A warped product splitting theorem
type: Item
instance of: scholarly article
title: A warped product splitting theorem
zbMATH Open document ID: 0772.53030
DOI: 10.1215/S0012-7094-92-06723-8
author: Christopher B. Croke
author: Bruce Kleiner
published in: Duke Mathematical Journal
publication date: 1992-12-14T00:00:00Z
review text: Let \(M\) be a Riemannian \(n\)-manifold with \(\text{Ric} \geq k(n-1)\), and \(N \subset M\) an oriented hypersurface. To fix signs, let \(d = d(. ,N)\) be the distance from \(N\) and call \(H(t) = {1\over n-1}\text{div }\nabla d|_{N_ t}\) the mean curvature of the equidistant hypersurface \(N_ t = \{d=t\}\) with respect to the normal vector \(\nabla d\). This satisfies the Riccati inequality \(dH/dt + H^ 2 + k \leq 0\) with equality iff all \(N_ t\) are umbilic and the sectional curvature equals \(k\) on all planes containing the normal vector \(\nabla d\). Consequently, \(H\) is bounded from above by the corresponding mean curvature \(H_ k\) in a space \(M_ k\) of constant curvature \(k\), with equality only in the above mentioned case. If \(k = -\delta^ 2 \leq 0\), the horospheres in \(M_ k\) form such a family with \(H_ k = \delta\). Thus if \(H(0) \leq \delta\) on \(M\), we must have \(H(t) \leq \delta\) for all \(t\). This is used in the present paper to show the following theorem: Suppose that \(M\) has two boundary components \(N\) and \(N'\) where \(N\) is compact with mean curvature \(H \geq \delta\) with respect to the inward normal while the mean curvature of \(N'\) is \(H' \leq \delta\) with respect to the outward normal. Then \(M\) is isometric to a warped product: \(M = [0,L] \times N\) with metric \(ds^ 2 = dt^ 2 + e^{2\delta t}\cdot g_ N\). (Note the different sign convention for \(H\) in the paper.) This result is obtained similarly as the Cheeger-Gromoll splitting theorem by applying the maximum principle to the sum of the two distance functions \(d + d'\). Replacing \(d'\) by a Busemann function, the authors get an analogous result when \(N'\) is removed to \(\infty\). A similar theorem also holds for any \(k\) on regions between two concentric balls if again the mean curvatures are optimal.
reviewed by: J.-H. Eschenburg
Mathematics Subject Classification ID: 53C20
zbMATH DE Number: 78572
zbMATH Keywords: lower Ricci bounds
zbMATH Keywords: distance of hypersurfaces
zbMATH Keywords: boundary mean curvature
zbMATH Keywords: equidistant hypersurface
zbMATH Keywords: maximum principle
zbMATH Keywords: distance functions
zbMATH Keywords: Busemann function
MaRDI profile type: Publication
Wikidata QID: Q125755937
cites work: An elementary proof of the Cheeger-Gromoll splitting theorem
cites work: An extension of E. Hopf's maximum principle with an application to Riemannian geometry
cites work: Eigenvalue comparison theorems and its geometric applications
cites work: The splitting theorem for manifolds of nonnegative Ricci curvature
cites work: An elementary method in the study of nonnegative curvature
full work available at URL: s0012-7094-92-06723-8
OpenAlex ID: W1965229345
description: scientific article; zbMATH DE number 78572