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A note on Fischer-Marsden's conjecture

Author: Ying Shen
Journal: Proc. Amer. Math. Soc. 125 (1997), 901-905
MSC (1991): Primary 53C21, 53C42
MathSciNet review: 1353399
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Abstract: In this paper, we borrowed some ideas from general relativity and find a Robinson-type identity for the overdetermined system of partial differential equations in the Fischer-Marsden conjecture. We proved that if there is a nontrivial solution for such an overdetermined system on a 3-dimensional, closed manifold with positive scalar curvature, then the manifold contains a totally geodesic 2-sphere.

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  • [Be-Eb] M. Berger and D. Ebin, Some decompositions of the space of symmetric tensors on a Riemannian manifold, J. Diff. Geom. 3 (1969), 379-392. MR 42:993
  • [Bes] A. Besse, Einstein Manifolds, Springer-Verlag, Berlin, 1987. MR 88f:53087
  • [Bo] J.P. Bourguignon, Une stratification de l'espace des structures riemanniennes, Compositio Math. 30 (1975), 1-41. MR 54:6189
  • [B-Mas] G. L. Bunting and A.K.M. Masood-ul-Alam, Nonexistence of Multiple Black Holes in Asymptotically Euclidean Static Vacuum Space-Time, General Relativity and Gravitation 19 (1987), 147-154. MR 88e:83031
  • [F-M] A.E. Fischer and J.E. Marsden, Linearzation stability of nonlinear partial differential equations, Proc. Symp. Pure Math. 27 (1975), 219-262. MR 53:4337
  • [I] W. Israel, Event horizons in static vacuum space-times, Phys. Rev. 164 (1967), 1776-1779.
  • [Ko] O. Kobayashi, A differential equation arising from scalar curvature function, J. Math. Soc. Japan 34, No. 4 (1982), 665-675. MR 84a:53046
  • [K] J.P. Kunzle, On the Spherical Symmetry of a Static Perfect Fluid, Comm. Math. Phys. 20 (1971), 85-100. MR 43:1586
  • [Laf] J. Lafontaine, Sur la geometrie d'une generalisation de l'equation d'Obata, J. Math. Pures Appliquees 62 (1983), 63-72. MR 84i:53047
  • [Lind] L. Lindblom, Some properties of static general relativistic stellar models, J. Math. Phys. 21, No. 6 (1980), 1455-1459.
  • [Mas] A.K. M. Masood-ul-Alam, On spherical symmetry of static perfect fluid spacetimes and positive-mass theorem, Class. Quantum Grav. 4 (1987), 625-633. MR 88f:83054
  • [Ob] M. Obata, Certain conditions for a riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340. MR 25:5479
  • [Rob] D. C. Robinson, A simple proof of the generalization of Israel's Theorem, General Relativity and Gravitation 8, No. 8 (1977), 695-698.

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Additional Information

Ying Shen
Affiliation: Department of Mathematics, Texas A & M University, College Station, Texas 77843
Address at time of publication: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755

Received by editor(s): June 12, 1995
Communicated by: Peter Li
Article copyright: © Copyright 1997 American Mathematical Society

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