Local boundary rigidity of a compact Riemannian manifold with curvature bounded above
Authors:
Christopher B. Croke, Nurlan S. Dairbekov and Vladimir A. Sharafutdinov
Journal:
Trans. Amer. Math. Soc. 352 (2000), 39373956
MSC (2000):
Primary 53C65, 53C22, 53C20; Secondary 58C25, 58C40
Published electronically:
May 22, 2000
MathSciNet review:
1694283
Fulltext PDF Free Access
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Additional Information
Abstract: This paper considers the boundary rigidity problem for a compact convex Riemannian manifold with boundary whose curvature satisfies a general upper bound condition. This includes all nonpositively curved manifolds and all sufficiently small convex domains on any given Riemannian manifold. It is shown that in the space of metrics on there is a neighborhood of such that is the unique metric with the given boundary distancefunction (i.e. the function that assigns to any pair of boundary points their distance  as measured in ). More precisely, given any metric in this neighborhood with the same boundary distance function there is diffeomorphism which is the identity on such that . There is also a sharp volume comparison result for metrics in this neighborhood in terms of the boundary distancefunction.
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Additional Information
Christopher B. Croke
Affiliation:
Department of Mathematics, University of Pennsylvania, Philadelphia, Pennsylvania 19104
Email:
ccroke@math.upenn.edu
Nurlan S. Dairbekov
Affiliation:
Sobolev Institute of Mathematics, Universitetskii pr., 4, Novosibirsk, 630090, Russia
Email:
dair@math.nsc.ru
Vladimir A. Sharafutdinov
Affiliation:
Sobolev Institute of Mathematics, Universitetskii pr., 4, Novosibirsk, 630090, Russia
Email:
sharaf@math.nsc.ru
DOI:
http://dx.doi.org/10.1090/S0002994700025320
PII:
S 00029947(00)025320
Received by editor(s):
August 10, 1998
Published electronically:
May 22, 2000
Additional Notes:
The first and third authors were supported by CRDF, Grant RM2–143.
The first author was also supported by NSF grants #MDS9505175 and #MDS9626232.
The third author was also supported by INTAS — RFBR, Grant 95–0763.
Article copyright:
© Copyright 2000 American Mathematical Society
