I'd like to know whether a Riemannian symmetric space of compact type admits a non-flat totally geodesic surface. I've found an article by Mashimo on the classification of these surfaces for certain symmetric spaces but can't find a general existence result.



Non-flat totally geodesic surfaces
Actually, I just remembered that your question is addressed in Helgason's *Differential Geometry, Lie Groups, and Symmetric Spaces*. Look at Theorem 11.1 of Chapter VII, which implies what you want, since $m\_{\bar\delta}\ge1$ by definition. Once you know that there's a totally geodesic, constant curvature sphere of dimension $1+m\_{\bar\delta}\ge2$ in an irreducible symmetric space $M$ of compact type, you know that there is a totally geodesic $2$-sphere of constant curvature in $M$. Thus, the classification of symmetric spaces is not needed (which is what I was relying on in my comment above).


