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 $\mathcal {E}$ 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 $f\sim g$ if and only if $f=g\circ \phi$ 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.