Wiedersehen metrics and exotic involutions of Euclidean spheres

Abresch, Uwe; Durán, Carlos E.; Püttmann, Thomas; Rigas, A.

doi:10.1515/crelle.2007.025

Cited by 13 publications

(23 citation statements)

References 31 publications

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“…, y n ) is the normal coordinate around m + . By the diagram introducing the diffeomorphism η and the formulae of the metric around m + and m − , we conclude that η : S n−1 (1) → S n−1 (1) is an isometry. It follows at once that Σ n is diffeomorphic to S n (1), which completes the proof.…”

Section: Non-existence Results On Exotic Spheresmentioning

confidence: 86%

Isoparametric functions on exotic spheres

Qian

Tang

2015

Advances in Mathematics

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This paper extends widely the work in [11]. Existence and non-existence results of isoparametric functions on exotic spheres and Eells-Kuiper projective planes are established. In particular, every homotopy n-sphere (n > 4) carries an isoparametric function (with certain metric) with 2 points as the focal set, in strong contrast to the classification of cohomogeneity one actions on homotopy spheres [26] (only exotic Kervaire spheres admit cohomogeneity one actions besides the standard spheres). As an application, we improve a beautiful result of Bérard-Bergery [2] (see also pp. 234-235 of [3]).

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Section: Non-existence Results On Exotic Spheresmentioning

confidence: 86%

Isoparametric functions on exotic spheres

Qian

Tang

2015

Advances in Mathematics

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“…Note that our action is essentially the Gromoll-Meyer action on the first row when (a) = 0, and thus the denomination. The Gromoll-Meyer action on the set (a) = 0 also produces exotic phenomena [1].…”

Section: Extended Cartan Embeddingsmentioning

confidence: 99%

Suspending the Cartan Embedding of $${{\mathbb H}P^n}$$ Through Spindles and Generators of Homotopy Groups

2011

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We provide an equivariant suspension of the Cartan embedding of the symmetric space S 4n+3 → HP n → Sp(n + 1); this construction furnishes geometric generators of the homotopy group of π4n+6Sp(n + 1). We study the topology and geometry of the image of this generator; in particular we show that it is a spindle, minimal with respect to the biinvariant metric from Sp(n + 1). This spindle also admits a different metric of positive curvature away from the cone singular point.Mathematics Subject Classification (2010). 53C30, 57T20.

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“…Remark 1.8. Explicit formulas for exotic involutions on S 6 and S 14 are given in [1], where it is shown, on pages 13-17, that the corresponding fake RP 6 is diffeomorphic to the Hirsch-Milnor RP 6 that corresponds to Σ 7 3 .…”

Section: Introductionmentioning

confidence: 99%

Almost Nonnegative Curvature on Some Fake 6- And 14-Dimensional Projective Spaces

Rajan

Wilhelm

2016

Bull. Aust. Math. Soc.

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We apply the lifting theorem of Searle and the second author to put metrics of almost nonnegative curvature on the fake RP 6 's of Hirsch and Milnor and on the analogous fake RP 14 's.One of the great unsolved problems of Riemannian geometry is to determine the structure of collapse with a lower curvature bound. An apparently simpler, but still intractable problem, is to determine which closed manifolds collapse to a point with a lower curvature bound. Such manifolds are called almost nonnegatively curved. Here we construct almost nonnegative curvature on some fake RP 6 s and RP 14 s.Theorem A. The Hirsch-Milnor fake RP 6 s and the analogous fake RP 14 s admit Riemannian metrics that simultaneously have almost nonnegative sectional curvature and positive Ricci curvature.Remark. By considering cohomogeneity one actions on Brieskorn varieties, Schwachhöfer and Tuschmann observed in [15] that in each odd dimension of the form, 4k + 1, there are at least 4 k oriented diffeomorphism types of homotopy RP 4k+1 s that admit metrics that simultaneously have positive Ricci curvature and almost nonnegative sectional curvature.The Hirsch-Milnor fake RP 6 s are quotients of free involutions on the images of embeddings ι of the standard 6-sphere, S 6 , into some of the Milnor exotic 7-spheres, Σ 7 k ([12], [14]). Our proof begins with the observation that the SO (3)-actions that Davis constructed on the Σ 7 k s in [5] leave these Hirsch-Milnor S 6 s invariant and commute with the Hirsch-Milnor free involution. Next we compare the Hirsch-Milnor/Davis (SO (3) × Z 2 )-action on ι (S 6 ) ⊂ Σ 7 k with a very similar linear action of (SO (3) × Z 2 ) on S 6 ⊂ R 7 and apply the following lifting result of Searle and the second author. Theorem B. (See Proposition 8.1 and Theorems B and C in [17]) Let (M e , G) and (M s , G) be smooth, compact, n-dimensional G-manifolds with G a compact Lie group. Suppose that the orbit spaces M e /G and M s /G are equivalent, and M s /G has almost nonnegative curvature.Then M e admits a G-invariant family of metrics that has almost nonnegative sectional curvature. Moreover, if the principal orbits of (M e , G) have finite fundamental group and the quotient of the principal orbits of M s has Ricci curvature ≥ 1, then every metric in the almost nonnegatively curved family on M e can be chosen to also have positive Ricci curvature. Date: October 15, 2015. 1991 Mathematics Subject Classification. 53C20.

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Wiedersehen metrics and exotic involutions of Euclidean spheres

Cited by 13 publications

References 31 publications

Isoparametric functions on exotic spheres

Isoparametric functions on exotic spheres

Suspending the Cartan Embedding of $${{\mathbb H}P^n}$$ Through Spindles and Generators of Homotopy Groups

Almost Nonnegative Curvature on Some Fake 6- And 14-Dimensional Projective Spaces

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