Book Chapter10.1016/S1874-5741(00)80011-4
Chapter 8 Sphere theorems
Katsuhiro Shiohama
- 01 Jan 2000
- Vol. 1, pp 865-903
15
TL;DR: In this paper, the uniqueness and finiteness of topological types of certain classes of Riemannian manifolds determined by geometric quantities are discussed. And the differentiable pinching problem is discussed.
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Abstract: The study of curvature and topology of Riemannian manifolds is one of the main streams in differential geometry. This chapter discusses the uniqueness and finiteness of topological types of certain classes of Riemannian manifolds determined by geometric quantities. The basic facts on Riemannian geometry, the (pointed) Hausdorff convergence, and the Alexandrov geometry are discussed. The Toponogov comparison theorem for geodesic triangles and the Bishop–Gromov volume comparison theorem for concentric metric balls are very important. The chapter discusses the classic sphere theorems and finiteness theorems within bounded geometry where the range of sectional curvature is bounded from both sides. Recent results concerning with the pinching numbers less than 1/4 are stated. The differentiable pinching problem is discussed. The chapter discusses the structure theorems of complete noncompact manifolds with (asymptotically) nonnegative curvature and presents ideas of providing sphere theorems for manifolds without a restricted range of sectional curvature.
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Citations
The sphere theorems for manifolds with positive scalar curvature
Juan-Ru Gu,Hong-Wei Xu +1 more
TL;DR: In this paper, the Ricci flow and stable currents were used to obtain differentiable sphere theorems via Ricci curvatures and the stable currents on a Riemannian manifold.
Topological and differentiable sphere theorems for complete submanifolds
Hongwei Xu,Entao Zhao +1 more
TL;DR: In this paper, the authors investigated topological and differentiable structures of submanifolds under extrinsic restrictions, and they obtained a topological sphere theorem for compact submaniffolds in a Riemannian manifold.
48
•Posted Content
An Optimal Differentiable Sphere Theorem for Complete Manifolds
TL;DR: In this paper, a differentiable sphere theorem is obtained from the view of submanifold geometry, where the Hamilton-Brendle-Schoen convergence result for Ricci flow and the Lawson-Simons-Xin formula for the nonexistence of stable currents are used.
28
•Posted Content
Topological and differentiable rigidity of submanifolds in space forms
Hongwei Xu,Juan-Ru Gu +1 more
TL;DR: In this paper, it was shown that if M n (n ≥ 4) is a compact submanifold in F n+p (c), and if RicM > (n − 2)(c + H 2 ), where H is the mean curvature of M, then M is homeomorphic to a sphere.
24
Differentiable sphere theorems for submanifolds of positive k-th ricci curvature
Hongwei Xu,Fei Ye +1 more
TL;DR: In this paper, the authors investigated the differentiable pinching problem for compact immersed submanifolds of positive k-th Ricci curvature, and proved that if Mn is simply connected, and the kth curvature of Mn is bounded below by a quantity involving the mean curvatures of Mn and the curvature in the ambient manifold, then Mn is diffeomorphic to the standard sphere.
20
References
•Book
Comparison theorems in Riemannian geometry
Jeff Cheeger,David G. Ebin +1 more
- 01 Jan 2008
TL;DR: In this article, Toponogov's theorem and its generalizations are studied for complete manifolds of nonnegative curvature and compact manifold of nonpositive curvature, respectively.
1.2K
•Book
Manifolds all of whose Geodesics are Closed
Arthur L. Besse
- 13 Oct 2011
TL;DR: Weinstein and Sturmouasville as discussed by the authors introduced the concept of projective spaces as base spaces of the Hopf Fibrations and showed that they can be used to define the topology of the Cayley Projective Plane.
1K