What is the free loop space $\mathcal{L}M$ of a manifold of a manifold $M$ for which the energy functional has no critical points?

Question

What is the free loop space $\mathcal{L}M$ of a manifold $M$ for which $E:LM\to\mathbb{R}$ for $E:\gamma\mapsto\int_{S^1}\|\dot\gamma(t)\|^2dt$ has no non-degenerate critical points? Is it simply the empty set (is this possible)?

Thanks in advance!

Answer 1

The energy functional on the free loop space $LM$ only has degenerate critical points if $M$ is not zero dimensional. Of course all the constant loops are degenerate critical points (the critical set is a copy of $M$)

Here is the reason why all non-constant closed geodesic are degenerate:

There is a non-trivial action of the circle $S^1$ on the free loop space that leaves the energy functional invariant. If $\gamma:S^1\rightarrow M$ is a loop we define a new loop $s\cdot \gamma:S^1\rightarrow M$ by $c\cdot \gamma(t)=\gamma(t+s)$. If $\gamma$ is a non-constant loop then $c\cdot \gamma$ is also a non-constant loop (and different for most values of $c$!).

The functional $E$ is invariant under this action. Thus for any closed geodesic $\gamma$ for almost all $c$ the loop $c\cdot \gamma$ is also a closed geodesic (this is just a reparametrization). This shows that there cannot be isolated critical points. The best one can hope for is that the critical points come in $S^1$ families. I think it is true that this is true for a generic set of metrics, although I cannot give a reference from the top of my head.

You might wonder if there are manifolds without non-trivial closed geodesics. There is a Theorem of Lyusternik and Fet that this can only occur for non-compact manifolds (I take my manifolds without boundary here). Euclidean space is a simple example of a manifold without closed geodesics.