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

Let (M) and (N) be two connected smooth manifolds, where (M) is compact and oriented and (N) is Riemannian. Let (mathcal {E}) be the Fréchet manifold of all embeddings of (M) in (N) , endowed with the canonical weak Riemannian metric. Let (sim ) be the equivalence relation on (mathcal {E}) defined by (fsim g) if and only if (f=gcirc phi ) for some orientation preserving diffeomorphism (phi ) of (M) . The Fréchet manifold (mathcal {S}= mathcal {E}/_{sim }) 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 ( mathcal {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 (mathcal {C}) be the set of submanifolds of (N) which are congruent to (M) . We prove that the natural inclusion of (mathcal {C}) in (mathcal {S}) is totally geodesic.