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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Spherical points in Riemannian manifolds


Author: Benjamin Schmidt
Journal: Proc. Amer. Math. Soc. 139 (2011), 305-308
MSC (2010): Primary 53B21; Secondary 53C20, 53C22, 53C24, 53C45
Published electronically: August 5, 2010
MathSciNet review: 2729092
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Abstract: A point $ p$ in a Riemannian manifold $ M$ is weakly spherical if for each point $ q \neq p$ there is either exactly one or at least three minimizing geodesic segments joining $ p$ to $ q$. In this note, it is shown that round 2-dimensional spheres are the only Riemannian surfaces with a weakly spherical point realizing the injectivity radius.


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Additional Information

Benjamin Schmidt
Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
Email: schmidt@math.msu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10521-X
PII: S 0002-9939(2010)10521-X
Received by editor(s): December 10, 2009
Received by editor(s) in revised form: March 30, 2010
Published electronically: August 5, 2010
Additional Notes: The author was supported in part by NSF Grant DMS-0905906.
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.