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

Full-text PDF Free Access

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.

**[ADN]**S. Agmon, A. Douglis, and L. Nirenberg,*Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II*, Comm. Pure Appl. Math.**17**(1964), 35–92. MR**0162050**, https://doi.org/10.1002/cpa.3160170104**[An]**Michael T. Anderson,*Remarks on the compactness of isospectral sets in low dimensions*, Duke Math. J.**63**(1991), no. 3, 699–711. MR**1121151**, https://doi.org/10.1215/S0012-7094-91-06329-5**[BPP]**Robert Brooks, Peter Perry, and Peter Petersen V,*Compactness and finiteness theorems for isospectral manifolds*, J. Reine Angew. Math.**426**(1992), 67–89. MR**1155747****[Cr1]**Christopher B. Croke,*Rigidity and the distance between boundary points*, J. Differential Geom.**33**(1991), no. 2, 445–464. MR**1094465****[Cr2]**Christopher B. Croke,*Rigidity for surfaces of nonpositive curvature*, Comment. Math. Helv.**65**(1990), no. 1, 150–169. MR**1036134**, https://doi.org/10.1007/BF02566599**[Cr3]**C. B. Croke,*A sharp four-dimensional isoperimetric inequality*, Comment. Math. Helv.**59**(1984), 187-192. MR**83f:53060****[Cr-Sh]**Christopher B. Croke and Vladimir A. Sharafutdinov,*Spectral rigidity of a compact negatively curved manifold*, Topology**37**(1998), no. 6, 1265–1273. MR**1632920**, https://doi.org/10.1016/S0040-9383(97)00086-4**[Gr]**Mikhael Gromov,*Filling Riemannian manifolds*, J. Differential Geom.**18**(1983), no. 1, 1–147. MR**697984****[GK]**Victor Guillemin and David Kazhdan,*Some inverse spectral results for negatively curved 𝑛-manifolds*, Geometry of the Laplace operator (Proc. Sympos. Pure Math., Univ. Hawaii, Honolulu, Hawaii, 1979) Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, R.I., 1980, pp. 153–180. MR**573432****[KA]**L. V. Kantorovich and G. P. Akilov,*Functional analysis*, 2nd ed., Pergamon Press, Oxford-Elmsford, N.Y., 1982. Translated from the Russian by Howard L. Silcock. MR**664597****[Ml]**René Michel,*Sur la rigidité imposée par la longueur des géodésiques*, Invent. Math.**65**(1981/82), no. 1, 71–83 (French). MR**636880**, https://doi.org/10.1007/BF01389295**[Ot]**Jean-Pierre Otal,*Sur les longueurs des géodésiques d’une métrique à courbure négative dans le disque*, Comment. Math. Helv.**65**(1990), no. 2, 334–347 (French). MR**1057248**, https://doi.org/10.1007/BF02566611**[Sh1]**V. A. Sharafutdinov,*Integral geometry of a tensor field on a manifold with upper-bounded curvature*, Sibirsk. Mat. Zh.**33**(1992), no. 3, 192–204, 221 (Russian, with Russian summary); English transl., Siberian Math. J.**33**(1992), no. 3, 524–533 (1993). MR**1178471**, https://doi.org/10.1007/BF00970902**[Sh2]**V. A. Sharafutdinov,*Integral geometry of tensor fields*, Inverse and Ill-posed Problems Series, VSP, Utrecht, 1994. MR**1374572****[Sh3]**V. A. Sharafutdinov,*Some questions of integral geometry on Anosov manifolds*, to appear.**[Sl]**Luis A. Santaló,*Integral geometry and geometric probability*, Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of Mathematics and its Applications, Vol. 1. MR**0433364****[SU]**Yu. A. Aminov and M. L. Rabelo,*On toroidal submanifolds of constant negative curvature*, Mat. Fiz. Anal. Geom.**2**(1995), no. 3-4, 275–283 (English, with English, Russian and Ukrainian summaries). MR**1484324**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
53C65,
53C22,
53C20,
58C25,
58C40

Retrieve articles in all journals with MSC (2000): 53C65, 53C22, 53C20, 58C25, 58C40

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