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A synthetic characterization of the hemisphere
Author(s):
Christopher
B.
Croke
Journal:
Proc. Amer. Math. Soc.
136
(2008),
1083-1086.
MSC (2000):
Primary 53C22
Posted:
November 23, 2007
MathSciNet review:
2361884
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References |
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Additional information
Abstract:
We show that round hemispheres are the only compact two-dimensional Riemannian manifolds (with or without boundary) such that almost every pair of complete geodesics intersect once and only once. We prove this by establishing a sharp isoperimetric inequality for surfaces with boundary such that every pair of geodesics has at most one interior intersection point.
References:
-
- [Ba]
- V. Bangert, Manifolds with geodesic chords of constant length, Math. Ann. 265 (1983), no. 3, 273-281. MR 721397 (85d:53016)
- [Be]
- A. Besse, Manifolds all of Whose Geodesics are Closed, Ergebisse Grenzgeb. Math., no. 93, Springer, Berlin, 1978. MR 496885 (80c:53044)
- [Bu-Kn]
- K. Burns and G. Knieper, Rigidity of surfaces with no conjugate points, J. Diff. Geom. 34 (1991), no. 3, 623-650. MR 1139642 (92j:53015)
- [Cr]
- C. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Scient. Ec. Norm. Sup., 4e serie, t.13(1980), 419-435. MR 608287 (83d:58068)
- [C-D]
- C. Croke and N. Dairbekov, Lengths and volumes in Riemannian manifolds, Duke Mathematical Journal 125 (2004), no. 1, 1-14. MR 2097355 (2005k:53045)
- [We]
- A. Weil, Sur les surfaces à courbure negative, C.R.A.S. 182 (1926), 1069-1071.
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Additional Information:
Christopher
B.
Croke
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6395
Email:
ccroke@math.upenn.edu
DOI:
10.1090/S0002-9939-07-09196-4
PII:
S 0002-9939(07)09196-4
Received by editor(s):
January 23, 2007
Posted:
November 23, 2007
Additional Notes:
Supported by NSF grants DMS 02-02536 and 07-04145
Communicated by:
Jon G. Wolfson
Copyright of article:
Copyright
2007,
American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.
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