Involutions of compact Riemannian 4-symmetric spaces. (English) Zbl 1170.53029

Kurihara, Hiroyuki; Tojo, Koji

Osaka J. Math. 45, No. 3, 643-689 (2008).

A homogeneous space \((G/H,\langle,\rangle)\), where \(G\) is a Lie group, \(H\) a compact subgroup of \(G\) and \(\langle,\rangle\) a \(G\)-invariant Riemannian metric, is called a Riemannian \(k\)-symmetric space if there exists an automorphism \(\sigma\) on \(G\) such that \(G_o^{\sigma}\subset H\subset G^{\sigma}\), where \(G^{\sigma}\) and \(G_o^{\sigma}\) is the set of fixed points of \(\sigma\) and its identity component, respectively; \(\sigma^k=Id\) and \(\sigma^l\neq Id\) for any \(l<k\); the transformation of \(G/H\) induced by \(\sigma\) is an isometry. Such a space is denoted by \((G/H,\langle,\rangle,\sigma)\).
Riemannian \(3\)-symmetric spaces were classified in [A. Gray, J. Differ. Geom. 7, 343–369 (1972; Zbl 0275.53026)] and compact Riemannian \(4\)-symmetric spaces in [J. A. Jiménez, Trans. Am. Math. Soc. 306, 715–734 (1988; Zbl 0647.53039)]. Then classification results were obtained for the involutions of a compact Riemannian \(3\)-symmetric in [K. Tojo, Tohoku Math. J. 53, 131–143 (2001; Zbl 1026.53030) and J. Math. Soc. Japan 58, 17–53 (2006; Zbl 1097.53039)].
In this paper the authors consider the compact, irreducible Riemannian \(4\)-symmetric spaces \((G/H,\langle,\rangle,\sigma)\) such that the dimension of the centre of \(H\) is \(0\), or \(H\) is a centralizator of a toral subgroup of \(G\), and then they give all involutions \(\tau\) of such a space, with the property that \(\tau(H)=H\). Finally, they find the conjugations between these involutions and use them in order to obtain the classification of all equivalence classes of involutions preserving \(H\).

Reviewer: Cezar Dumitru Oniciuc (Iaşi)

MSC:

53C30	Differential geometry of homogeneous manifolds
17B20	Simple, semisimple, reductive (super)algebras
53C35	Differential geometry of symmetric spaces

Keywords:

symmetric spaces; Lie groups; involutions

Citations:

Zbl 0275.53026; Zbl 0647.53039; Zbl 1026.53030; Zbl 1097.53039

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