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), 3937-3956
MSC (2000):
Primary 53C65, 53C22, 53C20; Secondary 58C25, 58C40
DOI:
https://doi.org/10.1090/S0002-9947-00-02532-0
Published electronically:
May 22, 2000
MathSciNet review:
1694283
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Abstract | References | Similar Articles | Additional Information
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 distance-function (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 distance-function.
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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:
https://doi.org/10.1090/S0002-9947-00-02532-0
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 #MDS-9505175 and #MDS-9626232.
The third author was also supported by INTAS — RFBR, Grant 95–0763.
Article copyright:
© Copyright 2000
American Mathematical Society