Generalization of Myers' theorem on a contact manifold
type: Item
instance of: scholarly article
title: Generalization of Myers' theorem on a contact manifold
zbMATH Open document ID: 0705.53020
author: Ramesh Sharma
author: David E. Blair
published in: Illinois Journal of Mathematics
publication date: 1990-01-01T00:00:00Z
review text: \textit{S. B. Myers} [Duke Math. J. 8, 401-404 (1941; Zbl 0025.22704)] has proved that a complete Riemannian manifold for which \(Ric\geq \delta >0\), is compact. \textit{I. Hasegawa} and \textit{M. Seino} [J. Hokkaido Univ. Ed. Sect. II A 32, 1-7 (1981)] have generalized Myers' theorem for a Sasakian manifold by proving that a complete Sasakian (normal contact metric) manifold for which \(Ric\geq -\delta >-2\), is compact. In the present paper, the authors generalize Myers' theorem as follows: Let M be a complete \((2n+1)\)-dimensional contact metric manifold and (\(\phi\),\(\xi\),\(\eta\),g) its contact metric structure [cf. the first author, Contact manifolds in Riemannian geometry, Lect. Notes Math. 509 (1976; Zbl 0319.53026)]. Assume that div \(\tau\) \(=\sigma \eta\), \(\tau\) being the strain tensor defined by \(g(\tau X,Y)=({\mathcal L}_{\xi}g)(X,Y),\) X,Y\(\in {\mathfrak X}(M)\), and \(\sigma\) a function. Then, M is compact if \(Ric\geq -\delta >-2\) and the sectional curvatures of plane sections containing \(\xi\) are \(\geq \epsilon >\delta '\geq 0\), where \[ \delta '=2\sqrt{n(\delta -2\sqrt{2\delta}+n+2)}-(\delta - 2\sqrt{2\delta}+1+2n), \] or \[ \delta '=-\frac{1}{4}\delta^ 2+\sqrt{2}\delta^{3/2}-3\delta +2\sqrt{2}\delta^{1/2}\quad if\quad 2n+1=3. \]
Mathematics Subject Classification ID: 53C15
zbMATH DE Number: 4157416
zbMATH Keywords: Myers' theorem
zbMATH Keywords: contact metric manifold
zbMATH Keywords: sectional curvatures
reviewed by: Zbigniew Olszak
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 4157416