Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-19T03:57:33.238Z Has data issue: false hasContentIssue false
  • English
  • Français

On a theorem of Ambrose

Part of: Global differential geometry

Published online by Cambridge University Press:  09 April 2009

David J. Wraith
David J. Wraith
Affiliation:
Department of Mathematics, National University of Ireland Maynooth, Ireland, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A Riccati inequality involving the Ricci curvature can be used to deduce many interesting results about the geometry and topology of manifolds. In this note we use it to present a short alternative proof to a theorem of Ambrose.

MSC classification

Secondary: 53C20: Global Riemannian geometry, including pinching
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 81 , Issue 2 , October 2006 , pp. 149 - 152
Copyright
Copyright © Australian Mathematical Society 2006

References

[1]Ambrose, W., ‘A theorem of Myers’, Duke Math. J. 24 (1957), 345–348.CrossRefGoogle Scholar
[2]Cheeger, J., Critical points of distance functions and applications to geometry, Lecture Notes in Math. 1504 (Springer, Berlin, 1991) PP. 1–38.Google Scholar
[3]Lohkamp, J., ‘Metrics of negative Ricci curvature’, Ann. of Math. (2) 140 (1994), 655–683.CrossRefGoogle Scholar
[4]Myers, S., ‘Riemannian manifolds with positive mean curvature’, Duke Math. J. 8 (1941), 401–404.CrossRefGoogle Scholar
[5]Sprouse, C., ‘Integral curvature bounds and bounded diameter’, Comm. Anal. Geom. 8 (2000), 531–543.CrossRefGoogle Scholar
[6]Stolz, S., ‘Simply connected manifolds of positive scalar curvature’, Bull. Amer. Math. Soc. (N.S.) 23 (1990), 427–432.CrossRefGoogle Scholar
[7]Wraith, D., ‘Ricci curvature decay on open manifolds’, Bull. London Math. Soc. 35 (2003), 72–78.CrossRefGoogle Scholar