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Transactions of the American Mathematical Society

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Homogeneous manifolds with negative curvature. I


Authors: Robert Azencott and Edward N. Wilson
Journal: Trans. Amer. Math. Soc. 215 (1976), 323-362
MSC: Primary 53C30; Secondary 22E25, 17B30
DOI: https://doi.org/10.1090/S0002-9947-1976-0394507-4
MathSciNet review: 0394507
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Abstract: This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature. Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra $ \mathfrak{s}$ satisfying these conditions and any member of an easily constructed family of inner products on $ \mathfrak{s}$, a metric deformation argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group.


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  • [1] R. Azencott and E. Wilson, Variétés homogènes à courbure négative, C. R. Acad. Sci. Paris Sér. A 278 (1974), 561-562. MR 0350658 (50:3150)
  • [2] A. Borel, Lectures on symmetric spaces, dittoed notes, Massachusetts Institute of Technology, 1958.
  • [3] É. Cartan, Leçons sur la géométrie des espaces de Riemann, Gauthier-Villars, Paris, 1928.
  • [4] E. Heintze, On homogeneous manifolds of negative curvature (pre-publication manuscript).
  • [5] S. Helgason, Differential geometry and symmetric spaces, Pure and Appl. Math., vol. 12, Academic Press, New York, 1962. MR 26 #2986. MR 0145455 (26:2986)
  • [6] J. E. Humphreys, Introduction to Lie algebras and representation theory, Graduate Texts in Math., vol. 9, Springer-Verlag, New York and Berlin, 1972. MR 48 #2197. MR 0323842 (48:2197)
  • [7] K. Iwasawa, On some types of topological subgroups, Ann. of Math. (2) 50 (1949), 507-558. MR 10, 679. MR 0029911 (10:679a)
  • [8] G. R. Jensen, Homogeneous Einstein spaces of dimension four, J. Differential Geometry 3 (1969), 309-349. MR 41 #6100. MR 0261487 (41:6100)
  • [9] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. I, Interscience, New York, 1963. MR 27 #2945. MR 0152974 (27:2945)
  • [10] A. Malcev, On the theory of Lie groups in the large, Mat. Sb. 16 (58) (1945), 163-190. MR 7, 115. MR 0013165 (7:115c)
  • [11] S. Myers and N. Steenrod, The group of isometries of a Riemannian manifold, Ann. of Math. 40 (1939), 400-416. MR 1503467
  • [12] N. Wallach, Harmonic analysis on homogeneous spaces, Dekker, New York, 1973. MR 0498996 (58:16978)
  • [13] J. A. Wolf, Homogeneity and bounded isometries in manifolds of negative curvature, Illinois J. Math. 8 (1964), 14-18. MR 29 #565. MR 0163262 (29:565)

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1976-0394507-4
Keywords: Simply transitive isometry groups, solvable Lie groups, negative curvature Lie algebras, almost normal operators, Lie algebra deformations
Article copyright: © Copyright 1976 American Mathematical Society

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