Some totally geodesic submanifolds of the nonlinear Grassmannian of a compact symmetric space

Abstract

Let M and N be two connected smooth manifolds, where M is compact and oriented and N is Riemannian. Let E be the Fréchet manifold of all embeddings of M in N, endowed with the canonical weak Riemannian metric. Let ∼ be the equivalence relation on E defined by f ∼ g if and only if f = g ◦ φ for some orientation preserving diffeomorphism φ of M. The Fréchet manifold S = E/∼ of equivalence classes, which may be thought of as the set of submanifolds of N diffeomorphic to M and is called the nonlinear Grassmannian (or Chow manifold) of N of type M, inherits from E a weak Riemannian structure. We consider the following particular case: N is a compact irreducible symmetric space and M is a reflective submanifold of N (that is, a connected component of the set of fixed points of an involutive isometry of N). Let C be the set of submanifolds of N which are congruent to M. We prove that the natural inclusion of C in S is totally geodesic.