Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The fine structure of transitive Riemannian isometry groups. I

Authors: Carolyn S. Gordon and Edward N. Wilson
Journal: Trans. Amer. Math. Soc. 289 (1985), 367-380
MSC: Primary 53C30
MathSciNet review: 779070
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ M$ be a connected homogeneous Riemannian manifold, $ G$ the identity component of the full isometry group of $ M$ and $ H$ a transitive connected subgroup of $ G$. $ G = HL$, where $ L$ is the isotropy group at some point of $ M$. $ M$ is naturally identified with the homogeneous space $ H/H \cap L$ endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of $ M$ as a Riemannian homogeneous space of a connected Lie group $ H$, describe the structure of the full connected isometry group $ G$ in terms of $ H$. This problem has already been studied in case $ H$ is compact, semisimple of noncompact type, or solvable. We use the fact that every Lie group is a product of subgroups of these three types in order to study the general case.

References [Enhancements On Off] (What's this?)

  • [G] Carolyn Gordon, Riemannian isometry groups containing transitive reductive subgroups, Math. Ann. 248 (1980), no. 2, 185–192. MR 573347,
  • [GW] C. S. Gordon and E. N. Wilson, Isometry groups of Riemannian solvmanifolds, in preparation.
  • [H] 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
  • [J] Nathan Jacobson, Lie algebras, Interscience Tracts in Pure and Applied Mathematics, No. 10, Interscience Publishers (a division of John Wiley & Sons), New York-London, 1962. MR 0143793
  • [OT] 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 0412354,
  • [On] A. L. Oniščik, Inclusion relations among transitive compact transformation groups, Amer. Math. Soc. Transl. (2) 50 (1966), 5-58.
  • [Oz] Hideki Ozeki, On a transitive transformation group of a compact group manifold, Osaka J. Math. 14 (1977), no. 3, 519–531. MR 0461377

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 53C30

Retrieve articles in all journals with MSC: 53C30

Additional Information

Keywords: Isometry groups, homogeneous Riemannian manifolds, Levi decompositions
Article copyright: © Copyright 1985 American Mathematical Society

American Mathematical Society