A gap phenomenon on Riemannian manifolds with reverse volume pinching

P. Wang

doi:10.1007/s10474-006-0536-4

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Acta Mathematica Hungarica

Volume/Issue:: Volume 115: Issue 1-2

A gap phenomenon on Riemannian manifolds with reverse volume pinching

Author:

P. Wang

P. Wang Qufu Normal University School of Mathematics Qufu Shandong 273165 P.R. China Qufu Shandong 273165 P.R. China

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Pages:: 133–144

Online Publication Date:: 30 Nov 2006

Publication Date:: 01 Apr 2007

Article Category:: Research Article

DOI:: https://doi.org/10.1007/s10474-006-0536-4

Keywords:: diameter; volume comparison theorem; Hausdorff convergence; 53C20

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Abstract

Let Mⁿ be an n(≧ 3)-dimensional compact, simply connected Riemannian manifold without boundary and Sⁿ be the unit sphere of the Euclidean space Rⁿ⁺¹. By two different means we derive an estimate of the diameter whenever the manifold considered satisfies that the sectional curvature K_M ≦ 1, while Ric (M) ≧

\documentclass{aastex} \usepackage{amsbsy} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{bm} \usepackage{mathrsfs} \usepackage{pifont} \usepackage{stmaryrd} \usepackage{textcomp} \usepackage{upgreek} \usepackage{portland,xspace} \usepackage{amsmath,amsxtra} \usepackage{bbm} \pagestyle{empty} \DeclareMathSizes{10}{9}{7}{6} \begin{document} $$\tfrac{{n + 2}}{4}$$ \end{document}

and the volume V (M) ≦

(1 + η)V (Sⁿ) for some positive number η depending only on n. Consequently, a gap phenomenon of the manifold will be given according to the estimate of the diameter.

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Acta Mathematica Hungarica

Print ISSN:: 0236-5294

Online ISSN:: 1588-2632

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Acta Mathematica Hungarica
Language	English
Size	B5
Year of Foundation	1950
Volumes per Year	3
Issues per Year	6
Founder	Magyar Tudományos Akadémia
Founder's Address	H-1051 Budapest, Hungary, Széchenyi István tér 9.
Publisher	Akadémiai Kiadó Springer Nature Switzerland AG
Publisher's Address	H-1117 Budapest, Hungary 1516 Budapest, PO Box 245. CH-6330 Cham, Switzerland Gewerbestrasse 11.
Responsible Publisher	Chief Executive Officer, Akadémiai Kiadó
ISSN	0236-5294 (Print)
ISSN	1588-2632 (Online)