On totally geodesic submanifolds

We give a proof of Cartan's Theorem on totally geodesic submanifolds for real analytic manifolds endowed with a real analytic, torsion-free, affine connection. We apply the theorem to real analytic Hadamard manifolds and, more generally, to real analytic manifolds with a torsion-free, analytic, affine connection, such that at a manifold point $p\in M$, the exponential map is a real analytic diffeomorphism from the tangent space $T_p(M)$ to $M$. Examples of manifolds with this property are statistical manifolds with a cubic form divisible by the metric, as was recently proven. We also give examples of totally geodesic submanifolds obtained as fixed points of affine transformations of $M$ and, moreover, as certain submanifolds of connected Lie groups with the $0$-connection of Cartan--Schouten. Finally, we also determine all connected complete totally geodesic surfaces of the Riemannian manifold $(\mathcal{P}_2, g)$ of symmetric positive definite $2\times 2$ \,real matrices, endowed with the trace metric $g$.