Exotic spheres with lots of positive curvatures

Wilhelm, Frederick

doi:10.1007/bf02921960

Cited by 15 publications

(32 citation statements)

References 17 publications

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“…When both S 7 's in the diagram have the same constant curvature, then the pullback metric of Sp(2) is biinvariant (see [19] and proposition 2.1 in [22]). Some more sophisticated examples will be considered in Sect.…”

Section: Geometry Of Pullback Bundlesmentioning

confidence: 96%

“…• The first example is the family of bundles studied by Wilhelm [22] (5.1). This family is a ladder of bundles, which extends Sp(2) as presented in the Example 1; there the fibers are totally geodesic (just the Hopf fibers) and Sp (2) with the pullback metric has non-negative curvature, whereas for the higher elements Sp(2, m) one must carefully chose the metric in order to satisfy the hypothesis of Theorem 3.…”

Section: Applicationsmentioning

confidence: 98%

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Rigidity of flat sections on non-negatively curved pullback submersions

Durán

Sperança

2015

manuscripta math.

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Section: Geometry Of Pullback Bundlesmentioning

confidence: 96%

Section: Applicationsmentioning

confidence: 98%

Rigidity of flat sections on non-negatively curved pullback submersions

Durán

Sperança

2015

manuscripta math.

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“…They showed that in fact all exotic spheres in dimension seven which are S 3 -bundles over S 4 (the Milnor spheres) admit metrics with non-negative sectional curvature. It should be noted that previously, Wilhelm [Wi1] had shown the existence of a sequence of metrics g i on every Milnor sphere for which the diameter is bounded above by 1, the sectional curvature is strictly positive at some point, and the lower bound for the sectional curavture is −1/i. Like the GromollMeyer metric, these also admit an effective isometric O(2) × SO(3) action.…”

Section: Exotic Spheres and Curvature 601mentioning

confidence: 96%

Exotic spheres and curvature

Michael

Wraith

2008

Bull. Amer. Math. Soc.

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“…Further motivation for this problem is provided by the many interesting metrics discovered on these spaces in [GromMey], [GrovZil], [PetWil], [Wil1],and [Wil2].…”

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confidence: 99%

“…This is easy to check using the explicit formula for h in [Wil1] 1,1 h (0, 1) , we have constructed another diffeomorphism that takes a fiber to a fiber.…”

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confidence: 99%

The diffeomorphism type of certain 𝑆³-bundles over 𝑆⁴

Sanchez¹,

Wilhelm

2001

Proc. Amer. Math. Soc.

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Abstract. In this note we show that the unit tangent bundle of S 4 is diffeomorphic to the total space of a certain principal S 3 -bundle over S 4 , solving a problem of James and Whitehead.For more than 50 years it has been known that the S 3 -bundles over S 4 are classified by Z ⊕ Z. The bundle that corresponds to (m, n) ∈ Z ⊕ Z is obtained by gluing two copies ofwhere we have identified R 4 with H and S 3 with {v ∈ H | |v| = 1} ([Hat], [Steen]). We will call the bundle obtained from g m,n "the bundle of type (m, n)", and we will denote it by E m,n .The problem of classifying the total spaces of these bundles up to homotopy, homeomorphism, and diffeomorphism type is still open. It has led to a revolution in topology that began with Milnor's discovery that most of the bundles of type (m, −m + 1) are exotic spheres [Mil].Further motivation for this problem is provided by the many interesting metrics discovered on these spaces in [GromMey], [GrovZil], [PetWil], [Wil1],and [Wil2].In 1953 James and Whitehead gave the homotopy classification for (total space, fiber) pairs, except that among the bundles whose third homology group is Z/2Z they were only able to assert that there are at most 2 homotopy types. We will complete this classification here by proving Theorem 1. The total spaces of E 1,1 and E 2,0 are diffeomorphic, via a diffeomorphism that takes a fiber of E 1,1 to a fiber of E 2,0 .The complete classification of these total spaces is given independently in [CrowEsc], where it will be shown that there is no orientation preserving homotopy equivalence between these two total spaces.

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Exotic spheres with lots of positive curvatures

Cited by 15 publications

References 17 publications

Rigidity of flat sections on non-negatively curved pullback submersions

Rigidity of flat sections on non-negatively curved pullback submersions

Exotic spheres and curvature

The diffeomorphism type of certain 𝑆³-bundles over 𝑆⁴

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