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The fine structure of transitive Riemannian isometry groups. I. (English) Zbl 0565.53030

Let (M,g) be a connected Riemannian manifold. Let \(G=Iso(M,g)\) be the connected identity component of the group of all isometries of (M,g). We assume G acts transitively on M. Let H be a transitive connected subgroup of G. The authors study the following two problems: (i) describe G in terms of H; (ii) modify H to obtain a ”nicer” transitive group of isometries of (M,g). In this paper as part I, some preliminary results are proved such as ”We show that semisimple Levi factors \(H_ 1\) of H and \(G_ 1\) of G can be chosen so that \(H_ 1\subset G_ 1\) and \(G_ 1\) are ”compatible” with the isotropy subgroups \(L_ 0=L\cap H\) of H and L of G respectively”. Here L is the isotropy group at some point of M.
Reviewer: T.Ochiai

MSC:

53C30 Differential geometry of homogeneous manifolds
22E15 General properties and structure of real Lie groups
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