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)



Volume growth and holonomy
in nonnegative curvature

Author: Kristopher Tapp
Journal: Proc. Amer. Math. Soc. 127 (1999), 3035-3041
MSC (1991): Primary 53C20; Secondary 57S20
Published electronically: April 23, 1999
MathSciNet review: 1605945
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between the codimension of the soul and the maximal dimension of an orbit of the action of the normal holonomy group of the soul. Additionally, an example of a simply-connected soul with a non-compact normal holonomy group is constructed.

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

  • 1. J. Cheeger, Some Examples of Manifolds of Nonnegative Curvature, J. Differential Geom. 8(1972), 623-628. MR 49:6085
  • 2. J. Cheeger and D. Gromoll, On the structure of compete open manifolds of nonnegative curvature, Ann. of Math. 96 (1972), 413-443. MR 46:8121
  • 3. J.H. Eschenburg, Comparison theorems and hypersurfaces, Manuscripta Math. 59 (1987), 295-323. MR 89f:53062
  • 4. L. Guijarro and G. Walschap, The metric projection onto the soul, Trans. Amer. Math. Soc. (to appear). CMP 98:06
  • 5. S. Kobayashi and K. Nomizu, Foundations of differential geometry I, Interscience Publishers (1963), J. Wiley and Sons. MR 27:2945
  • 6. G. Perelman, Proof of the soul conjecture of Cheeger and Gromoll, J. Differential Geom. 40(1994), 209-212. MR 95d:53037
  • 7. V. Schroeder and M. Strake, Volume growth of open manifolds with nonnegative curvature, Ann. Global Anal. Geom., 8, no.2 (1990), 159-165. MR 92e:53052
  • 8. M. Strake and G. Walschap, $\Sigma$-flat manifolds and Riemannian submersions, Manuscripta Math., 64(1989), 213-226. MR 90g:53054
  • 9. J.W. Yim, Space of souls in a complete open manifold of nonnegative curvature, J. Differential Geom. 32 (1990), 429-455. MR 91j:53023

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 53C20, 57S20

Retrieve articles in all journals with MSC (1991): 53C20, 57S20

Additional Information

Kristopher Tapp
Affiliation: Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395

Keywords: Volume growth, holonomy, nonnegative curvature, soul
Received by editor(s): December 11, 1997
Published electronically: April 23, 1999
Communicated by: Christopher Croke
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society