Totally geodesics submanifolds

Question

Determine all submanifolds totally geodesics to a hiperbolic space $\mathbb{H}^{n}$. I have no idea how i prove this fact. Any tips?

Thanks

Answer 1

Work in the hyperboloid model: $\mathbb{H}^n$ is the set $\{x \colon~\langle x, x\rangle = -1\}$ in $\mathbb{R}^{n,1}$ (or rather the upper sheet of this hyperboloid).

A geodesic in $\mathbb{H}^n$ is the intersection of $\mathbb{H}^n$ with a 2-dimensional linear subspace of $\mathbb{R}^{n,1}$. Use this fact to show that a totally geodesic submanifold of $\mathbb{H}^n$ of dimension $k$ is the intersection of $\mathbb{H}^n$ with a $k+1$-dimensional linear subspace of $\mathbb{R}^{n,1}$. In particular, notice that any totally geodesic submanifold of $\mathbb{H}^n$ of dimension $k$ is isometric to $\mathbb{H}^k$ (also, notice that there are "a lot" of totally geodesic manifolds through any point).