A differentiable sphere theorem by curvature pinching. II
type: Item
instance of: scholarly article
title: A differentiable sphere theorem by curvature pinching. II
zbMATH Open document ID: 0828.53032
author: Yoshihiko Suyama
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1995-07-19T00:00:00Z
review text: [For the first part see J. Math. Soc. Japan 43, No. 3, 527-553 (1991; Zbl 0790.53041).] Answers to the \(\delta ( > 1/4)\)-pinched problem are improved here to the extent of \(\delta = 0.654\), namely, the following theorem is proved: Let \((M,g)\) be a complete, simply connected and 0.654-pinched Riemannian manifold. Then \(M\) is diffeomorphic to the standard sphere. The proof proceeds with the aid of the following two propositions together with their numerical outcomes. 1. Let \(f: \mathbb{Z}^{n - 1} \to S^{n - 1}\) be a diffeomorphism, and assume that there exists a differentiable map \(F : [0,1] \times S^{n - 1} \to S^{n - 1}\) satisfying the conditions: (1) \(F(0, \cdot) = f\); (2) \(F_1 = F(1, \cdot)\) is an isometry of \(S^{n-1}\); (3) \(F_t = F(t,\cdot): S^{n-1} \to S^{n-1}\) is a diffeomorphism for each \(t \in [0,1]\). Then \(M\) is diffeomorphic to \(S^n\). 2. Let \(f\) be a diffeomorphism of \(S^{n-1}\) and \(\alpha\) a differentiable map of \(S^{n-1}\) into \(\text{SO} (n;\mathbb{R})\) with \(f(x) = \alpha_X x\). Choose numbers \(N_1\), \(\theta_1\) and \(\theta_2\) such that \(|(d_X \alpha)V|< N_1\), \(d_s(\alpha_x V, \alpha_{-x} V) < \theta_1\), \(d_s (\alpha_x X, (df)_x X/|(df)_x X|) < \theta_2\) for any \(x \in S^{n - 1}\) and any unit vector \(X \in T_X (S^{n -1})\) and \(V \in \mathbb{R}^n\). Then one has that, if \(N_1 \pi + \theta_1 + 2\theta_2 < 2\pi\), there exists a diffeotopy \(F\) constructed from \(f\). This may be referred to as a diffeotopy theorem.
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 772419
zbMATH Keywords: sphere theorem
zbMATH Keywords: pinching
zbMATH Keywords: diffeotopy theorem
reviewed by: Tanjiro Okubo
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225633
OpenAlex ID: W1991245941
cites work: An equivariant pinching theorem
cites work: Krümmung und differenzierbare Struktur auf Sphären. II
cites work: On the differentiable pinching problem. Improved by H. Karcher
cites work: Differentiable sphere theorem by curvature pinching
cites work: Q3138494
DOI: 10.2748/TMJ/1178225633
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description: scientific article; zbMATH DE number 772419