Burstall and Guest have given a classification of harmonic maps of the 2‐sphere with values in Lie groups and inner symmetric spaces. We extend their result to outer symmetric spaces G/K, using the pointed Cartan embedding into G. We show that in this case the number of classes can be reduced from 2r to 2s where r = rank G and s = rank K. Moreover we replace the 2‐sphere by a simply connected compact Kähler manifold and ‘harmonic’ by ‘pluriharmonic’.

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Abstract. We show that every reflective submanifold of a symmetric R-space is (geodesically) convex.

We show that every isometry of an extrinsic symmetric space extends to an isometry of its ambient euclidean space. As a consequence, any isometry of a real form of a hermitian symmetric space extends to a holomorphic isometry of the ambient hermitian symmetric space. Moreover, every fixed point component of an isometry of a symmetric R-space is a symmetric R-space itself.

We introduce a new technique to the study and identification of submanifolds of simply-connected symmetric spaces of compact type based upon an approach computing k-positive Ricci curvature of the ambient manifolds and using this information in order to determine how highly connected the embeddings are.This provides codimension ranges in which the Cartan type of submanifolds satisfying certain conditions which generalize being totally geodesic necessarily equals the one of the ambient manifold. Using results by Guijarro-Wilhelm our approach partly generalizes recent work by Berndt-Olmos on the index conjecture.