves. Let ϕ ˜ c t be the ˜ F -holonomy trans-formation defined by ˜ c . Claim 9.4. ϕ ˜ c t ( h ) = ( dφ c t ) τ ( h ) h .Proof. Given q ∈ L c (0) , denote by c q the F -horizontal curve induced by φ c t , i.e., c q ( t ) = φ c t ( q ). Recall that the holonomy field ξ along c q is given by ( dφ c t ) q ( ξ (0)),therefore ( b c q ) h = ( dφ c t ) q h . In particular, ϕ ˜ c t ( h ) = ( [ c τ ( h ) ) h = ( dφ c t ) τ ( h ) h , sincethere is only one ˜ F -horizontal curve starting at h that τ p -projects to c τ ( h ) . (cid:3) Since F has totally geodesic leaves, φ c t , the holonomy transformation defined by c , is an isometry. Let γ be a vertical geodesic in G . According to our definition of ω , the ˜ F -horizontal lift of γ at h ∈ Aut( V ) p is given by the curve ˜ γ h ( s ) = P γ ( s ) h ,where P γ ( s ) : V γ (0) → V γ ( s ) is the parallel transport along γ . Since φ c is an isometry, P φ ct γ ( s ) dφ c t = dφ c t P γ ( s ). In particular, ϕ ˜ c t sends the ω -horizontal lift of γ at h to the ω -horizontal lift of φ c t γ at dφ c t h = ϕ ˜ c t ( h ). Since ˜ V M is spanned by thevelocities of curves ˜ γ h , we have shown that infinitesimal holonomy transformationsin ˜ F preserves ˜ V M . The Lemma follows since holonomy tranformations preserveaction fields. (cid:3) From now on, we assume F as in Theorem 1.1. Lemma 9.5.
Let X ∈ ˜ H h , ζ ∈ ˜ V h and { ξ i } be an orthonormal basis of V τ ( h ) . Then dτ ( A ∗ X ζ ) = A ∗ X ζ M + P i R ( ζξ i , ξ i ) X .Proof. Using Lemma 3.8 and Corollary 8.6, we conclude that ω ( A X Y ) ξ = A A ξ X Y + A X A ξ Y. On the other hand: hA X Y, ζ i τ = h A X Y, dτ ζ i τ + X h ω ( A X Y ) ξ i , ζξ i i τ = (cid:10) A ∗ X ζ M , Y (cid:11) τ + X (cid:10) A A ξi X Y + A X A ξ i Y, ζξ i (cid:11) τ = (cid:10) A ∗ X ζ M , Y (cid:11) τ + X (cid:10) [ A ζξ i , A ξ i ] X, Y (cid:11) τ where [ A ζξ i , A ξ i ] = A ζξ i A ξ i − A ξ i A ζξ i . Recalling that R ( V , V ) H ⊂ H , Corollary8.6 and O’Neill formulas (see [12, page 44]) gives [ A ζξ i , A ξ i ] X = R ( ζξ i , ξ i ) X. (cid:3) Lemma 9.6.
Assume that F is irreducible and that the leaves of F are locallyisometric to round 7-spheres. Suppose ζ ∈ ˜ V h satisfies A ∗ X ζ = 0 for all X ∈ ˜ H h .Then A ∗ X ζ M = 0 for all X ∈ H τ ( p ) and P i R ( ζξ i , ξ i ) = 0 .Proof. By possibly translating ζξ i , ξ i to id, we have R ( ζξ i , ξ i ) X = − ad [ ζξ i ,ξ i ] g X .Therefore, A ∗ X ζ = 0 for all X if and only if(31) A ∗ X ζ M = 12 ad P [ ζξ i ,ξ i ] g X (recall that on totally geodesic foliations, h R ( V , V ) H , Vi = 0, thus ad P [ ζξ i ,ξ i ] g X ∈H .) We argue that (31) is only possible if ζ M = P [ ζξ i , ξ i ] g = 0. According to [4,Lemma 7], ([ V id , V id ] g + V id , [ V id , V id ] g ) = ( k + m , k ) must be isomorphic to thesymmetric pair ( so (8) , so (7)), satisfying the identities[ k , k ] ⊂ k , [ k , m ] ⊂ m , [ m , m ] = k . IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 37
Therefore, there is a Lie algebra isomorphism of the pair ( k + m , k ) to ( so (8) , so (7)),sending k to the set of skew-symmetric matrices with vanishing first column and m = V id to its orthogonal complement, so (7) ⊥ = { e ∧ w | w ∈ e ⊥ } , where { e , ..., e } is a basis for R and, given v, w ∈ R , v ∧ w is the endomorphism v ∧ w ( z ) = h v, z i w − h w, z i v. For simplicity we denote ζ M = ξ ∈ V id and P [ ζξ i , ξ i ] g = A ∈ k . Putting togetherequations (15) and (31), we have ad ξ B = ad A B for all B ∈ so (7), since so (7) =[ V id , V id ] g is horizontal. We assume that ξ = 0 and show that ad ξ | so (7) can not berealized by an endomorphism of the form ad A | so (7) , A ∈ k . Assuming ξ = 0, up toisomorphism (and possibly dividing ξ by its norm), we can assume ξ = e ∧ e . Forevery B ∈ so (7), we get(32) [ ξ, [ ξ, B ]] = ξ B + Bξ = − e ∧ ( Be ) . Which is minus the orthogonal projection from so (7) to the space V = span { e ∧ e , ..., e ∧ e } , with respect to the Cartan-Killing metric on so (7). From now one,we always assume w unitary and orthogonal to e and e . We have[ A, [ A, e ∧ w ]]( w ) = − A e + 2 h Ae , w i Aw + (cid:10) A e , w (cid:11) w − (cid:10) A w, w (cid:11) e . (33)Equaling (33) to (32), we conclude that(34) e = − A e + 2 h Ae , w i Aw + (cid:10) A e , w (cid:11) w − (cid:10) A w, w (cid:11) e . Equation (34) implies:(1) (1 + (cid:10) A w, w (cid:11) ) e = − A e for all w orthogonal to { Ae , A e } ,(2) in particular, A e ∈ span { e } and (cid:10) A e , w (cid:11) = 0 for all w ⊥ Ae . Thus, span { e , Ae } is A -invariant and (cid:10) A w, w (cid:11) is a constant function on w , forunitary w orthogonal to Ae .Items (1) and (2) can only be satisfied in the following two situations (recall that A | R must have non-trivial kernel): A e = − e and Aw = 0 for w ⊥{ e , Ae } ; Ae = 0 and A w = − w for all w ∈ ⊥ e . Both cases contradicts equation (32): inthe first case, A = e ∧ ( Ae ), therefore ad ξ A = − A , contradicting the fact thatad A A = 0; in the second case we get a contradiction by observing that ad ξ B = 0for every B in the subalgebra so (6), spanned by e i ∧ e j , 2 ≤ i, j ≤
7. However(ad A B ) | R = − B + ABA ) which is not always zero for non-zero A . (cid:3) Lemma 9.7. If ξ i , η i are holonomy fields along a horizontal curve γ , then P J ξ i , η i K has constant length.Proof. Note that || P J ξ i , η i K || = 4 P i,j R ( ξ i , η i , η j , ξ j ). But ddt R ( ξ i , η i , η j , ξ j ) = R ( A ξ i X, η i , η j , ξ j ) + R ( ξ i , A η i X, η j , ξ j )+ R ( ξ i , η i , A η j X, ξ j ) + R ( ξ i , η i , η j , A ξ j X )Which vanishes since R h ( V , V ) V = 0. (cid:3) Proof of Proposition 9.2.
Let ζ ∈ ( A (Λ ˜ H id )) ⊥ . As in the proof of Theorem 1.6,it is sufficient to show that the holonomy field ζ ( t ) with initial condition ζ (0) = ζ satisfies A Y ( t ) ζ ( t ) = 0, for every ˜ F -horizontal geodesic c and horizontal field Y ( t ).Let ζ = ζ M + ζ ω and take { ξ i ( t ) } an orthonormal base of holonomy fields along c . From Lemma 9.6, we know that ζ M = 0. Since dτ ( ζ M ( t )) is a holonomy field(Lemma 9.3), ζ M ( t ) vanishes identically. Lemma 9.5 now gives A ∗ Y ( t ) ζ = X R ( ζξ i ( t ) , ξ i ( t )) Y ( t ) = −
14 ad P J ζξ i ( t ) ,ξ i ( t ) K Y ( t ) . Since P J ζξ i (0) , ξ i (0) K = 0 (Lemma 9.6) and ζξ i ( t ) , ξ i ( t ) are holonomy fields (Lemma9.3, item ( ii )), we have P J ζξ i ( t ) , ξ i ( t ) K = 0 for all t (Lemma 9.7). (cid:3) Foliations whose leaves are (locally) isometric to subgroups.
In section9.2 we prove that the only irreducible factors of ˜ L id are abelian or compact simpleLie groups with bi-invariant metrics. Given a symmetric space L , a subspace CK ofKilling fields is called Clifford-Killing if for any two elements
Z, W ∈ CK , h Z, W i is constant. In particular, the elements of CK are constant length Killing fieldswhose integral flows are Clifford-Wolf translations ([4, Proposition 3]). On onehand, Proposition 1.9 shows that A (Λ H ) (seem as in equation 30) is a Clifford-Killing space. On the other hand, constant length Killing-fields on compact simpleLie groups are either left or right invariant fields. Here we use Theorem 1.6 to verifythe hypothesis in Corollary 1.4 when ˜ L id is a subgroup. In this context, the fieldsconstructed in section 4.2 forms a CK space, concluding the proof.We again assume that G is simply connected and that the foliation is irreducible(in the light of Theorem 7.17). For simplicity, we implicitly identify the space ofKilling fields on ˜ L id with the germs of Killing fields on L id around id.Let L be a leaf and let a L denote the space in (30), i.e., a L is the space spannedby the (local) fields A X Y , where X, Y are basic horizontal.
Lemma 9.8.
Let ˜ L = L × ... × L s be the decomposition of ˜ L onto an abelian group L and simple compact groups. Then, a L = ⊕ a iL , where a iL is either the set of rightinvariant fields or of left invariant fields on L i . In particular, a L is isomorphic tothe Lie algebra of ˜ L id .Proof. Observe that a Clifford-Killing space on a compact simple Lie group thattrivializes its tangent bundle must be either a the set of left invariant fields or theset of right invariant fields. Furthermore, using [4, Theorem 4 and Proposition3] we conclude that every element η ∈ a L is the sum of constant length Killingfields η i ∈ X ( L i ). From [4, Propoition 7] and Theorem 1.6, we conclude that theprojection of the elements of a L to each component L i must trivialize their tangentbundle, concluding the proof. (cid:3) Let c be a horizontal curve, c (0) = id. Decompose L ( t ) = ˜ L c ( t ) = Π L i ( t )according to L i ( t ) = φ c t ( L i ). Consider then the decomposition of a L ( t ) given inLemma 9.8: a L c ( t ) = ⊕ a iL c ( t ) . According to Lemma 9.8, a iL ( t ) must be a smoothbundle with constant rank along c . Furthermore, at every t , it must be either all leftor all right invariant fields, if i >
0, therefore the property of being right invariant(respectively, left invariant) must be constant with respect to t . Keeping the proofof Theorem 1.6 in mind, we just have proved: IEMANNIAN FOLIATIONS ON COMPACT LIE GROUPS 39
Lemma 9.9. hol id ( F ) = a L id . In particular, if G is simply connected and F isirreducible, H id ( F ) = { id } .Proof. According to Corollary 4.3, it is sufficient to show that ˆ c (1) − ( a L c ) ⊂ a L id for every horizontal curve c , c (0) = id. This condition holds from the discussionabove. (cid:3) We are in position to apply Corollary 1.4. We recall that the fields p χ ( ξ , p )in section 4.2 were constructed using holonomy transportation. In particular, theyhave constant length. We now use Proposition 4.8 to prove that they are Killingfields. Lemma 9.10.
Let F be irreducible and G simply connected. Identify E id with G .The set of vector fields { h hξ | ξ ∈ V id } is a Clifford-Killing space on G .Proof. We already know that the Hol id ( F )-action is transitive and commutes leaf-wise with a L . Therefore, its restriction to each leaf must be component-wise byeither right or left invariant fields (opposing a L ). Now, Proposition 4.8 guaranteesthat the Hol id ( F )-action fields are Killing, and the arguments in the present sectionguarantee they are Clifford-Killing, completing the proof of Theorem 1.1. (cid:3) References
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