Higher rank curved Lie triples. (Q1396373)

A curved Lie triple of a Riemannian symmetric space \(S\) is an immersion \(f\colon M\to S\), such that each tangent space \(df_{p}(T_pM)\subset T_{f(p)}S\) is a Lie triple, i.e. preserved by the curvature tensor of \(S\). The present paper is concerned with curved Lie triples of higher rank. These correspond to semisimple Lie triples all of whose irreducible factors contain flat subspaces of dimension \(\geq 2\). The author conjectures that, besides the obvious case of totally geodesic immersions, all higher rank curved Lie triples in an irreducible compact symmetric space are extrinsically symmetric isotropy orbits. In support of this, the holonomy theorem of Berger-Simons is applied to prove that a curved Lie triple \(M\) of higher rank is locally symmetric with its induced metric. The conjecture is verified directly for Lie triples of symmetric R-spaces and, using the classification of \textit{H. Naitoh} [J. Math. Soc. Japan, 50, 557--592 (1998; Zbl 0911.53033)], for normal curved Lie triples.