Non-flat totally geodesic surfaces in symmetric spaces of classical type (Q2631628)

The goal of the paper is to provide a classification of non-flat totally geodesic surfaces in compact irreducible Riemannian symmetric spaces. More specifically, the author considers the spaces of classical type by considering each type (AI, AII, AIII, etc.). The author points out that given the duality between these spaces, the corresponding problem is also solved for symmetric spaces of noncompact type. Some partial results (especially for the type AI) had been obtained previously. The classification is given by providing a basis of the Lie triple system \(\mathfrak{m}\) corresponding to the tangent space of the totally geodesic surface of the symmetric space \(G/K\) at \(eK\) which satisfies two simple equations. From \(\mathfrak{m}\), the Lie algebra \(\mathfrak{u}=[\mathfrak{m},\mathfrak{m}]+\mathfrak{m}\) which is isomorphic to \(\mathfrak{su}(2)\) is formed. The proofs follow the same pattern for each classical type with notable differences. Properties of irreducible representations of \(\operatorname{SU}(2)\) are used in these proofs. Examples follow each case. The author concludes by answering in the positive the following question ``Is there a classical irreducible Riemannian symmetric space in which there exist two totally geodesic surfaces which have the same value of the curvature, but which are not congruent to each other?''.