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The isometry groups of manifolds of nonpositive curvature with finite volume. (English) Zbl 0554.53029

Let M be a complete Riemannian manifold with finite volume, sectional curvature \(-\Lambda^ 2\leq K\leq 0\) and negative definite Ricci tensor at some point. Then the isometry group I(M) of M is finite; see G. Avérous and S. Kobayashi [Differ. Geom. Relativ., Vol. Honour A. Lichnerowicz 60th Birthday, 19-26 (1976; Zbl 0366.53025)]. Assume that M’ is a bounded open submanifold of M with finite diameter d(M’) which is a deformation retract of M. Then the order of I(M) is estimated from above in terms of \(\Lambda\), d(M’), dim(M) and the infimum of the injectivity radius \(i_ M(p)\) on M’.
Reviewer: G.Thorbergsson

MSC:

53C20 Global Riemannian geometry, including pinching

Citations:

Zbl 0366.53025
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References:

[1] Adachi, T., Sunada, T.: Energy spectrum of certain harmonic mappings, Preprint, Nagoya University · Zbl 0578.58009
[2] Averous, G., Kobayashi, S.: On automorphisms of spaces of nonpositive curvature with finite volume, Differential geometry and Relativity. Dortrecht: D. Reidel Publishing, Co., 19-26 (1976) · Zbl 0366.53025
[3] Bishop, R.L., Crittenden, R.J.: Geometry of manifolds. New York: Academic Press 1964 · Zbl 0132.16003
[4] Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc.145, 1-49 (1969) · Zbl 0191.52002 · doi:10.1090/S0002-9947-1969-0251664-4
[5] Buser, P., Karcher, H.: Gromov’s almost flat manifolds, Astérisque81 (1981) · Zbl 0459.53031
[6] Cheeger, J., Ebin, D.: Comparison theorems in Riemannian geometry. Amsterdam: North-Holland 1975 · Zbl 0309.53035
[7] Eberlein, P.: Lattices in spaces of nonpositive curvature, Ann. Math.111, 435-476 (1980) · Zbl 0432.53023 · doi:10.2307/1971104
[8] Frankel, T.: On theorems of Hurwitz and Bochner. J. Math. Mech.15, 373-377 (1966) · Zbl 0139.39103
[9] Fukaya, K.: A finiteness theorem for negatively curved manifolds. Preprint, Tokyo University · Zbl 0542.53023
[10] Gromov, M.: Manifolds of negative curvature. J. Differ. Geom.13, 223-230 (1978) · Zbl 0433.53028
[11] Gromov, M.: Structures metriques pour les variétés riemanniennes, rédigé par J. Lafontaine et P. Pansu. Paris: Cedic-Fernand Nathan 1981
[12] Heintze, E.: Mannigfaltigkeiten negativer Krümmung. Preprint, Bonn, 1976 · Zbl 1032.53024
[13] Huber, H.: Über die Isometriegruppe einer kompakten Mannigfaltigkeit mit negativer Krümmung. Helv. Phys. Acta45, 277-288 (1972)
[14] Im Hof, H.C.: Über die Isometriegruppe bei kompakten Mannigfaltigkeiten negative Krümmung. Comment. Math. Helv.,48, 14-30 (1973) · Zbl 0258.53040 · doi:10.1007/BF02566108
[15] Lawson, H.B., Yau, S.T.: Compact manifolds of nonpositive curvature. J. Differ. Geom.7, 211-288 (1972) · Zbl 0266.53035
[16] Lemaire, L.: Harmonic mappings of uniform bounded dilatation. Topology16, 199-201 (1977) · Zbl 0343.53029 · doi:10.1016/0040-9383(77)90021-0
[17] Maeda, M.: The isometry groups of compact manifolds with nonpositive curvature. Proc. Japan Acad., Ser.A51, 790-794 (1975) · Zbl 0341.53026 · doi:10.3792/pja/1195518435
[18] Thurston, W.: The geometry and topology of 3-manifolds. Lecture notes, Princeton University, 1978
[19] Wolf, J.A.: Homogeneity and bounded isometries in manifolds of negative curvature. III. J. Math.8, 14-18 (1964) · Zbl 0126.17702
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