Totally real totally geodesic submanifolds of compact 3-symmetric spaces (Q5937398)

It is well-known that a totally geodesic submanifold of a Riemannian symmetric space is an orbit of a Lie subgroup of the isometry group, but it is not true in general. The authors study Riemannian 3-symmetric spaces \((G/K, \langle\;,\;\rangle)\) [cf. \textit{A. Gray}, J. Differ. Geom. 7, 343-369 (1972; Zbl 0275.53026)], and present a class of totally geodesic submanifolds which are orbits of Lie subgroups of the isometry group. The main result states that a half-dimensional, totally real and totally geodesic submanifold of \((G/K, \langle\;,\;\rangle)\) is an orbit of a Lie subgroup of \(G\), (\(G/K\) carries the canonical almost complex structure). The last section contains a classification of half-dimensional totally real and totally geodesic submanifolds of a compact Riemannian 3-symmetric space \((G/K, \langle\;,\;\rangle)\) in the case of \(rkG = rk K\) and \(\dim\operatorname {center} K \neq 0\).