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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)

MSC:
 53C30 Differential geometry of homogeneous manifolds