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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: May 22, 2000
MathSciNet review: 1694283
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This paper considers the boundary rigidity problem for a compact convex Riemannian manifold $(M,g)$ with boundary $\partial M$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 $g'$ on $M$ there is a $C^{3,\alpha }$-neighborhood of $g$ such that $g$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 $M$). More precisely, given any metric $g'$ in this neighborhood with the same boundary distance function there is diffeomorphism $\varphi $which is the identity on $\partial M$such that $g'=\varphi ^{*}g$. 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

Nurlan S. Dairbekov
Affiliation: Sobolev Institute of Mathematics, Universitetskii pr., 4, Novosibirsk, 630090, Russia

Vladimir A. Sharafutdinov
Affiliation: Sobolev Institute of Mathematics, Universitetskii pr., 4, Novosibirsk, 630090, Russia

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

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