# If $M$ is a globally symmetric space, do we need it to be compact to prove that all the critical submanifolds in $\Lambda M$ are nondegenerate?

In "The free loop space of globally symmetric spaces" by Ziller, he proves the following theorem:

Theorem 2. For a globally symmetric space the critical submanifolds in $\Lambda M$ are all nondegenerate.

Here, $\Lambda M$ denotes the free loop space. He heavily implies that compactness of $M$ is not needed on his remark:

Remarks. 1) Theorem 2 has been proved by Bott for the loop space with fixed end points $\Omega (M,p,q)$ of a compact manifold $M$. (...) This fact also follows from our proof but Bott takes a slightly different argument which uses compactness of $M$.

The fact is: for Theorem $2$, he (apparently) uses the fact that all critical submanifolds are of type $I_0(M).c%$, where $c$ is a closed geodesic. This (again, apparently) follows from the observations on section $1.3.$ But in this section, he mentions clearly that : "We will assume $M$ is compact".

Since the arguments used to prove are based on Lie Algebra/Lie Groups theory, which I don't have much acquaintance with, I can't say for sure that he does not use compactness of $M$ to arrive at the fact needed. Hence, my question is:

Isn't compactness necessary for the proof of Theorem $2$?