aa r X i v : . [ m a t h . DG ] O c t METRICS WITH NONNEGATIVE CURVATURE ON S × R KRISTOPHER TAPP
Abstract.
We study nonnegatively curved metrics on S × R . First, we proverigidity theorems for connection metrics; for example, the holonomy group ofthe normal bundle of the soul must lie in a maximal torus of SO(4). Next, weprove that Wilking’s almost-positively curved metric on S × S extends to anonnegatively curved metric on S × R (so that Wilking’s space becomes thedistance sphere of radius 1 about the soul). We describe in detail the geometryof this extended metric. Introduction and Background
The nonnegatively curved metrics on S × R were classified in [2]. Rigidityresult for nonnegatively curved metrics on S × R were obtained in [6], including aclassification of the connection metrics. Aside from these results, very little is knownabout the family of nonnegatively curved metrics on S n × R k . The significance ofthis problem derives in part from its relationship to the generalized Hopf conjecture.In fact, there are general relationships between the nonnegatively curved metricson a vector bundle and on its unit sphere bundle, which we will now review.Suppose that M is an open manifold with nonnegative curvature. Accordingto [1], M is diffeomorphic to the total space of the normal bundle of its soul,Σ ⊂ M . We will denote this normal bundle as ν (Σ), and its fiber at p ∈ Σ as ν p (Σ). According to [3], a tube of sufficiently small radius about Σ is convex,so the tube’s boundary (which can be identified with the total space of the unitsphere bundle, ν (Σ), of ν (Σ)) inherits nonnegative curvature. The following “soulinequality” for the curvature tensor, R , of M is found in [5]: Proposition 1.1 ([5]) . For all p ∈ Σ , X, Y ∈ T p Σ and V, W ∈ ν p (Σ) , we have: ( D X R )( X, Y, W, V ) ≤ (cid:18) | R ( W, V, X ) | + 23 ( D X D X R )( W, V, W, V ) (cid:19) · R ( X, Y, X, Y ) . In this inequality, we are considering R sometimes as a function from ( T p M ) → T p M and sometimes from ( T p M ) → R . The following can help decide whether apoint of ν (Σ) has strictly positive curvature: Definition 1.2.
A non-zero vector V ∈ ν p (Σ) is called “good” if the inequality ofProposition 1.1 is strictly satisfied for all X, Y ∈ T p Σ with | X ∧ Y | 6 = 0 and all W ∈ ν p (Σ) with | V ∧ W | 6 = 0. Proposition 1.3 ([5]) . If V is good, then for sufficiently small ǫ , exp( ǫ · V ) is apoint of M at which all planes tangent to the ǫ -sphere about Σ have strictly positivecurvature. The soul inequality was originally expressed in [5] in a manner which more ex-plicitly distinguished the three different curvatures which it relates: h ( D X R ∇ )( X, Y ) W, V i ≤ (cid:18) | R ∇ ( W, V ) X | + 23 ( D X D X k f )( W, V ) (cid:19) · k Σ ( X, Y ) . Here, R ∇ denotes the curvature tensor of the induced connection, ∇ , in ν (Σ), and k Σ and k f denote respectively the unnormalized intrinsic sectional curvatures of Σand of the Sharafutinov fiber, exp( ν p (Σ)). This point of view leads to the idea ofputting positive curvature on a sphere bundle by finding structures on the vectorbundle which make the inequality strict: Proposition 1.4 ([5]) . If structures on a Euclidean vector bundle (a metric on thebase, a connection compatible with the Euclidean structure, and a smoothly varyingcurvature tensor on each fiber) can be found such that all unit-vectors are good,then its sphere bundle admits a metric with positive curvature.
Unfortunately, no new examples of sphere bundles with positive curvature haveyet been constructed from this theorem. The problem is a lack of existing toolsfor constructing structures (particularly connections) on vector bundles to satisfythis inequality. Towards improving this situation, we believe that it is importantto explicitly understand how the inequality is satisfied by known examples withnonnegative curvature. The majority of this paper is devoted to understanding theinequality for a particular metric on S × R – a metric for which we’ll prove thatgood vectors exist.When the soul is two dimensional, R ∇ ( X, Y ) W does not depend on the choiceof oriented orthonormal basis { X, Y } of T p Σ, so we’ll shorthand this as R ∇ ( W ).With this shorthand, the inequality becomes: h ( D X R ∇ )( W ) , V i ≤ (cid:18) h R ∇ ( W ) , V i + 23 ( D X D X k f )( W, V ) (cid:19) · k Σ . which is valid for all p ∈ Σ, all
V, W ∈ ν p (Σ) and all unit-length X ∈ T p Σ.Our paper is organized as follows. In Section 2, we prove rigidity results forconnection metric with nonnegative curvature, including:
Proposition 1.5.
For any connection metric with nonnegative curvature on an R bundle over S , the holonomy group of the normal bundle of the soul must liein a maximal torus of SO (4) . In other words, ν (Σ) globally decomposes as twoorthogonal ∇ -invariant R -bundles over S . Furthermore, we prove that the restriction of the curvature tensor of ∇ to oneof the R -bundles is a multiple of its restriction to the other.In Section 3, we review Wilking’s construction from [8] of an almost positivelycurved metric on S × S . In Section 4, we prove that his metric extends to anonnegatively curved metric on S × R . In the remaining sections, we prove thatthis extended metric has the following list of geometric features. Identifying thefiber with H = the quaternions, and the base with S \ S (where S ⊂ H is thegroup of unit quaternions), we have: Proposition 1.6. (Summary of metric properties of M ∼ = ( S \ S ) × H ) The soul of M is Σ = { ([ p ] , | p ∈ S } . Let q = ([1] , ∈ Σ , X ∈ T q Σ ∼ = span { j , k } and V ∈ ν q (Σ) ∼ = H (extended to a constant section of ν (Σ) ). ETRICS WITH NONNEGATIVE CURVATURE ON S × R (1) The distance sphere of radius about Σ is isometric to Wilking’s metricwith almost positive curvature on S × S . (2) S acts isometrically on M , with s ∈ S acting as s ⋆ ([ a ] , v ) = ([ as − ] , svs − ) . (3) Σ has constant curvature . (4) The connection, ∇ , in ν (Σ) is the unique connection that is invariant underthe above S -action and such that: ∇ X V = 34 XV − V X. (5)
The parallel extension of V along t ([ e tX ] ,
0) = e − tX ⋆ q equals: V ( t ) = e − tX · V · e tX = e − tX ⋆ (cid:16) e tX · V · e − tX (cid:17) . (6) The curvature tensor, R ∇ , of ∇ is determined by: R ∇ ( V ) = 72 V i − i V. (7) The covariant derivative, DR ∇ , of R ∇ is determined by: ( D X R ∇ )( V ) = 158 ( X i − i X ) V − V ( X i − i X ) . (8) The holonomy group of ν (Σ) is isomorphic to SO (4) . (9) If V is not perpendicular to or i , then V is a good vector. (10) There exists an orthogonal pair of 2-dimensional subspaces, σ , σ ⊂ ν q (Σ) (depending on X ), whose parallel extensions, σ ( t ) and σ ( t ) , along thegeodesic γ ( t ) := ([ e tX ] , , satisfy R ∇ ( σ ( t )) ⊂ σ ( t ) for all t ∈ R . If X = j , then σ = span { , j } and σ = span { i , k } . The existence of good vectors (Property 9) means that the almost positive cur-vature of the sphere bundle (Property 1) can be detected by second derivativeinformation at the soul. The good vectors are exactly those which exponentiateto points which have positive curvature in Wilking’s metric. Property (10) reflectsthe reason that the sphere bundle does not have strictly positive curvature. If
V, W are both chosen from σ (or both from σ ), and V ( t ) , W ( t ) denote their parallelextensions along γ ( t ), then Property (10) implies: h R ∇ ( V ( t )) , W ( t ) i = 0 and h ( D γ ′ ( t ) R ∇ )( V ( t )) , W ( t ) i = 0for all t ∈ R . Furthermore, ( D γ ′ ( t ) D γ ′ ( t ) k f )( W ( t ) , V ( t )) = 0 because R f ( V ( t ) , W ( t ))is periodic with nonnegative second derivative, and is therefore constant. Thus, allterms of the soul inequality vanish for the triple { γ ′ ( t ) , V ( t ) , W ( t ) } . In fact, σ ( t )and σ ( t ) exponentiate to the totally geodesic flat tori in the sphere bundle whichprevent the sphere bundle from having positive curvature.We mention that Wilking’s metric also extends to the vector bundle S × R ,which is clear from the re-description of Wilking’s metric found in [7]. In the finalsection of this paper, we prove that no structures on S × R (or on any vectorbundle over an odd-dimensional base space) could ever satisfy the soul inequality strictly (that is, such that all vectors are good). The analogous statement for S × R is not known, which originally motivated our interest in understanding theextension of Wilking’s metric to S × R . KRISTOPHER TAPP
The author is pleased to thank Marius Munteanu for discovering the proof ofCorollary 9.2, Sam Smith for helping with the proof of Lemma 2.2, and WolfgangZiller for helpful conversations about this work.2.
Connection Metrics
In this section, we prove Proposition 1.5 and other rigidity results for connectionmetrics. Suppose that M is the total space of an R -bundle over S with a connec-tion metric of nonnegative curvature. Let Σ ⊂ M be a soul, let ν (Σ) denote thenormal bundle, let ∇ denote the induced connection in ν (Σ), and let R ∇ denote itscurvature.Since the metric is a connection metric, ( D X D X k f )( W, V ) = 0. Regarding R ∇ at p as an endomorphisms of ν p (Σ), the inequality becomes:(2.1) h ( D X R ∇ )( V ) , W i ≤ h R ∇ ( V ) , W i · k Σ , which is valid for all p ∈ Σ, all
V, W ∈ ν p (Σ) and all unit-length X ∈ T p Σ. Proof of Proposition 1.5.
Let p ∈ Σ. Choose vectors
V, W ∈ ν p (Σ) for which h R ∇ ( V ) , W i = 0. Let V ( t ) and W ( t ) denote their parallel extensions along apiecewise geodesic, γ ( t ), in Σ. The soul inequality implies that the function f ( t ) := h R ∇ ( V ( t )) , W ( t ) i satisfies f ′ ( t ) ≤ f ( t ) · k Σ ( γ ( t )) for all t ∈ R . Since f (0) = 0, it isa simple calculus exercise to conclude that f ( t ) = 0 for all t ∈ R . Piecewise geodesicloops at p generate the normal holonomy group at p . Thus, if h R ∇ ( V ) , W i = 0then h R ∇ (Φ( V )) , Φ( W ) i = 0 for every element, Φ, in the normal holonomy group.Since R ∇ at p is a skew-symmetric endomorphism of ν p (Σ), we can decompose ν p (Σ) = σ ⊕ σ , where σ and σ are R ∇ -invariant 2-dimensional subspaces, sothat h R ∇ ( V ) , W i = 0 for all V ∈ σ and W ∈ σ . This splitting is unique unless R ∇ p = 0 (in which case the inequality implies that R ∇ = 0 at every point, so theconnection is flat). In any case, we can conclude that σ and σ extend via paralleltransport to well-defined global ∇ -invariant sub-bundles of ν (Σ). (cid:3) The soul inequality forces additional rigidity beyond Proposition 1.5. Write ν (Σ) = ν (Σ) ⊕ ν (Σ)for the ∇ -invariant splitting of ν (Σ) into a pair of R -bundles. For i ∈ { , } , define F i : Σ → R as F i ( p ) = h R ∇ ( V i ) , W i i , where { V i , W i } is an oriented orthonormalbasis of ( ν i ) p (Σ). Since every R -bundle over S is oriented, these functions areglobally well-defined.If F vanished at a single point of Σ, then the previous proof implies that it wouldvanish everywhere, so ν (Σ) would be the trivial bundle with a flat connection.Assuming this is not the case, we prove now that F is a multiple of F . Proposition 2.1. If ν (Σ) is not flat, then there exists a constant λ ∈ R such that F = λF .Proof. Let p ∈ Σ, let
V, W ∈ ν p (Σ) and let X ∈ T p Σ. Assume X is unit-length.Decompose V = V + V and W = W + W , where V i , W i ∈ ( ν i ) p (Σ). Inequality 2.1says: h ( D X R ∇ )( V + V ) , W + W i ≤ h R ∇ ( V + V ) , W + W i · k Σ , which simplifies to: (cid:0) h ( D X R ∇ )( V ) , W i + h ( D X R ∇ )( V ) , W i (cid:1) ≤ (cid:0) h R ∇ ( V ) , W i + h R ∇ ( V ) , W i (cid:1) · k Σ . ETRICS WITH NONNEGATIVE CURVATURE ON S × R Letting c i = V i ∧ W i , this becomes:( c ( XF ) + c ( XF )) ≤ ( c F + c F ) · k Σ . This inequality is valid for all c , c , since the vectors V , W , V , W were arbitrary.In particular, the choice c = − c F F makes the right side vanish, and thus mustmake the left side vanish as well, which implies that XF F = XF F . The function λ := F F must therefore be constant on Σ, since its derivative is Xλ = F ( XF ) − F ( XF ) F = 0 F = 0 . (cid:3) Since R Σ F i is a topological invariant of ν i (Σ), the constant λ is completely de-termined by the Euler classes of ν (Σ) and ν (Σ). For example, if ν (Σ) and ν (Σ)have the same Euler classes, then λ = 1.No further restrictions can be obtained from the soul inequality. Given a metricon S and a connection on S × R which satisfy the conclusions of Propositions 1.5and Proposition 2.1, the soul inequality will be satisfied, provided it is separatelysatisfied in ν (Σ) and ν (Σ).It was observed in [4] that the total space of each nontrivial R -bundle over S admits a large family of nonnegatively curved connection metrics. One obtains thesimplest metric on the k th topological bundle type as a submersion metric of theform: M k = (cid:0) ( S , round) × ( C , g f ) (cid:1) /S , where S acts on S × C as: e iθ ⋆ ( p, V ) = ( e iθ · p, e iθk · V ) , and g f is an S -invariant (that is, rotationally symmetric) metric on C ∼ = R . Theinteger k determines the Euler class of the resulting bundle. For this submersionmetric, the connection in the normal bundle of the soul has a parallel curvature ten-sor, so DR ∇ = 0. One obtains a larger family of nonnegatively curved connectionmetrics from this one by performing (sufficiently small) arbitrary perturbations tothis connection.The only known examples of connection metrics with nonnegative curvature on R -bundles over S are the submersion metrics of the form: M ( k ,k ) = (cid:16) ( S , round) × ( C , g f ) × ( C , g f ) (cid:17) /S , where S acts on S × C × C as: e iθ ⋆ ( p, V , V ) = ( e iθ · p, e iθk · V , e iθk · V ) , and g f and g f are S -invariant (that is, rotationally symmetric) metrics on C ∼ = R . The integers k and k determine the Euler classes of the resulting bundles ν (Σ) and ν (Σ). For this submersion metric, F and F are constant functions.Topologically, M ( k ,k ) equals the Whitney sum of M k and M k . It is useful toobserve: Lemma 2.2. M ( k ,k ) is the trivial bundle if and only if k ≡ k ( mod . KRISTOPHER TAPP
Proof.
This R -bundle over S is topologically classified by the homotopy class ofits “clutching map” α : S → SO (4). The domain of α is the equator of the basespace, the range of α is the space of orthogonal maps from the fiber over the northpole to the fiber over the south pole, and the definition of α is α ( p ) = paralleltransport along the great half-circle through p . The image of α lies in the standardmaximal torus of SO (4), and is homotopic to the standard ( k , k )-torus knot in T ∼ = S × S defined as α ( t ) = ( e ik t , e ik t ) with t ∈ [0 , π ].We claim that α is nulhomotopic in SO (4) if and only if k ≡ k (mod 2). To seethis, let α be a lift of α to the universal cover, S × S , of SO (4), which lies in thestandard maximal torus of S × S . The derivative at the identity of the coveringmap between the two standard maximal tori is: ( a, b ) ( a + b, a − b ). Since α ′ (0) =( k , k ), we know that α ′ (0) = ( M, N ) with M = ( k + k ) / N = ( k − k ) / α ( t ) = ( e iMt , e iNt ), which has period 2 π (and hence closes up) if and only if M and N are integers, which happens if and only if k ≡ k (mod 2). (cid:3) These known examples of connection metrics with nonnegative curvature on S × R all have parallel curvature tensors. From the above discussion, there arelarge families of connection metrics with non-parallel curvature tensors which satisfythe soul inequality, and it is not known whether these can be constructed to havenonnegative curvature. 3. Wilking’s metric on S × S In this section, we summarize Wilking’s construction of metrics with almostpositive curvature on certain homogeneous spaces, particularly on S × S .In general, a homogeneous space M = H \ G can always be re-described as abi-quotient of G × G as follows: M = G \ H = ∆( G ) \ ( G × G ) / (1 × H ) , where ∆( G ) = { ( g, g ) | g ∈ G } ⊂ G × G denotes the diagonal. In other words, M is the quotient of G × G under the action of ∆( G ) × (1 × H ) ∼ = G × H defined as:( g, h ) ⋆ ( g , g ) = ( g · g , g · g · h − ) . The diffeomorphism from ∆( G ) \ ( G × G ) / (1 × H ) to H \ G sends:(3.1) [ g , g ] [ g − g ] , with brackets denoting equivalence classes (orbits).The advantage of this biquotient description of M is the large variety of Riemann-ian submersion metrics which it induces. Any metric on G × G which is invariantunder this action of G × H induces a Riemannian submersion metric on M . Gen-erally, there is a large family of such metrics on G × G which have nonnegativecurvature. Wilking discovered many examples for which the induced Riemanniansubmersion metric on M has positive curvature almost everywhere.His simplest such example used G = S × S and H = S (embedded diagonallyin G ), so that topologically,(3.2) M = G/H ∼ = T S ∼ = S × S . The biquotient description is: M = ∆( S × S ) \ (cid:0) ( S × S ) × ( S × S ) (cid:1) / × S . ETRICS WITH NONNEGATIVE CURVATURE ON S × R The metric Wilking chose on ( S × S ) × ( S × S ) was a product metric, g L × g L ,where g L is the left-invariant and right-∆( S )-invariant metric on G = S × S constructed as follows:(3.3) ( S × S , g L ) = (cid:0) ( S , g ) × ( S , g ) × ( S , g ) (cid:1) /S . Here, g is bi-invariant, and S acts by right multiplication on each of the threefactors. Notice that this quotient is diffeomorphic to S × S via:(3.4) [ p, v, a ] ( p · a − , v · a − ) . Thus, g L is defined as the push-forward via this diffeomorphism of the Riemanniansubmersion metric on the above quotient.In summary, M is defined as the quotient of ˆ M := ( S × S , g L ) × ( S × S , g L )under the action of ˆ G := S × S × S defined as follows:( g , g , σ ) ⋆ ( a, v, b, c ) = ( g · a, g · v, g · b · σ − , g · c · σ − ) , with the induced Riemannian submersion metric.An explicit diffeomorphism from M = ˆ M / ˆ G to S × S is obtained using Equa-tion 3.1 and an explicit formula for the identification in Equation 3.2 as follows:ˆ M / ˆ G ∼ = S \ ( S × S ) ∼ = ( S \ S ) × S [ x, y ] ([ x ] , x − y )[ a, v, b, c ] [ b − · a, c − v ] ([ b − · a ] , a − · b · c − · v ) . (3.5)Since ( S × S , g L ) is itself a quotient of S × S × S , we can re-express thespace M as a quotient of the space M = ( S × S × S ) × ( S × S × S )(with the product metric in which each S has a unit-round metric). More precisely, M is the quotient of M by the free action of the Lie group G = S × S × S × S × S defined as follows:( g , g , s, t, σ ) ⋆ ( a, v, x, b, c, y ) = ( g as − , g vs − , xs − , g bt − , g ct − , σyt − ) , with the induced Riemannian submersion metric. This is called the “normal biquo-tient” description of M . An explicit diffeomorphism from M /G to S × S isobtained by combining Equation 3.4 and 3.5, as follows: M /G ∼ = ˆ M / ˆ G ∼ = ( S \ S ) × S (3.6)[ a, v, x, b, c, y ] [ ax − , vx − , by − , cy − ] ([ yb − ax − ] , xa − bc − vx − ) . Extending Wilking’s metric to S × H In this section, we extend Wilking’s metric on S × S to a metric on the trivialvector bundle S × H .We first establish notation related the quaternions, H = span { , i , j , k } . We willalways consider S as the group of unit-length elements of H , with circle-subgroup S = { e i t | t ∈ R } ⊂ S ⊂ H . If v ∈ H , then L v , R v : H → H will denote the leftand right multiplication maps. If v ∈ S , then we denote Ad v = L v ◦ R v − , whichrestricts to Im( H ) ∼ = sp (1) as the usual adjoint action. The real and imaginaryparts of v ∈ H will be denoted as Re( v ) and Im( v ). The conjugate will be denotedas v := Re( v ) − Im( v ). Finally, recall that the (real) inner product of q , q ∈ H is:(4.1) h q , q i = Re( q q ) = Re( q q ) = Re( q q ) = Re( q q ) . KRISTOPHER TAPP
We will modify the definition of M from the previous chapter by replacing oneoccurrence of “ S ” with “ H ”. The non-normal description of this modification is: M = ∆( S × S ) \ (cid:0) ( S × H , g ′ L ) × ( S × S , g L ) (cid:1) / × S , where g ′ L is defined like g L by replacing one occurrence of “( S , g )” with “( H , flat)”in Equation 3.3However, all future calculations will instead be done using the equivalent modi-fication of the normal description of M . That is, we define M = ( S × H × S ) × ( S × S × S ) , with the product metric in which each S has a unit-round metric and H ∼ = R has a flat metric. Then we define M to be the quotient of M by the action of G = S × S × S × S × S defined as follows:(4.2)( g , g , s, t, σ ) ⋆ ( a, v, x, b, c, y ) = ( g as − , g vs − , xs − , g bt − , g ct − , σyt − ) , with the induced metric which makes the projection π : M → M become a Rie-mannian submersion.Define Σ := { [ a, v, x, b, c, y ] ∈ M | v = 0 } , which is the soul of M . TheSharafutdinov projection, sh : M → Σ, is the map which send [ a, v, x, b, c, y ] [ a, , x, b, c, y ]. Notice that Wilking’s space (described in the previous chapter) isisometric to the sphere of radius 1 about Σ in M . Let ν (Σ) denote the normalbundle of Σ in M . If p ∈ Σ, then let ν p (Σ) denote its fiber at p .Exactly as in Equation 3.6, an explicit diffeomorphism from M = M /G to( S \ S ) × H ∼ = S × H is given by:(4.3) [ a, v, x, b, c, y ] ([ yb − ax − ] , xa − bc − vx − ) . Let V and H denote the vertical and horizontal distributions of the Riemanniansubmersion π : M → M . It is straightforward to verify that: Lemma 4.1.
At the point q = (1 , , , , , ∈ M , the horizontal space is: H q = { ( A, B, − A, − A, , A ) | A ∈ span { j , k } ⊂ H , B ∈ H } . Later, we will describe H at an arbitrary point of M , but first we will establishseveral consequences of this special case.5. Isometries of M The group S acts by isometries on M , with s ∈ S acting as:(5.1) s ⋆ [ a, v, x, b, c, y ] = [ a, v, sx, b, c, y ] . If M is identified with ( S \ S ) × H via diffeomorphism 4.3, then this isometric S -action on M looks like:(5.2) s ⋆ ([ a ] , v ) = ([ as − ] , svs − ) . This action restricts to the soul, Σ ∼ = { ([ q ] , | q ∈ S } , as the standard transitiveaction of S on S , so the soul is homogeneous, and therefore round. In fact, Lemma 5.1.
The soul, Σ , of M has constant curvature . ETRICS WITH NONNEGATIVE CURVATURE ON S × R Proof. If X ∈ span { j , k } ⊂ H has unit-length (with respect to inner product 4.1),then the geodesic in M given by γ ( t ) = ( e ( t/ X , , e − ( t/ X , e − ( t/ X , , e ( t/ X )is initially horizontal by Lemma 4.1, and is therefore everywhere horizontal. Thisgeodesic has constant speed equal to | γ ′ (0) | = , so the path γ := π ◦ γ in Σ ⊂ M also has constant speed . The image of γ in ( S \ S ) × H under diffeomorphism 4.3is the path t ([ e tX ] , γ equals the period of this image, which is π . Thus, Σ has a homogeneous metric with a geodesic of speed and period π . (cid:3) The normal connection
In this section, we describe the connection, ∇ , in normal bundle, ν (Σ), of Σ in M . We continue to identify M ∼ = ( S \ S ) × H and Σ ∼ = { ([ q ] , | q ∈ S } and q = ([1] , ∈ Σ. Since this identification provides an explicit trivialization of thebundle, we can identify ∇ with its “connection difference form,” compared to thetrivial flat connection in ( S \ S ) × H . In other words, for each q ∈ S , X ∈ T ([ q ] , Σand V ∈ ν ([ q ] , (Σ) ∼ = H , the expression ∇ X V will denotes the covariant derivativeat ([ q ] ,
0) of the constant extension of V along a path in the direction of X . Proposition 6.1. ∇ is the unique connection on the trivial bundle ( S \ S ) × H with the following two properties: (1) For any X ∈ T q Σ = span { j , k } and any V ∈ H ∼ = ν q (Σ) , ∇ X V = 34 XV − V X. (2)
The S -action of Equation 5.2 leaves ∇ invariant, which means that ∇ ( s⋆X ) ( s ⋆ V ) = s ⋆ ( ∇ X V ) for all q ∈ S , X ∈ T ([ q ] , Σ , V ∈ H ∼ = ν ([ q ] , (Σ) , and s ∈ S (with s actingon vectors via the derivative of the isometry it represents).Proof. Property (2) is obvious because the S -action is by isometries. It is straight-forward to check that the equation in property (1) is isotropy-invariant, and there-fore that it extends to a well-defined connection, which is clearly unique. So itremains to establish property (1).For this, let X ∈ T ([1] , Σ = span { j , k } . As in the proof of Lemma 5.1, thehorizontal geodesic γ ( t ) := (cid:16) e ( t/ X , , e ( − t/ X , e ( − t/ X , , e ( t/ X (cid:17) in M is such that the geodesic γ = π ◦ γ in M is identified with the geodesic γ ( t ) ∼ = ([ e tX ] ,
0) in ( S \ S ) × H .Next, let v ∈ H , and consider the following path in M : v ( t ) := (cid:16) e ( t/ X , e (3 t/ X ve ( − t/ X , e ( − t/ X , e ( − t/ X , , e ( t/ X (cid:17) Notice that v ( t ) := π ( v ( t )) is a path in M which is identified in ( S \ S ) × H withthe path v ( t ) ∼ = ([ e tX ] , v ).Let V ( t ) be the vector field along γ ( t ) in M which exponentiates to v ( t ); namely, V ( t ) := (cid:16) , e (3 t/ X ve ( − t/ X , , , , (cid:17) , so that V ( t ) := π ∗ ( V ( t )) is the vector field along γ ( t ) in M which exponentiates to v ( t ). Since V ( t ) is a horizontal vector field along a horizontal geodesic, we have: V ′ (0) = π ∗ (cid:16) V ′ (0) (cid:17) = π ∗ (cid:18) , Xv − vX, , , , (cid:19) . Under the identification M ∼ = ( S \ S ) × H , this vector V ′ (0) ∈ T q M is identifiedwith: V ′ (0) ∼ = (cid:18) , Xv − vX (cid:19) ∈ T q (cid:0) ( S \ S ) × H (cid:1) . (cid:3) Corollary 6.2. If X ∈ span { j , k } and V ∈ ν q (Σ) ∼ = H , then parallel extension of V along the path γ ( t ) = ([ e tX ] , is: V ( t ) = e − tX · V · e tX = e − tX ⋆ (cid:16) e tX · V · e − tX (cid:17) . Corollary 6.3.
The holonomy group of ν (Σ) is isomorphic to SO (4) .Proof. Let X ∈ { j , k } , let Y denote the orthogonal compliment of X in span { j , k } and let γ ( t ) = ([ e tX ] , π . The paralleltransport map, P γ , along one iteration of γ , is the endomorphism of ν q (Σ) ∼ = H defined by P γ ( V ) = e − πX · V · e πX . It is straightforward to check that P γ (1) = − X, P γ ( X ) = 1 , P γ ( i ) = − i , P γ ( Y ) = − Y. In particular, span { , X } and span { i , Y } are the irreducible subspaces of H .The holonomy group, Φ, is a subgroup of SO (4). It is not contained in a maximaltorus of SO (4) because the irreducible subspaces for P γ vary with X . Further, Φis not isomorphic to SO (3) or S acting non-transitively on H because P γ doesnot have any fixed vectors. Finally, Φ is not isomorphic to SO (3) or S actingtransitively on H because the isotropy groups are too large; for example, there areinfinitely many different holonomy elements sending i
7→ − i (namely, P γ for all X ).The only remaining possibility is that Φ is all of SO (4). (cid:3) Next we study the curvature, R ∇ , of ∇ at q . For V ∈ H ∼ = ν q (Σ), the expression R ∇ ( V ) := R ∇ ( X, Y ) V does not depend on the choice of oriented orthonormal basis, { X, Y } , of T q Σ ∼ =span { j , k } . By the proof of Lemma 5.1, the basis { j , k } is orthonormal, so R ∇ ( V ) = 4 R ∇ ( j , k ) V. Lemma 6.4.
For all V ∈ H , R ∇ ( V ) = V i − i V. Proof.
Over a neighborhood of q in Σ, extend V to the constant section, and let2 X and 2 Y denote the extensions of 2 j and 2 k to coordinate vector fields of a normal coordinate patch at q . We must compute the following at q : R ∇ ( V ) = 4 R ∇ ( j , k ) V = 4( ∇ X ∇ Y V − ∇ Y ∇ X V ) . Consider the one parameter group a ( t ) = e tX in S , and the geodesic γ ( t ) = a ( − t ) ⋆ q = ([ a ( t )] ,
0) in Σ. let Y ( t ) := Y ( γ ( t )) and V ( t ) := V ( γ ( t )) denote the ETRICS WITH NONNEGATIVE CURVATURE ON S × R restrictions of Y and V to γ . The value of ∇ Y V at the point γ ( t ) equals:( ∇ Y V )( γ ( t )) = a ( − t ) ⋆ (cid:0) ∇ ( a ( t ) ⋆Y ( t )) ( a ( t ) ⋆ V ( t )) (cid:1) = a ( − t ) ⋆ (cid:0) ∇ g ( t ) Y ( a ( t ) V a ( − t )) (cid:1) , where g (0) = 1 and g ′ (0) = 0= g ( t ) · a ( − t ) ⋆ (cid:18) Y a ( t ) V a ( − t ) − a ( t ) V a ( − t ) Y (cid:19) = g ( t ) (cid:18) a ( − t ) Y a ( t ) V − V a ( − t ) Y a ( t ) (cid:19) . The covariant derivative of the section t ( ∇ Y V )( γ ( t )) along γ ( t ) at γ (0) = q isnow found by adding its covariant derivative with respect to the flat connection tothe connection difference form:( ∇ X ∇ Y V ) q = ddt (cid:12)(cid:12)(cid:12) t =0 (cid:18) g ( t ) (cid:18) a ( − t ) Y a ( t ) V − V a ( − t ) Y a ( t ) (cid:19)(cid:19) + 34 X (cid:18) Y V − V Y (cid:19) − (cid:18) Y V − V Y (cid:19) X = −
34 ( XY − Y X ) V + 14 V ( XY − Y X )+ 916 i V −
316 (
XV Y + Y V X ) − V i = − i V −
316 (
XV Y + Y V X ) + 716 V i . The result follows by similarly computing ( ∇ Y ∇ X V ) q and subtracting. (cid:3) Corollary 6.5.
Let { X, Y } be an orthonormal basis of span { j , k } , and considerthe splitting: H = σ ⊕ σ , where σ = span { , X } and σ = span { i , Y } . Let σ ( t ) and σ ( t ) denote the parallel extensions of these planes along γ ( t ) = ([ e tX ] , . (1) For any t ∈ R , R ∇ ( σ ( t )) ⊂ σ ( t ) and ( D γ ′ ( t ) R ∇ )( σ ( t )) ⊂ σ ( t ) . (2) If V ( t ) , W ( t ) ∈ σ ( t ) (or V ( t ) , W ( t ) ∈ σ ( t ) ), then all terms of the soulinequality vanish for the triple { V ( t ) , W ( t ) , γ ′ ( t ) } .Proof. Lemma 6.4 implies that span { , i } and span { j , k } are the invariant subspacesfor R ∇ at q . Therefore, R ∇ ( σ ) ⊂ σ . Furthermore, Corollary 6.2 implies that e − tx ⋆ σ i = σ i ( t ) for i = 1 ,
2. In other words, there is an isometry taking σ i to σ i ( t ),so R ∇ ( σ ( t )) ⊂ σ ( t ) for all t ∈ R . It follows that ( D γ ′ ( t ) R ∇ )( σ ( t )) ⊂ σ ( t ) for all t ∈ R as well.To prove part (2), let V, W ∈ σ (or V, W ∈ σ ) and let V ( t ) , W ( t ) denote theirparallel extensions along γ ( t ). Since h R ∇ ( V ( t )) , W ( t ) i = 0 for all t ∈ R , the soulinequality implies that the periodic function t k f ( V ( t ) , W ( t )) has nonnegativesecond derivative, and must therefore be constant. Thus, all terms of the soulinequality vanish. (cid:3) If the metrics on the Sharafutdinov fibers were modified in any manner whichmaintained nonnegative curvature, the above proof would remain valid, so we wouldstill have, for every closed geodesic in the soul, a pair of parallel planes along alongwhich all terms of the soul inequality would be forced to vanish.Finally, we compute the covariant derivative, DR ∇ , of the tensor R ∇ . Lemma 6.6.
For all V ∈ H and all X ∈ T q Σ = span { j , k } , ( D X R ∇ )( V ) = 158 ( X i − i X ) V − V ( X i − i X ) . Proof.
Consider the one parameter group a ( t ) = e tX in S , and the correspondinggeodesic γ ( t ) = a ( − t ) ⋆ q = ([ a ( t )] ,
0) in Σ. Extend V to a section along γ asfollows: V ( t ) = a ( − t ) ⋆ V = a ( − t ) V a ( t ) . The covariant derivative of V ( t ) equals its covariant derivative with respect to theflat connection plus the connection difference form; that is, Ddt (cid:12)(cid:12)(cid:12) t =0 V ( t ) = ( − XV + V X ) + (cid:18) XV − V X (cid:19) = − XV + 34 V X.
So we have:( D X R ∇ )( V ) = Ddt (cid:12)(cid:12)(cid:12) t =0 (cid:0) R ∇ ( V ( t )) (cid:1) − R ∇ (cid:18) Ddt (cid:12)(cid:12)(cid:12) t =0 V ( t ) (cid:19) = Ddt (cid:12)(cid:12)(cid:12) t =0 (cid:0) a ( − t ) ⋆ R ∇ ( V ) (cid:1) − R ∇ (cid:18) − XV + 34 V X (cid:19) = − XR ∇ ( V ) + 34 R ∇ ( V ) X − R ∇ (cid:18) − XV + 34 V X (cid:19)
Using Lemma 6.4, this simplifies to the desired formula. (cid:3) The horizontal distribution and the O’Neill A -tensor In this section, we derive a formula for the horizontal space of π : M → M at anarbitrary point of M , and then use this formula to compute the O’Neill A -tensorat the point q = (1 , , , , , Proposition 7.1.
Let a, b, c, v, x, y ∈ S , r > , and q := ( a, rv, x, b, c, y ) ∈ M . V q = { ( R a X, R rv Y, , R b X, R c Y, | X, Y ∈ sp (1) }⊕{ ( L a S, L rv S, L x S, L b T, L c T, L y T ) | S, T ∈ sp (1) }⊕ span { (0 , , , , , R y i ) } , and H q = { X q ( A, B ) | A ∈ span { j , k } , B ∈ H } , where X q ( A, B ) is defined as follows: X q ( A, B ) := R a ( Ad by − ( A − r · Im ( B ))) , R rv (cid:18) r Ad cy − B (cid:19) ,L x (cid:0) Ad a − by − ( − A + r · Im ( B )) − r · Ad v − cy − Im ( B ) (cid:1) ,L b (cid:0) Ad y − ( − A + r · Im ( B )) (cid:1) , − r · L c (cid:0) Ad y − Im ( B ) (cid:1) , R y A ! . Proof.
The formula for the vertical space is easily verified, because it is spannedby the action fields. It is straightforward to verify that each vector in the allegedhorizontal space is orthogonal to the vertical space. Sincedim( H q ) = dim( M ) − dim( V q ) = 19 −
13 = 6 , the alleged horizontal space must equal the entire horizontal space. (cid:3) ETRICS WITH NONNEGATIVE CURVATURE ON S × R For fixed vectors A ∈ span { j , k } and B ∈ H , we can regard q X q ( A, B ) asa smooth horizontal vector field on the r = 0 portion of M , but this field doesnot extend continuously to the r = 0 portion of M (although Lemma 4.1 can bere-proven by considering the limit as r → π at the point q = (1 , , , , , ∈ M . Proposition 7.2.
Let q = (1 , , , , , ∈ M . Suppose X, A ∈ span { j , k } and V, W ∈ H , so that X = ( X, V, − X, − X, , X ) and Y = ( A, W, − A, − A, , A ) bothlie in the horizontal space H q . (1) X = 0 , then: A ( X , Y ) = (cid:16) − ˆ R + S, , ˆ R − ˆ L − S, ˆ R − S, − ˆ R, S (cid:17) , where ˆ R := Im ( W V ) , ˆ L := Im ( V W ) and S := (cid:16) h R − ˆ L, j i j + h R − ˆ L, k i k (cid:17) . (2) If V = 0 then, A ( X , Y ) = (cid:18) −
52 [
X, A ] , ,
72 [
X, A ] ,
32 [
X, A ] , , −
12 [
X, A ] (cid:19) . Proof.
The most efficient way to explicitly project a vector W = ( W , W , W , W , W , W ) ∈ T q M onto the vertical space V q is to subtract its horizontal component as follows: W V = W − (0 , W , , , , − hW , H i H − hW , H i H , where H = 12 ( j , , − j , − j , , j ) and H = 12 ( k , , − k , − k , , k ) . We first assume that X = 0 and prove part (1). We lose no generality in assumingthat | V | = 1, in which case V − = V , so ˆ R = Im( R V − W ) and ˆ L = Im( L V − W ).The path r γ ( r ) := (1 , r · V, , , ,
1) in M goes through γ (0) = q with initialdirection γ ′ (0) = X . The following is a differentiable extension of the horizontalvector Y to a horizontal field, Y ( r ), along this path: Y ( r ) = X γ ( r ) ( A, R V − W ) = (cid:16) A − r ˆ R, W, − A + r ( ˆ R − ˆ L ) , − A + r ˆ R, − r ˆ R, A (cid:17) . So we have: A ( X , Y ) = (cid:18) ddr (cid:12)(cid:12)(cid:12) r =0 Y ( r ) (cid:19) V = (cid:16) − ˆ R, , ˆ R − ˆ L, ˆ R, − ˆ R, (cid:17) V . The formula in part (1) of the proposition is obtained by explicitly computing the V -projection in the manner described above.Next, we assume that V = 0, and prove part (2). In this case, the path t γ ( t ) := (cid:0) e tX , , e − tX , e − tX , , e tX (cid:1) in M goes through γ (0) = q with initialdirection γ ′ (0) = X . The following is a differentiable extension of the horizontalvector Y to a horizontal field, Y ( t ), along this path: Y ( t ) = lim r → X ( e tX ,r · ,e − tX ,e − tX , ,e tX ) ( A, B )= ( R e tX (Ad e − tX A ) , Ad e − tX B, L e − tX (Ad e − tX ( − A )) , L e − tX (Ad e − tX ( − A )) , , R e tX A ) . So we have: A ( X , Y ) = (cid:18) ddt (cid:12)(cid:12)(cid:12) t =0 Y ( t ) (cid:19) V = (cid:18) −
52 [
X, A ] , − [ X, B ] ,
72 [
X, A ] ,
32 [
X, A ] , , −
12 [
X, A ] (cid:19) V . The formula in part (2) of the proposition is obtained by explicitly computing the V -projection in the manner described above, using that [ X, A ] ∈ span { i } . (cid:3) Vertical curvatures
In this section, we use O’Neill’s formulas to study the sectional curvature of anarbitrary plane at q spanned by two normal vectors to the soul.As in the previous section, define q = (1 , , , , , ∈ M and q = π ( q ) ∈ Σ.Let
V, W ∈ H be orthogonal and unit-length, set V = (0 , V, , , ,
0) and set W = (0 , W, , , , { V := π ∗ V , W := π ∗ W } is a general orthonormal pairin ν q (Σ).Let R denote the curvature tensor of M and let R denote the curvature tensorof M . Denote the un-normalized sectional curvature as k ( A, B ) := h R ( A, B ) B, A i and k ( A, B ) := h R ( A, B ) B, A i . Denote the restriction of k to ν q (Σ) × ν q (Σ) as k f . Since a Sharafudtinov fiber is always totally geodesic at a point of the soul, k f could also be interpreted as the curvature at q of the intrinsic metric on theSharafutinov fiber through q . Notice that k f must be invariant under the actionof the isotropy group, S , on ν q (Σ).O’Neill’s formula gives k f ( V , W ) = k ( V , W ) + 3 | A ( V , W ) | = 3 | A ( V , W ) | (8.1) = 3 (cid:12)(cid:12)(cid:12)(cid:16) − ˆ R + S, , ˆ R − ˆ L − S, ˆ R − S, − ˆ R, S (cid:17)(cid:12)(cid:12)(cid:12) where ˆ R := Im( W V ), ˆ L := Im( V W ) and S := (cid:16) h R − ˆ L, j i j + h R − ˆ L, k i k (cid:17) .Equation 8.1 yields the following explicit vertical curvature formula: Proposition 8.1.
Identifying M ∼ = ( S \ S ) × H and Σ ∼ = { ([ q ] , | q ∈ S } and q ∼ = ([1] , as before, the unnormalized sectional curvature of the vectors V = a + b i + c j + d k and W = x + y i + z j + w k in ν q (Σ) ∼ = H equals: k f ( V, W ) = 6 x c + 6 z a + 6 x d + 6 w a + 21 d z + 21 c w z b + 9 w b + 9 y d + 9 y c + 9 b x + 9 a y − dzcw − wbyd − yczb − bxay − aydz + 12 aycw − bxcw + 12 bxdz − xcza − xdwa. Verifying the soul inequality
As reviewed in the introduction, nonnegative curvature implies that the soulinequality is satisfied; that is: h ( D X R ∇ )( V ) , W i ≤ (cid:18) h R ∇ ( V ) , W i + 23 ( D X D X k f )( W, V ) (cid:19) · , for all p ∈ Σ, all unit-length X ∈ T p Σ and all
V, W ∈ ν p (Σ).Denote the right side minus the left side of this inequality as IN( X, V, W ). Itsuffices to understand this inequality when p = q = ([1] ,
0) and X = 2 j , since anyother ( p, X ) can be taken to ( q , j ) by the isometric S action on M . ETRICS WITH NONNEGATIVE CURVATURE ON S × R Proposition 9.1. If q = ([1] , ∈ Σ , X = 2 j ∈ T q Σ and V, W ∈ ν q (Σ) ∼ = H have components V = a + b i + c j + d k and W = x + y i + z j + w k , thenIN (2 j , V, W ) = 188 b z + 55 d x + 512 b x + 512 a y + 1104 d z +1104 c w + 55 a w + 188 c y +1300 bxdz − bxcw − aydz + 1300 aycw − dzcw − bxay − cybz + 108 cydx + 108 awbz − awdx. Proof.
Lemma 6.4 gives: R ∇ ( V ) = 72 V i − i V = 4 b − a i + 11 d j − c k . Lemma 6.6 gives:( D X R ∇ )( V ) = 158 ( X i − i X ) V − V ( X i − i X ) = − d + 18 c i − b j + 3 a k . It remains to compute the expression ( D X D X k f )( W, V ). Consider the path V ( t ) = e − tX · V · e − tX = (cid:18) − tX + 132 t X + · · · (cid:19) · V · (cid:18) − tX + 932 t X + · · · (cid:19) = V − t (cid:18) XV + 34 V X (cid:19) + t (cid:18) V X + 132 X V + 316 XV X (cid:19) + · · · (the terms of order greater than 2 are not exhibited here because they will noteffect the final answer). Define the path W ( t ) analogously. Corollary 6.2 impliesthat e − tX ⋆ V ( t ) and e − tX ⋆ W ( t ) are parallel along t exp( tX ). Define f ( t ) = k f ( V ( t ) , W ( t )), which can be explicitly computed using Proposition 8.1. Since the S action on M is by isometries, we have:( D X D X k f )( W, V ) = f ′′ (0)= − bxay + 156 dzcw − dxaw − cybz + 24 a y − d z − c w + 6 d x + 48 c y + 48 b z + 6 a w + 24 b x . The result follows by combining terms and simplifying. (cid:3)
We now use the above explicit formula to demonstrate that IN only equals 0when it is forced to do so by Corollary 6.5:
Corollary 9.2.
Let { X, Y } be an orthonormal basis of span { j , k } , and considerthe splitting: H = σ ⊕ σ , where σ = span { , X } and σ = span { i , Y } . If V, W ∈ ν q (Σ) ∼ = H , then IN (2 X, V, W ) ≥ and = 0 if and only if V and W are linearly dependent or span { V, W } equals σ or σ .Proof. Due to the isometry group, it suffices to verify the case X = j . For this,let A = bx − ay , B = dz − cw , C = bz − cy and D = dx − aw . The equation ofProposition 9.1 simplifies to:IN(2 j , V, W ) = 107 C + 19 D + (9 C − D ) + 28 A + 108 B + (22 A + 32 B ) . This expression is nonnegative and equals zero if and only if A = B = C = D = 0,which occurs if and only if V and W are linearly dependent or span { V, W } equalsspan { , j } or equals span { i , k } . (cid:3) Corollary 9.3.
The vector V ∈ ν q (Σ) ∼ = H is good if and only if V is perpendicularto neither nor i .Proof. If V is perpendicular to neither 1 nor i , then Corollary 9.2 implies thatIN( X, V, W ) > X and all W not parallel to V , which meansthat V is good. If V is perpendicular to 1, then we can choose X parallel to the( j , k )-component of V , and choose W so that span { V, W } = span { i , X } . It thenfollows from Corollary 9.2 (or from Corollary 6.5) that IN( X, V, W ) = 0, so V isnot good. Similarly, if V is perpendicular to i , then we can choose X parallel tothe ( j , k )-component of V and choose W so that span { V, W } = span { , X } , so thatIN( X, V, W ) = 0. (cid:3)
Wilking described explicitly the locus of point at which zero-curvature planesoccur for his metric. The previous corollary implies that the good vectors areexactly the vectors which exponentiate to positive curvature points of Wilking’smetric. Thus, the soul inequality (which is based on second derivative informationat the soul) contains complete information about which points of the sphere bundlehave positive curvature.10.
Metrics with Nonnegative Curvature on S × R Since Wilking’s metric on S × S also extends to the vector bundle S × R , wemention here that this extension could never be altered to make the soul inequalitybecome strict. Much more generally: Proposition 10.1.
No structures on a vector bundle over an odd dimensional basespace could strictly satisfy the soul inequality.Proof.
Let B denote the base space of the bundle. Chose p ∈ B and orthogonalunit-length vectors W, V in the fiber at p such that k f ( W, V ) is maximal (amongall such p, V, W ), which implies that ( D X D X k f )( W, V ) = 0 for all X ∈ T p B .Since X R ∇ ( W, V ) X is a skew-symmetric endomorphism of the odd-dimensionalvector space T p B , there exists a non-zero vector X ∈ T p B such that R ( W, V ) X = 0.For any Y ∈ T p B , the right side of the soul inequality vanishes for the vectors { X, Y, W, V } . (cid:3) References
1. J. Cheeger and D. Gromoll,
On the structure of complete manifolds of nonnegative curvature ,Ann. of Math. (1972), 413–443.2. D. Gromoll and K. Tapp, Nonnegatively curved metrics on S × R , Geometriae Dedicata. (2003), 127–136.3. L. Guijarro and G. Walschap, The metric projection onto the soul , Tans. Amer. Math. Soc. (2000), no. 1, 55-69.4. M. Strake and G. Walschap,
Connection metrics of nonnegative curvature on vector bundles ,Manuscripta Math. (1990), 309-318.5. K. Tapp, Conditions of nonnegative curvature on vector bundles and sphere bundles , DukeMath. J., to appear.6. K. Tapp
Rigidity for nonnegatively curved metrics on S × R , Annals of Global Analysis andGeometry. (2004), 43-58.7. K. Tapp, Quasi-positive curvature on homogeneous bundles , J. Diff. Geom, (2003), pp.273–287.8. B. Wilking, Manifolds with positive sectional curvature almost everywhere , Invent. Math.148