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The group of isometries of a compact Riemannian homogeneous space. (English) Zbl 0789.53032
Szenthe, J. (ed.) et al., Differential geometry and its applications. Proceedings of a colloquium, held in Eger, Hungary, August 20-25, 1989, organized by the János Bolyai Mathematical Society. Amsterdam: North- Holland Publishing Company. Colloq. Math. Soc. János Bolyai. 56, 597-616 (1992).
Let \(G\) be a connected simple compact Lie group and \(H\) be a closed Lie subgroup of \(G\). The Killing form of the Lie algebra of \(G\) induces a \(G\)-invariant Riemannian metric \(\gamma_ 0\) on \(M = G/H\) called the natural Riemannian metric. Let \(I(M) = I(M,\gamma_ 0)\) be the group of all isometries of \(\gamma_ 0\) and \(I(M)^ 0\) be its identity component. Let \(\text{Aut}_ GM\) be the group of automorphisms of \(M\) and \((\text{Aut}_ M)^ 0\) be its identity component, \(\text{Sim}_ GM\) the group of auto-similitudes of \(M\) and \(\text{Aut}(G,H)\) be the group of all automorphisms of \(G\) mapping \(H\) onto itself. The main result of this paper is the following: \(I(M) = G(\text{Aut}_ GM)^ 0\) (locally direct product), \(I(M) = \text{Sim}_ GM = GA\), \(A \simeq \text{Aut}(G,H)\), except for the following cases: a) \(M = G_ 2/SU_ 3 = S^ 6\), \(I(M) = O_ 7\); b) \(M =\text{Spin}_ 7/G_ 2 = S^ 7\), \(I(M) = O_ 8\); c) \(M = \text{Spin}_ 8/G_ 2 = S^ 7 \times S^ 7\), \(I(M) = (O_ 8 \times O_ 8) \rtimes \langle s\rangle\), \(s\) being the transposition of factors; d) \(M = G\) with the action \(l\) by left translations, \(\gamma_ 0\) being the bi-invariant Riemannian metric on \(G\), \(I(M) = (\text{Hol }G\rtimes \langle s\rangle\), \(s: g\to g^{-1}\) for \(g\in G\), \(\text{Hol }G = l(G)\cdot \text{Aut }G\).
For the entire collection see [Zbl 0764.00002].
Reviewer: F.Zhu (Hubei)

53C30 Differential geometry of homogeneous manifolds
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