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

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Abstract: Let be a connected homogeneous Riemannian manifold, the identity component of the full isometry group of and a transitive connected subgroup of . , where is the isotropy group at some point of . is naturally identified with the homogeneous space endowed with a suitable left-invariant Riemannian metric. This paper addresses the problem: Given a realization of as a Riemannian homogeneous space of a connected Lie group , describe the structure of the full connected isometry group in terms of . This problem has already been studied in case 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.

**[**Carolyn Gordon,**G**]*Riemannian isometry groups containing transitive reductive subgroups*, Math. Ann.**248**(1980), no. 2, 185–192. MR**573347**, 10.1007/BF01421956**[**C. S. Gordon and E. N. Wilson,**GW**]*Isometry groups of Riemannian solvmanifolds*, in preparation.**[**Sigurdur Helgason,**H**]*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****[**Nathan Jacobson,**J**]*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****[**Takushiro Ochiai and Tsunero Takahashi,**OT**]*The group of isometries of a left invariant Riemannian metric on a Lie group*, Math. Ann.**223**(1976), no. 1, 91–96. MR**0412354****[**A. L. Oniščik,**On**]*Inclusion relations among transitive compact transformation groups*, Amer. Math. Soc. Transl. (2)**50**(1966), 5-58.**[**Hideki Ozeki,**Oz**]*On a transitive transformation group of a compact group manifold*, Osaka J. Math.**14**(1977), no. 3, 519–531. MR**0461377**

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Additional Information

DOI:
http://dx.doi.org/10.1090/S0002-9947-1985-0779070-3

Keywords:
Isometry groups,
homogeneous Riemannian manifolds,
Levi decompositions

Article copyright:
© Copyright 1985
American Mathematical Society