Totally geodesic submanifolds of $\mathbb{R}P^n$

Question

Problem. I'm stuck on the following exercise from these notes on Riemannian geometry:

Determine the totally geodesic submanifolds of $\mathbb{R}P^n$.

Thoughts. I know that a submanifold $M\subseteq \mathbb{R}P^n$ is totally geodesic if and only if every geodesic in $M$ is also a geodesic in $\mathbb{R}P^n$.

I also know that every geodesic of $\mathbb{R}P^n$ can be thought of as the image of a geodesic in $S^n$ [i.e. a piece of a great circle!] under the map $\varphi\colon S^n\to \mathrm{Sym}(\mathbb{R}^{n+1})$ defined by $\varphi(p)=(2pp^\top-I)$.

This makes me think that we could perhaps somehow translate the problem into a question about the totally geodesic submanifolds of $S^n$, since $S^n$ feels more concrete than $\mathbb{R}P^n$. But at the same time, I'm not at all sure how to tackle $S^n$ either. Especially if we allow incomplete or disconnected submanifolds, it feels like there are a lot a different totally geodesic submanifolds of $S^n$ [for example any open subset of $S^n$, or unions of disjoint great circles when $n\geqslant 3$]. So perhaps thinking about $S^n$ is not at all the right approach...

Answer 1

If $N$ is $m$-dimensional complete totally geodesic submanifold in $\mathbb{R}P^n$, then let $p\in N$.

If $c:(-\infty,\infty)\rightarrow N$ is a geodesic in $N$, then there is a great circle $C$ in $S^n$ s.t. $c$ is a quotient of $C$.

If $X:=B(p,\epsilon)\cap N$, then consider a totally geodesic submanifold containing $X$ in $S^n$. It is $S^m$. Its quotient is totally geodesic $m$-dimensional submanifold contained in $N$. Hence it is $N$.

Answer 2

This is an attempt to combine HK Lee's approach with an approach suggested by a course mate. I might have missed something somewhere, so please comment if something seems unclear or plain wrong.

The idea will be to use the double-cover $\varphi\colon S^n\to \mathbb{R}P^n$ given by $\varphi(p)=[p]$, which is locally an isometry. (It is well-known that this is locally a diffeomorphism, and since it is oftentimes used to construct the metric on $\mathbb{R}P^n$, see for instance this thread, it is locally an isometry more or less by construction.) Since geodesics minimize distances locally, and $\varphi$ preserves distances locally, it is (at least intuitively) clear that $\varphi$ and $\varphi^{-1}$ will locally take geodesics to geodesics.

(1) $S^k$ (sitting in $S^n$) is a totally geodesic submanifold of $S^n$ for any $1\leqslant k\leqslant n$.

This is easy to verify. See for instance the end of Ch. 7 here.

(2) If $M^m$ is a totally geodesic submanifold of $S^n$, then $M^m$ is locally equal to $S^m$ (sitting in $S^n$).

Let $p\in M$ and let $V:=T_pM\cong \mathbb{R}^m$. It is now easy to embed $S^m$ in $S^n$ in such a way that $T_pS^m=V$. By this result, that must mean that $M$ and $S^m$ locally coincide.

(3) Let $1\leqslant k\leqslant n$ and let $S^k$ be sitting in $S^n$. Then $\varphi(S^k)$ is locally isometric to $\mathbb{R}P^k$ sitting in $\mathbb{R}P^n$.

Just choose $k$ linearly independent vectors $p_1,\ldots,p_k\in S^k\subseteq S^m\subseteq \mathbb{R}^{n+1}$ and form $\mathbb{R}P^k=\{[v]:v\in\mathrm{span}\{p_1,\ldots,p_k\}\setminus\{0\}\}$.

(4) For every $1\leqslant k\leqslant n$, it holds that $\mathbb{R}P^k$ (sitting in $\mathbb{R}P^n$) is a totally geodesic submanifold of $\mathbb{R}P^n$.

Every geodesic $\gamma$ in $\mathbb{R}P^k$ will (locally) correspond to a geodesic $\varphi^{-1}\circ \gamma$ in $S^k$ (sitting in $S^n$). By (1), $\varphi^{-1}\circ\gamma$ will also be a geodesic in $S^n$, and hence, $\gamma$ will be a geodesic in $\mathbb{R}P^n$.

(5) Every totally geodesic submanifold $M^m$ of $\mathbb{R}P^n$ is locally equal to $\mathbb{R}P^m$ sitting in $\mathbb{R}P^n$ .

Suppose that $M$ is a totally geodesic submanifold of $\mathbb{R}P^n$, and let $p\in M$. Then there exists some $\epsilon>0$ such that $B_\epsilon(p)\cap M$ is isometric under $\varphi^{-1}$ to a submanifold of $S^n$. Since $M$ is totally geodesic, $\varphi^{-1}(B_\epsilon(p)\cap M)$ must also be totally geodesic, and hence, by (2), locally be some embedding of $S^m$ in $S^n$. By (3), this means that $B_\epsilon(p)\cap M$ is locally $\mathbb{R}P^m$.

(4) and (5) together mean that the connected, complete totally geodesic submanifolds of $\mathbb{R}P^n$ are excactly the natural embeddings of $\mathbb{R}P^k$ in $\mathbb{R}P^n$.