A synthetic characterization of the hemisphere

Croke, Christopher

doi:10.1090/S0002-9939-07-09196-4

A synthetic characterization of the hemisphere
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by Christopher B. Croke

Proc. Amer. Math. Soc. 136 (2008), 1083-1086

DOI: https://doi.org/10.1090/S0002-9939-07-09196-4

Published electronically: November 23, 2007

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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

Victor Bangert, Manifolds with geodesic chords of constant length, Math. Ann. 265 (1983), no. 3, 273–281. MR 721397, DOI 10.1007/BF01456020
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, DOI 10.1007/978-3-642-61876-5
Keith Burns and Gerhard Knieper, Rigidity of surfaces with no conjugate points, J. Differential Geom. 34 (1991), no. 3, 623–650. MR 1139642
Christopher B. Croke, Some isoperimetric inequalities and eigenvalue estimates, Ann. Sci. École Norm. Sup. (4) 13 (1980), no. 4, 419–435. MR 608287, DOI 10.24033/asens.1390
Christopher B. Croke and Nurlan S. Dairbekov, Lengths and volumes in Riemannian manifolds, Duke Math. J. 125 (2004), no. 1, 1–14. MR 2097355, DOI 10.1215/S0012-7094-04-12511-4
A. Weil, Sur les surfaces à courbure negative, C.R.A.S. 182 (1926), 1069-1071.