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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On the number of geodesic segments
connecting two points
on manifolds of non-positive curvature


Author: Paul Horja
Journal: Trans. Amer. Math. Soc. 349 (1997), 5021-5030
MSC (1991): Primary 53C22; Secondary 53C70
MathSciNet review: 1401773
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Abstract: We prove that on a complete Riemannian manifold $M$ of dimension $n$ with sectional curvature $K_M < 0$, two points which realize a local maximum for the distance function (considered as a function of two arguments) are connected by at least $2n+1$ geodesic segments. A simpler version of the argument shows that if one of the points is fixed and $K_M \leq 0$ then the two points are connected by at least $n+1$ geodesic segments. The proof uses mainly the convexity properties of the distance function for metrics of negative curvature.


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Additional Information

Paul Horja
Affiliation: Department of Mathematics, Duke University, Durham, North Carolina 27708-0320
Email: horja@math.duke.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-97-01847-3
PII: S 0002-9947(97)01847-3
Received by editor(s): October 27, 1995
Received by editor(s) in revised form: June 6, 1996
Article copyright: © Copyright 1997 American Mathematical Society