Isometry groups of Riemannian solvmanifolds
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- by Carolyn S. Gordon and Edward N. Wilson
- Trans. Amer. Math. Soc. 307 (1988), 245-269
- DOI: https://doi.org/10.1090/S0002-9947-1988-0936815-X
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Abstract:
A simply connected solvable Lie group $R$ together with a left-invariant Riemannian metric $g$ is called a (simply connected) Riemannian solvmanifold. Two Riemannian solvmanifolds $(R, g)$ and $(R’ , g’ )$ may be isometric even when $R$ and $R’$ are not isomorphic. This article addresses the problems of (i) finding the "nicest" realization $(R, g)$ of a given solvmanifold, (ii) describing the embedding of $R$ in the full isometry group $I(R, g)$, and (iii) testing whether two given solvmanifolds are isometric. The paper also classifies all connected transitive groups of isometries of symmetric spaces of noncompact type and partially describes the transitive subgroups of $I(R, g)$ for arbitrary solvmanifolds $(R, g)$.References
- D. V. Alekseevskiĭ, The conjugacy of polar decompositions of Lie groups, Mat. Sb. (N.S.) 84 (126) (1971), 14–26 (Russian). MR 0277662
- D. V. Alekseevskiĭ, Homogeneous Riemannian spaces of negative curvature, Mat. Sb. (N.S.) 96(138) (1975), 93–117, 168 (Russian). MR 0362145
- Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. I, Trans. Amer. Math. Soc. 215 (1976), 323–362. MR 394507, DOI 10.1090/S0002-9947-1976-0394507-4
- Robert Azencott and Edward N. Wilson, Homogeneous manifolds with negative curvature. II, Mem. Amer. Math. Soc. 8 (1976), no. 178, iii+102. MR 426002, DOI 10.1090/memo/0178 E. D. Deloff, Naturally reductive metrics and metrics with volume preserving geodesic symmetries on $NC$ algebras, Thesis, Rutgers Univ., New Brunswick, N. J., 1979.
- Carolyn Gordon, Riemannian isometry groups containing transitive reductive subgroups, Math. Ann. 248 (1980), no. 2, 185–192. MR 573347, DOI 10.1007/BF01421956
- Carolyn Gordon, Transitive Riemannian isometry groups with nilpotent radicals, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 2, vi, 193–204 (English, with French summary). MR 617247
- Carolyn S. Gordon and Edward N. Wilson, Isospectral deformations of compact solvmanifolds, J. Differential Geom. 19 (1984), no. 1, 241–256. MR 739790
- Carolyn S. Gordon and Edward N. Wilson, The fine structure of transitive Riemannian isometry groups. I, Trans. Amer. Math. Soc. 289 (1985), no. 1, 367–380. MR 779070, DOI 10.1090/S0002-9947-1985-0779070-3
- Sigurdur Helgason, Differential geometry, Lie groups, and symmetric spaces, Pure and Applied Mathematics, vol. 80, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. MR 514561
- James E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Mathematics, Vol. 9, Springer-Verlag, New York-Berlin, 1972. MR 0323842
- Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons, Inc.), New York-London, 1962. MR 0143793
- Takushiro Ochiai and Tsunero Takahashi, The group of isometries of a left invariant Riemannian metric on a Lie group, Math. Ann. 223 (1976), no. 1, 91–96. MR 412354, DOI 10.1007/BF01360280 A. L. Oniscik, Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl. (2) 50 (1966), 5-58.
- Hideki Ozeki, On a transitive transformation group of a compact group manifold, Osaka Math. J. 14 (1977), no. 3, 519–531. MR 461377
- Edward N. Wilson, Isometry groups on homogeneous nilmanifolds, Geom. Dedicata 12 (1982), no. 3, 337–346. MR 661539, DOI 10.1007/BF00147318
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 307 (1988), 245-269
- MSC: Primary 53C30
- DOI: https://doi.org/10.1090/S0002-9947-1988-0936815-X
- MathSciNet review: 936815