# Index bounded Riemannian metrics

Let $$L$$ be a closed simply-connected smooth manifold with a Riemannian metric $$g$$. We say $$g$$ is index bounded if the energy functional (which is assumed to be Morse/Morse-Bott) $$E: C^k(L,g) \rightarrow \mathbb{R}$$ is a bounded function for each $$k$$. Here $$C^k(L,g)$$ is the set of closed geodesics with Morse index $$k$$.

Question 1: When does $$L$$ admit an index bounded metric? (Examples I know are spheres.)

Question 2: If $$L$$ admits an index bounded metric, then under what kind of perturbations the index bounded property will be preserved?

My motivation: Morse theory of the energy functional computes the homology of the loop space of $$L$$. Suppose we know that on the homology level it is finite dimensional in each degree, then when is there a chain level analogue? Question 2 maybe related to the homotopy theory between different minimal model DGA's for the loop space, which I am not very familiar with. There are also parallel questions in the contact/symplectic setting, replacing closed geodesics by Reeb orbits.

• Do you have an example of a metric that is not index bounded? Jul 8, 2020 at 19:56
• @RBega2 Not yet. I tried for some time to show any simply-connected manifold admits such a metric, but there is not much progress. Jul 8, 2020 at 20:23