Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Improving the metric in an open manifold
with nonnegative curvature


Author: Luis Guijarro
Journal: Proc. Amer. Math. Soc. 126 (1998), 1541-1545
MSC (1991): Primary 53C20
DOI: https://doi.org/10.1090/S0002-9939-98-04287-7
MathSciNet review: 1443388
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The soul theorem states that any open Riemannian manifold $(\!M\!,\!g\!)$ with nonnegative sectional curvature contains a totally geodesic compact submanifold $S$ such that $M$ is diffeomorphic to the normal bundle of $S$. In this paper we show how to modify $g$ into a new metric $g'$ so that:

  1. $g'$ has nonnegative sectional curvature and soul $S$.
  2. The normal exponential map of $S$ is a diffeomorphism.
  3. $(M,g')$ splits as a product outside of a compact set.
As a corollary we obtain that any such $M$ is diffeomorphic to the interior of a convex set in a compact manifold with nonnegative sectional curvature.


References [Enhancements On Off] (What's this?)

  • 1. Jeff Cheeger and Detlef Gromoll, On the structure of complete manifolds of nonnegative curvature, Annals of Mathematics 96 (1972), no. 3, 413-443. MR 46:8121
  • 2. Jose F. Escobar and Alexandre Freire, The spectrum of the Laplacian of manifolds of positive curvature, Duke Mathematical Journal 65 (1992), no. 1, 1-21. MR 93d:58174
  • 3. Stephen Kronwith, Convex manifolds of nonnegative curvature, Journal of Differential Geometry 14 (1979), 621-628. MR 82k:53063
  • 4. Grisha Perelman, Alexandrov's spaces with curvatures bounded from below, ii, Preprint.
  • 5. -, Proof of the soul conjecture of Cheeger and Gromoll, Journal of Differential Geometry 40 (1994), 209-212. MR 95d:53037
  • 6. V. A. Sharafutdinov, The Pogorelov-Klingenberg theorem for manifolds homeomorphic to ${\mathbb{R}}^n$, Siberian Mathematical Journal 18 (1977), 915-925. MR 58:7488
  • 7. Jin-Whan Yim, Distance nonincreasing retraction on a complete open manifold of nonnegative sectional curvature, Ann. Global Anal. Geom. 6 (1988), no. 2, 191-206. MR 90a:53049

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C20

Retrieve articles in all journals with MSC (1991): 53C20


Additional Information

Luis Guijarro
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email: guijarro@math.upenn.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04287-7
Received by editor(s): October 25, 1996
Communicated by: Christopher Croke
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society