Applications of geometry of symmetric spaces to submanifold theory

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A symmetric space is a Riemannian manifold which admits the geodesic symmetry at each point, which is an important and fundamental object in differential geometry. The principal investigator proved, in the joint work with Hiroyuki Tasaki, that the intersection of two real forms in a Hermitian symmetric space M of compact type is an antipodal set where M is not necessarily irreducible. She also proved, in the joint work with Peter Quast, that any reflective submanifold in a symmetric R-space is convex. And she also proved, in the joint work with Jost-Hinrich Eschenburg and Quast, that any isometry of a semisimple Hermitian symmetric space M can be extended to a linear isometry of the Lie algebra of G when M is realized as an adjoint orbit of a semisimple Lie group G.