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