Marco Mazzucchelli

Isometry invariant geodesics

(a question from Grove?)

Key idea: Marco is interested in geometrically distinct geodesic arcs. The key tool is to consider the local homology of a critical point/set. The idea is to look at the local homology of iterates.

(For instance, to find infinitely many geodesics from p to q, the variational problem doesn't have a $\mathbb{Z}/m$ symmetry anymore, so cannot use the symmetry anymore)

Suppose I is an isometry. Want the existence of $\tau$ so $I( \gamma(t) ) = \gamma( t + \tau)$.

If $I$ is the identity, then just looking for closed.

Question: when do we have infinitely many invariant geodesics?

Thm (Grove): If there exists a non-closed $I$ invariant geodesic then the closure of $\gamma(\mathbb R)$ contains uncountably infinitely many $I$-invariant geodesics.

It therefore remains to study periodic $I$ invariant geodesics.

Variational setting:
$$ \Lambda( M, I )
=
\{
\gamma \colon \mathbb R \to M
\, | \,
I ( \gamma(t) )
= \gamma (t+1)
\quad \forall t \in \mathbb R
\}
$$

Has $\mathbb R$ action by time shift.

Energy $E( \gamma ) = \int_0^1 | \dot \gamma |^2 dt$

The critical points of $E$ are $I$-invariant geodesics. The critical orbit of $\gamma$ is $\mathbb R$ if not closed, or $S^1$ if closed.

Warning: every such $\gamma$ comes with infinitely many critical points of $E$.

$\tau \in \mathbb R$. $\gamma^\tau(t) = \gamma( \tau t)$.
For each $\gamma$ that is a periodic critical point of $E$ of period $p \ge 1$, also have $\gamma^{mp + 1}$ for any $m \in \mathbb{N}$.

Study Morse-theoretic properties of this set $\{ \gamma^{mp+1} \}$.

Grove-Tanaka: show that $m \mapsto nul( \gamma^{mp+1})$ is bounded,
$m \mapsto index( \gamma^{mp+1} )$ groes linearly and finally $m \mapsto rank L_*( \gamma^{mp+1})$ is bounded.

Theorem: Assume $\pi_1(M) =0$ and $I$ is homotopic to the identity and $H^*(M, \mathbb Q)$ has more than 1 generator.

Then, there exist infinitely many $I$-invariant geodesics.

(RMK: I homotopic to the identity allows us to identify the homology of $\Lambda(M, I)$ with the loop space homology.)

(When $I$ is the identity, this recovers Gromoll-Meyer theorem.)

Theorem (Mazzucchelli):
Assume:

•  $M = M_1 \times M_2$ (as manifolds... do not require a product metric).
•  $rank H_1(M, \mathbb Z) > 0$ and $dim M_2 \ge 2$.
•  $I$ is homotopic to the identity.

Then, there exist infinitely many $I$-invariant geodesics.


Theorem (M):

in a general setting: if exist $\gamma$ closed $I$ invariant and $d \ge 2$ such that $L_*( \gamma^{mp+1})$ is non-trivial for infinitely many $m$ then there are infinitely many $I$ invariant geodesics.


Marco remarks that the ideas are closely related to Umberto's discussion of symplectically degenerate maxima.