Exotic spheres and curvature

Michael, Joachim; Wraith, David J.

doi:10.1090/s0273-0979-08-01213-5

Cited by 21 publications

(16 citation statements)

References 74 publications

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“…(See also Boyer, Galicki and Nakamaye [5]. For a general reference about the construction and curvature of homotopy spheres, see Joachim and Wraith [10].) Our main result is as follows:…”

Section: Introductionmentioning

confidence: 85%

On the moduli space of positive Ricci curvature metrics on homotopy spheres

Wraith

2011

Geom. Topol.

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“…(See also Boyer, Galicki and Nakamaye [5]. For a general reference about the construction and curvature of homotopy spheres, see Joachim and Wraith [10].) Our main result is as follows:…”

Section: Introductionmentioning

confidence: 85%

On the moduli space of positive Ricci curvature metrics on homotopy spheres

Wraith

2011

Geom. Topol.

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“…However, in dimension four, the question of whether an exotic 4-sphere exists remains open, which is the so called smooth Poincaré conjecture ( c.f. [JW08] ).…”

Section: Theorem 3 ([Gt14]) the Focal Sets M ± Of An Isoparametric Fmentioning

confidence: 99%

Recent Progress in Isoparametric Functions and Isoparametric Hypersurfaces

Qian

Tang

2014

Springer Proceedings in Mathematics &Amp; Statistics

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This paper gives a survey of recent progress in isoparametric functions and isoparametric hypersurfaces, mainly in two directions.(1) Isoparametric functions on Riemannian manifolds, including exotic spheres.The existences and non-existences will be considered. (2) The Yau conjecture on the first eigenvalues of the embedded minimal hypersurfaces in the unit spheres. The history and progress of the Yau conjecture on minimal isoparametric hypersurfaces will be stated.

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“…If n ≡ 1, 2 mod 8, there is a natural invariant α : Θ n → Z 2 . This invariant is described in detail in [57]. In particular, half of all smooth structures on S n have non-zero α-invariant.…”

Section: Preliminaries and The Main Theoremsmentioning

confidence: 99%

Riemannian Manifolds of Positive Curvature

Brendle¹,

Schoen²

2011

Proceedings of the International Congress of Mathematicians 2010 (ICM 2010)

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The study of positive sectional curvature is one of the oldest pursuits in Riemannian geometry, but despite the considerable efforts of many researchers, basic questions remain unanswered. In this lecture we will briefly summarize the state of knowledge in this area and outline the techniques which have had success. These techniques include geodesic and comparison methods, minimal surface methods, and Ricci flow. We will then describe our recent work (see [18], [21], [22]) which uses the Ricci flow to resolve the differentiable sphere theorem; that is, the complete classification of manifolds whose sectional curvatures are 1/4-pinched.Mathematics Subject Classification (2010). Primary 53C20; Secondary 53C21, 53C44, 35K55Keywords. Riemann curvature tensor, Ricci flow, strong maximum principle, symmetric space Preliminaries and the Main TheoremsWe let M denote a smooth manifold of dimension n. Recall that a Riemannian metric on M is a choice g of inner product on each tangent space which varies smoothly from point to point. Any manifold admits an infinite dimensional family of Riemannian metrics, but the question of whether a manifold admits metrics with desired geometric properties is one of the basic questions of global Riemannian geometry. Surfaces embedded in R 3 provide important examples of two dimensional Riemannian manifolds where the metric g is the restriction of the euclidean inner product to each tangent space. The geometry of surfaces was developed by Gauss in the early nineteenth century. Gauss understood

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Exotic spheres and curvature

Abstract: Abstract. Since their discovery by Milnor in 1956, exotic spheres have provided a fascinating object of study for geometers. In this article we survey what is known about the curvature of exotic spheres.

Cited by 21 publications

References 74 publications

On the moduli space of positive Ricci curvature metrics on homotopy spheres

On the moduli space of positive Ricci curvature metrics on homotopy spheres

Recent Progress in Isoparametric Functions and Isoparametric Hypersurfaces

Riemannian Manifolds of Positive Curvature

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