On locally convex hypersurfaces in Hadamard manifolds
type: Item
instance of: scholarly article
title: On locally convex hypersurfaces in Hadamard manifolds
zbMATH Open document ID: 0991.53035
DOI: 10.1007/BF02676397
published in: Mathematical Notes
publication date: 2000-11-16T00:00:00Z
review text: A complete simply connected Riemannian manifold of nonpositive sectional curvature is called here a Hadamard manifold. For an immersion \(f:N^n \to M^{n+1}\) of class \(C^2\) into a Riemannian space \(M^{n+1}\) the product of principal curvatures: \(K=k_1,k_2,\dots,k_n\) is called the Gaussian curvature. It is proved that if here \(N^n\) is compact connected, \(n\geq 2\), \(M^{n+1}\) is a Hadamard manifold, \(f\) is regular, and either 1) the Gaussian curvature of \(F=f(N^n)\) is positive at every point, or 2) the second quadratic form of \(F\) (at each point under a suitable choice of the unit normal) is nonpositive definite and of rank \(\geq 1\), or 3) the external Ricci curvature of \(F\) is nonnegative and the external scalar curvature is positive, then \(F\) is a convex hypersurface homeomorphic to the sphere. It is remarked that after this paper had been submitted, the author learned that the result under assumption 2) had in essence been proved by Alexander [see \textit{F. W. Warner}, Trans. Am. Math. Soc., 122, 341-356 (1966; Zbl 0139.15601)] based on a different idea that suggests an approach to solving the problem without constraints on the rank of the second quadratic form.
Mathematics Subject Classification ID: 53C40
Mathematics Subject Classification ID: 53C42
zbMATH DE Number: 1531350
zbMATH Keywords: convex hypersurface
zbMATH Keywords: Gaussian curvature
zbMATH Keywords: second quadratic form
zbMATH Keywords: external Ricci curvature
author: Alexander A. Borisenko
reviewed by: Ülo Lumiste
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1531350