A topological characterization of compact \(n\)-manifolds with \(C^ 2\)- boundary in \(R^ n\)
type: Item
instance of: scholarly article
title: A topological characterization of compact \(n\)-manifolds with \(C^ 2\)- boundary in \(R^ n\)
zbMATH Open document ID: 0749.52005
DOI: 10.1016/0166-8641(92)90132-J
published in: Topology and its Applications
publication date: 1992-09-26T00:00:00Z
review text: In this paper, an \(n\)-manifold \(M\) with \(C^ 2\)-boundary \(\partial M\) in \(\mathbb{R}^ n\) is called \(p^*\)-convex if each point on \(\partial M\) has at most \(p\) negative principal normal curvatures with respect to the inward normal vector and if the homology groups \(H_ i(M)\) of \(M\) of dimension \(i\geq p\) vanish. The author's main result is that a compact \(n\)-manifold \(M\) with \(C^ 2\)-boundary in \(\mathbb{R}^ n\) is \(p^*\)-convex if and only if for almost all \(x\) in \(\mathbb{R}^ n\), all \(r>0\) and all \(i\geq p\) it satisfies \(H_ i(M\cap B(x,r))=0\), where \(B(x,r)\) denotes the ball with centre \(x\) and radius \(r\). The proof uses Morse theory.
reviewed by: Rolf Schneider
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 53C42
Mathematics Subject Classification ID: 58E05
zbMATH DE Number: 55894
zbMATH Keywords: \(p^*\)-convex
zbMATH Keywords: focal point
zbMATH Keywords: principal normal curvatures
zbMATH Keywords: index
author: Y. D. Chai
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 55894