The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic in a compact Riemannian symmetric space necessarily a flat totally geodesic submanifold?



Tori in Compact Riemannian Symmetric Spaces
I think that this is treated in Helgason's *Differential Geometry, Lie Groups and Symmetric Spaces*. The point is that, for any Riemannian symmetric space $G/K$, one has the notion of the *rank* $r$ of the symmetric space, which is the dimension $r$ of a maximal abelian subspace $\frak{a}$ in $\frak{k}^\perp\subset \frak{g}$, where $\frak{k}\subset\frak{g}$ are the respective Lie algebras of $K\subset G$. The exponential of $\frak{a}$ gives a torus that then maps down to a totally geodesic torus in $G/K$. (The images of this torus under the action of $G$ are the so-called 'flats' in $G/K$.) Then, because the $K$-orbit of $\frak{a}$ in $\frak{k}^\perp$ contains everything in $\frak{k}^\perp$, it follows that every geodesic in $G/K$ is a geodesic in such a 'flat', and hence its closure is itself a closed geodesic sub-torus in a flat.


The case of a compact Lie group $K$ is the special case in which $G = K\times K$ and we regard $K$ as embedded diagonally in $G$.


