Spherical points in Riemannian manifolds

Schmidt, Benjamin

doi:10.1090/S0002-9939-2010-10521-X

Spherical points in Riemannian manifolds
HTML articles powered by AMS MathViewer

by Benjamin Schmidt PDF

Proc. Amer. Math. Soc. 139 (2011), 305-308 Request permission

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.

References

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
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
Jeff Cheeger and David G. Ebin, Comparison theorems in Riemannian geometry, North-Holland Mathematical Library, Vol. 9, North-Holland Publishing Co., Amsterdam-Oxford; American Elsevier Publishing Co., Inc., New York, 1975. MR 0458335
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, DOI 10.1007/978-1-4612-0963-8
A. Götz and A. Rybarski, Problem 102, Colloquium Mathematicum 2 (1951), 301-302.
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, DOI 10.1512/iumj.1982.31.31003
James J. Hebda, The local homology of cut loci in Riemannian manifolds, Tohoku Math. J. (2) 35 (1983), no. 1, 45–52. MR 695658, DOI 10.2748/tmj/1178229100
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
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
Sumner Byron Myers, Connections between differential geometry and topology II. Closed surfaces, Duke Math. J. 2 (1936), no. 1, 95–102. MR 1545908, DOI 10.1215/S0012-7094-36-00208-9
Frank W. Warner, The conjugate locus of a Riemannian manifold, Amer. J. Math. 87 (1965), 575–604. MR 208534, DOI 10.2307/2373064
Alan Weinstein, On the volume of manifolds all of whose geodesics are closed, J. Differential Geometry 9 (1974), 513–517. MR 390968
C. T. Yang, Odd-dimensional wiedersehen manifolds are spheres, J. Differential Geometry 15 (1980), no. 1, 91–96 (1981). MR 602442
Tudor Zamfirescu, On some questions about convex surfaces, Math. Nachr. 172 (1995), 313–324. MR 1330637, DOI 10.1002/mana.19951720122