A condition for positivity of curvature

Chaves, Lucas Monteiro; Derdziński, Andrzej; Rigas, A.

doi:10.1007/bf02584817

Cited by 12 publications

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“…Theorem B(1) in this case follows from Strake and Walschap's work and also appears explicitly in [4]. In this case, R ∇ can be identified with the 2-form on given by (X, Y ) = R ∇ (X, Y )W, J W , where |W | = 1 and J denotes the almost complex structure on E. The inequality of Theorem B becomes [20,Lem.…”

Section: Theorem Bmentioning

confidence: 83%

“…For part (4), first notice that (D W T ) W X = 0 because the Perelman flat through X and W is totally geodesic. Next, since…”

Section: Background: the Metric Near The Soulmentioning

confidence: 99%

“…This is a special case of part (4). To get the general case from this special case, we apply O'Neill's formula (see [3,Th.…”

Section: Background: the Metric Near The Soulmentioning

confidence: 99%

“…where R F , which we call the vertical curvature tensor, is just the restriction of the curvature tensor R of M to vectors in ν( ), so that (R F ) p : (ν p ( )) 4 → R. More generally, a tensor R F on a vector bundle R k → E π → such that, for each p ∈ , the map (R F ) p : (E p ) 4 → R has the symmetries of a curvature tensor (not necessarily including the Bianchi identity) is called a vertical curvature tensor on the bundle. We think of a vertical curvature tensor as prescribing the curvatures of vertical 2-planes at the zero-section.…”

mentioning

confidence: 99%

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Conditions for nonnegative curvature on vector bundles and sphere bundles

Tapp¹

2003

Duke Math. J.

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This paper addresses J. Cheeger and D. Gromoll's question about which vector bundles admit a complete metric of nonnegative curvature, and it relates their question to the issue of which sphere bundles admit a metric of positive curvature. We show that any vector bundle that admits a metric of nonnegative curvature must admit a connection, a tensor, and a metric on the base space, which together satisfy a certain differential inequality. On the other hand, a slight sharpening of this condition is sufficient for the associated sphere bundle to admit a metric of positive curvature. Our results sharpen and generalize M. Strake and G. Walschap's conditions under which a vector bundle admits a connection metric of nonnegative curvature. IntroductionA well-known question in Riemannian geometry is to what extent the converse of Cheeger and Gromoll's soul theorem holds. Their theorem states that any complete noncompact manifold, M, with nonnegative sectional curvature is diffeomorphic to the normal bundle of a compact totally geodesic submanifold, ⊂ M, called the soul of M (see [5]). The converse question is the classification problem: Which vector bundles over compact nonnegatively curved base spaces can admit complete metrics of nonnegative curvature? There are vector bundles that are known not to admit nonnegative curvature, but in all such examples the base space has an infinite fundamental group (see [13], [17], [1], [2]). Trivial positive results include all vector bundles over S 1 , S 2 , and S 3 , T S n for any n, and more generally all homogeneous vector bundles over homogeneous spaces. As for nontrivial positive results, D. Yang obtained nonnegatively curved metrics on rank 2 vector bundles over C P n #C P n (see [20]). More recently, K. Grove and W. Ziller constructed nonnegatively curved metrics on all vector bundles over S 4 and S 5 (see [7]).Our first result is a necessary condition for a vector bundle to admit a metric of nonnegative curvature. Suppose that M is an open (i.e., complete and noncompact)

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Section: Theorem Bmentioning

confidence: 83%

“…For part (4), first notice that (D W T ) W X = 0 because the Perelman flat through X and W is totally geodesic. Next, since…”

Section: Background: the Metric Near The Soulmentioning

confidence: 99%

“…This is a special case of part (4). To get the general case from this special case, we apply O'Neill's formula (see [3,Th.…”

Section: Background: the Metric Near The Soulmentioning

confidence: 99%

mentioning

confidence: 99%

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Conditions for nonnegative curvature on vector bundles and sphere bundles

Tapp¹

2003

Duke Math. J.

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“…Ẑ), Ω(X,Ŷ )] = g(g2 + g)[Q(Z), [Q(X), Q(Y )]].These expressions give the following simplification:Θ(Ẑ,X,Ŷ ) =ẐΩ(X,Ŷ ) (g 2 +g)Z[Q(X),Q(Y )] +[ω(Ẑ), Ω(X,Ŷ )] − Ω(∇ẐX,Ŷ ) − Ω(X, ∇ẐŶ ) = −(g 2 + g)(g + 1/2)[[Q(X), Q(Y )], Q(Z)] − g ′ (cos α)(sin α) ( Z, X Q(Y ) − Z, Y Q(X)) = 1 8 cos 2 (α) sin(α) 4 Z, X Q(Y ) − 4 Z, Y Q(X) − [[Q(X), Q(Y )], Q(Z)] .…”

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confidence: 99%

Radially symmetric connections over round spheres

Tapp

2018

Proc. Amer. Math. Soc.

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On the Geometry of Cohomogeneity One Manifolds with Positive Curvature

Ziller

2009

Riemannian Topology and Geometric Structures on Manifolds

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There are very few known examples of manifolds with positive sectional curvature. Apart from the compact rank one symmetric spaces, they exist only in dimensions 24 and below and are all obtained as quotients of a compact Lie group equipped with a biinvariant metric under an isometric group action. They consist of certain homogeneous spaces in dimensions 6, 7, 12, 13 and 24 The next natural case to study is therefore manifolds on which a group acts isometrically with one dimensional quotient, so called cohomogeneity one manifolds. L.Verdiani [V1, V2] showed that in even dimensions, positively curved cohomogeneity one manifolds are equivariantly diffeomorphic to an isometric action on a rank one symmetric space. In odd dimensions K.Grove and the author observed in 1998 that there are infinite families among the known non-symmetric positively curved manifolds which admit isometric cohomogeneity one actions, and suggested a family of potential 7 dimensional candidates P k . In [GWZ] a classification in odd dimensions was carried out and another family Q k and an isolated manifold R emerged in dimension 7. It is not yet known whether these manifolds admit a cohomogeneity one metric with positive curvature, although they all admit one with non-negative curvature as a consequence of the main result in [GZ].In [GWZ] the authors also discovered an intriguing connection that the manifolds P k and Q k have with a family of self dual Einstein orbifold metrics constructed by Hitchin [Hi1] on S 4 . They naturally give rise to 3-Sasakian metrics on P k and Q k , which by definition have lots of positive curvature already.The purpose of this survey is three fold. In Section 2 we study the positively curved cohomogeneity one metrics on known examples with positive curvature including the explicit functions that define the metric. In Section 3 we describe the classification theorem in [GWZ]. It is remarkable that among 7-manifolds where G = S 3 × S 3 acts by cohomogeneity one, one has the known positively curved Eschenburg spaces E p , the Berger space B 7 , the Aloff-Wallach space W 7 , and the sphere S 7 , and that the candidates P k , Q k and R all carry such an action as well. We thus carry out the proof in this most intriguing case where G = S 3 × S 3 acts by cohomogeneity one on a compact 7-dimensional simply connected manifold. In Section 4 we describe the relationship to Hitchin's self dual Einstein metrics. We also discuss some curvature properties of these Einstein metrics and the

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A condition for positivity of curvature

Abstract: Abstract.In this note we describe a necessary and sufficient condition, in order that a procedure of the type described in [D -R] yields metrics of positive sectional curvature in the total space of a principal bundle.

Cited by 12 publications

References 17 publications

Conditions for nonnegative curvature on vector bundles and sphere bundles

Conditions for nonnegative curvature on vector bundles and sphere bundles

Radially symmetric connections over round spheres

On the Geometry of Cohomogeneity One Manifolds with Positive Curvature

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