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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.

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    $\begingroup$ It suffices to consider the irreducible case. Certainly, all the Type II cases will have this property because they are the compact simple Lie groups, and they all contain either a $\mathrm{SO}(3)$ or a $\mathrm{SU}(2)$. Among the Type I cases, i.e., the $G/K$ with $G$ simple and $K$ the fixed subgroup of an involution $\sigma$, the inner ones will have this property ('inner' means $\sigma$ is an inner automorphism). That leaves only A-I, A-II, E-I, and E-IV to check. Now, A-I and A-II have totally geodesic $2$-spheres (by inspection). I'd have to think about E-I and E-IV, but, surely they do. $\endgroup$ – Robert Bryant Jul 16 '15 at 13:37
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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).

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  • $\begingroup$ Thanks very much Robert. I wasn't aware of that result in Helgason. $\endgroup$ – Oliver Jones Jul 16 '15 at 21:42

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