Let $S^n$ be the $n$-dimensional round sphere (i.e. with Riemannian metric of constant curvature +1). Is there any classification result of totally geodesic embedded submanifolds? Are they all round spheres? How about general case when the curvature only assumes to be positive?
edit: To make the second question more precise: Let $S^n$ be the standard sphere with Riemannian metric such that the sectional curvature $\ge 1$. Let $N\in M$ be a totally geodesic connected submanifold. (assume $N$ is not a point). What is the possible topology of $N$?