What are all the totally geodesic submanifolds of $\mathbb{R}P^N$? 
If I were to guess the answer, I would have thought that it is all $\mathbb{R}P^n$, where $1\leq n\leq N$, based on the analogy with what happens with the sphere in $\mathbb{R}^{N+1}$. Is this true, and is there an easy way to see this? 



Determining all totally geodesic submanifolds of $\mathbb{R}P^N$
I think Lemma 3, Section 3 of this paper of Wolf provides a positive answer to your question(s): 


* J.A. Wolf. Elliptic spaces in Grassmann manifolds. Illinois J. Math.
Volume 7, Issue 3 (1963), 447-462. [(link to journal website)](https://projecteuclid.org/euclid.ijm/1255644952)


