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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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  • 1. Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
  • 2. Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
  • 3. Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. North-Holland Mathematical Library, Vol. 9. MR 0458335
  • 4. Hallard T. Croft, Kenneth J. Falconer, and Richard K. Guy, Unsolved problems in geometry, Problem Books in Mathematics, Springer-Verlag, New York, 1991. Unsolved Problems in Intuitive Mathematics, II. MR 1107516
  • 5. A. Götz and A. Rybarski, Problem 102, Colloquium Mathematicum 2 (1951), 301-302.
  • 6. James J. Hebda, Conjugate and cut loci and the Cartan-Ambrose-Hicks theorem, Indiana Univ. Math. J. 31 (1982), no. 1, 17–26. MR 642612, 10.1512/iumj.1982.31.31003
  • 7. James J. Hebda, The local homology of cut loci in Riemannian manifolds, Tôhoku Math. J. (2) 35 (1983), no. 1, 45–52. MR 695658, 10.2748/tmj/1178229100
  • 8. James J. Hebda, Metric structure of cut loci in surfaces and Ambrose’s problem, J. Differential Geom. 40 (1994), no. 3, 621–642. MR 1305983
  • 9. Arthur L. Besse, Manifolds all of whose geodesics are closed, Ergebnisse der Mathematik und ihrer Grenzgebiete [Results in Mathematics and Related Areas], vol. 93, Springer-Verlag, Berlin-New York, 1978. With appendices by D. B. A. Epstein, J.-P. Bourguignon, L. Bérard-Bergery, M. Berger and J. L. Kazdan. MR 496885
  • 10. Sumner Byron Myers, Connections between differential geometry and topology II. Closed surfaces, Duke Math. J. 2 (1936), no. 1, 95–102. MR 1545908, 10.1215/S0012-7094-36-00208-9
  • 11. Frank W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math. 87 (1965), 575–604. MR 0208534
  • 12. Alan Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geometry 9 (1974), 513–517. MR 0390968
  • 13. C. T. Yang, Odd-dimensional wiedersehen manifolds are spheres, J. Differential Geom. 15 (1980), no. 1, 91–96 (1981). MR 602442
  • 14. Tudor Zamfirescu, On some questions about convex surfaces, Math. Nachr. 172 (1995), 313–324. MR 1330637, 10.1002/mana.19951720122

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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
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.