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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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.
 [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
(28 #5252)
 [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
(92m:58140), http://dx.doi.org/10.1215/S0012709491063295
 [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 (93f:53034)
 [Cr1]
Christopher
B. Croke, Rigidity and the distance between boundary points,
J. Differential Geom. 33 (1991), no. 2,
445–464. MR 1094465
(92a:53053)
 [Cr2]
Christopher
B. Croke, Rigidity for surfaces of nonpositive curvature,
Comment. Math. Helv. 65 (1990), no. 1, 150–169.
MR
1036134 (91d:53056), http://dx.doi.org/10.1007/BF02566599
 [Cr3]
C. B. Croke, A sharp fourdimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), 187192. MR 83f:53060
 [CrSh]
Christopher
B. Croke and Vladimir
A. Sharafutdinov, Spectral rigidity of a compact negatively curved
manifold, Topology 37 (1998), no. 6,
1265–1273. MR 1632920
(99e:58191), http://dx.doi.org/10.1016/S00409383(97)000864
 [Gr]
Mikhael
Gromov, Filling Riemannian manifolds, J. Differential Geom.
18 (1983), no. 1, 1–147. MR 697984
(85h:53029)
 [GK]
Victor
Guillemin and David
Kazhdan, Some inverse spectral results for negatively curved
𝑛manifolds, 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
(81i:58048)
 [KA]
L.
V. Kantorovich and G.
P. Akilov, Functional analysis, 2nd ed., Pergamon Press,
OxfordElmsford, N.Y., 1982. Translated from the Russian by Howard L.
Silcock. MR
664597 (83h:46002)
 [Ml]
René
Michel, Sur la rigidité imposée par la longueur des
géodésiques, Invent. Math. 65
(1981/82), no. 1, 71–83 (French). MR 636880
(83d:58021), http://dx.doi.org/10.1007/BF01389295
 [Ot]
JeanPierre
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 (91i:53054), http://dx.doi.org/10.1007/BF02566611
 [Sh1]
V.
A. Sharafutdinov, Integral geometry of a tensor field on a manifold
with upperbounded 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
(94d:53116), http://dx.doi.org/10.1007/BF00970902
 [Sh2]
V.
A. Sharafutdinov, Integral geometry of tensor fields, Inverse
and Illposed Problems Series, VSP, Utrecht, 1994. MR 1374572
(97h:53077)
 [Sh3]
V. A. Sharafutdinov, Some questions of integral geometry on Anosov manifolds, to appear.
 [Sl]
Luis
A. Santaló, Integral geometry and geometric
probability, AddisonWesley Publishing Co., Reading,
Mass.LondonAmsterdam, 1976. With a foreword by Mark Kac; Encyclopedia of
Mathematics and its Applications, Vol. 1. MR 0433364
(55 #6340)
 [SU]
Yu.
A. Aminov and M.
L. Rabelo, On toroidal submanifolds of constant negative
curvature, Mat. Fiz. Anal. Geom. 2 (1995),
no. 34, 275–283 (English, with English, Russian and Ukrainian
summaries). MR
1484324 (99c:53059)
 [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), 3592. MR 28:5252
 [An]
 M. Anderson, Remarks on the compactness of isospectral sets in low dimensions, Duke Math. J. 63 (1991), 699711. MR 92m:58140
 [BPP]
 R. Brooks, P. Perry, and P. Petersen, Compactness and finiteness theorems for isospectral manifolds, J. Reine Angew. Math. 426 (1992), 6789. MR 93f:53034
 [Cr1]
 C. B. Croke, Rigidity and the distance between boundary points, J. Differential Geometry 33 (1991), 445464. MR 92a:53053
 [Cr2]
 C. B. Croke, Rigidity for surfaces of nonpositive curvature, Comment. Math. Helv. 65 (1990), 150169. MR 91d:53056
 [Cr3]
 C. B. Croke, A sharp fourdimensional isoperimetric inequality, Comment. Math. Helv. 59 (1984), 187192. MR 83f:53060
 [CrSh]
 C. B. Croke, and V. A. Sharafutdinov, Spectral rigidity of a compact negatively curved manifold, Topology 37 (6) (1998), 12651273. MR 99e:58191
 [Gr]
 M. Gromov, Filling Riemannian manifolds, J. Differential Geometry 18 (1983), 1147. MR 85h:53029
 [GK]
 V. Guillemin and D. Kazhdan, Some inverse spectral results for negatively curved nmanifolds, Proc. Sympos. Pure Math., Amer. Math. Soc., Providence, RI (1980), 153180. MR 81i:58048
 [KA]
 L. V. Kantorovich and G. P. Akilov, Functional analysis (Translated from the Russian by Howard L. Silcock. Second edition), Pergamon Press, OxfordElmsford, N.Y., 1982. MR 83h:46002
 [Ml]
 R. Michel, Sur la rigidité imposée par lar longueur des géodésiques, Invent Math. 65 (1981), 7183. MR 83d:58021
 [Ot]
 J.P. Otal, Sur les longueur des géodésiques d'une métrique à courbure négative dans le disque, Comment. Math. Helv. 65 (1990), 334347. MR 91i:53054
 [Sh1]
 V. A. Sharafutdinov, Integral geometry of a tensor field on a manifold whose curvature is bounded above, Siberian Math. J. 33 (3) (1992), 524533. MR 94d:53116
 [Sh2]
 V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, the Netherlands, 1994. MR 97h:53077
 [Sh3]
 V. A. Sharafutdinov, Some questions of integral geometry on Anosov manifolds, to appear.
 [Sl]
 L. Santaló, Integral Geometry and Geometric Probability (Encyclopedia of Mathematics and its Applications. Vol. 1), AddisonWesley Publishing Co., Reading, Mass.LondonAmsterdam, 1976. MR 55:6340
 [SU]
 P. Stefanov and G. Uhlmann, Rigidity for metrics with the same lengths of geodesics, Mathematical Research Letters 5 (1998), 8396. MR 99c:53059
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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
