Why are we interested in closed geodesics?

Question

There's a lot of work about the existence, number and other properties of closed geodesics on a Riemannian manifold (belonging to some specific class of manifolds).

In the case of geodesics representing some non trivial homotopy class of closed curves on a (not simply connected) manifold, I understand the importance/usefullness of knowing that such a homotopy class (element of the fundamental group) can be represented by a geodesic. But what about simply connected manifolds? In dimension 2 the simply connected surfaces are $S^2, \mathbb{R}^2$ and $\mathbb{H}^2$; according to Lusternik-Fet, $S^2$, being compact, admits non trivial closed geodesics (whereas the other two do not), but they are all homotopically trivial; I don't get what these closed geodesic tell about the topology of $S^2$; my feeling is that maybe the whole situation is quite trivial in dimension 2, but I'm not much familiar with manifolds of dimension 3 or higher (spheres, euclidean and projective spaces apart)...so any interesting example is welcome!

I know that existence results are always of fundamental importance on their own, but here I can't figure out which are the implications of the existence of closed geodesic (especially in the simply connected case, as pointed out above).

To sum up, I'm interested in the following questions:

1) what if a manifold has a closed geodesic?

2) what if a manifold has no closed geodesic? (can I say that then the manifold is simply connected?...the punctured euclidean plane seems a counterexample...what if I restrict my question to complete manifolds?)

3) what if a manifold has more than one closed geodesic? or even infinite? (here I mean "distinct" geodesics, in some sense)

4) what if every geodesic is closed?

I would appreciate explicit examples as well as theorems or references on the subject.

Answer 1

I think closed geodesics are interesting because they place powerful constraints on the geometry and topology of a Riemannian manifold.

For instance, it is possible to place bounds on the volume and diameter of a manifold if you know the lengths of the longest or shortest closed geodesics. In fact there is sort of an entire subfield of Riemannian geometry called "systolic geometry" based on this principle; you might check out the wikipedia page for further discussion.

Moreover, there is a very nice theorem of Cartan which asserts that every free homotopy class of loops in a compact Riemannian manifold has a geodesic. This result answers your second question in the compact case, and there are counterexamples among complete manifolds (for instance, take an infinite cylinder whose radius monotonically decreases). It's useful because it can be used to relate topology and geometry; for instance this is the key ingredient in the proof of a theorem of Preissman which asserts that every nontrivial abelian subgroup of the fundamental group of a compact Riemannian manifold with negative curvature is infinite cyclic. This helps rule out metrics of negative curvature on a lot of manifolds.

Answer 2

There is a very strong relationship between the closed geodesics of $M$ and its topology.

Let $M$ be a manifold and fix a point $m\in M$. Let $P$ denote the based path space of $M$: $P =\{\gamma:[0,1]\rightarrow M: \gamma(0) = m$ and $\gamma$ is smooth $\}$. Let $\Omega \subseteq P$ be the based loop space, consisting of those $\gamma$ for which $\gamma(1) = m$ as well. (One can also just work with continuous curves if one wants. I believe the homotopy types of $E$ and $\Omega$ do not depend on this distinction. Also, one can work with free path spaces and free loop spaces, where the condition that $\gamma(0) = m$ is dropped.)

Then there is a natural projection $\pi:P\rightarrow M$ given by $\pi(\gamma) = \gamma(1)$. One can prove this map is a fibration with homotopy fiber given by $\Omega$. Even more importantly, one can show the $E$ is contractible (think of sucking spaghetti into your mouth).

Thus, one gets a long exact sequence $$\ldots\pi_k(\Omega)\rightarrow \pi_k(E)\rightarrow \pi_k(M)\rightarrow \pi_{k-1}(\Omega)\rightarrow \ldots$$

Using the fact that $E$ is contractible, this proves $\pi_k(M)\cong \pi_{k-1}(\Omega)$ for $k>1$.

Further, the fibration $\Omega\rightarrow E\rightarrow M$ also gives rise to a spectral sequence which relates the (co)homology of $M$ to that of $\Omega$.

So far, everything here is purely topological/differentiable. Where does the metric enter?

Define the energy to be a map $E:\Omega\rightarrow \mathbb{R}$by $E(\gamma) = \int_0^1 |\gamma'|^2 \; dt$. One can prove under fairly general hypothesis that $E$ is a Morse function. Further, the critical points are precisely the closed geodesics (I'm sweeping a lot under the rug here. For example, $P$ itself is not a finite dimensional manifold, but it can be approximated by a finite dimensional manifold. Alternatively, Morse theory has been developed on infinite dimensional things. Also, I may be misremembering - you may have to quotient out $\Omega$ by a natural $S^1$ action or something like that.)

Thus, Morse theory relates the number/type of closed geodesics to the topology of $\Omega$, and hence, to the topology of $M$.

Here's one of my favorite theorems which comes from studying all this in detail. It was proven in two parts by Vigué and Sullivan, and by Gromoll and Meyer.

(Vigué - Sullivan) Suppose $M$ is a compact simply connected manifold and $H^*(M;\mathbb{Q})$ requires at least two generators, then the free loop space has unbounded homology.

(Gromoll-Meyer) Suppose $M$ is a compact simply connected manifold and the free loop space of $M$ has unbounded rational homology. Then for any Riemannian metric on $M$, $M$ has infinitely many geometrically distinct closed geodesics.

(Geometrically distinct means the images are different, as opposed to just going around the same geodesics many times.)

References:

M. Vigué-Poirrier et D. Sullivan. The homology theory of the closed geodesic problem. J. Diff. Geo. 11 (1976), 633-644.

D. Gromoll and W. Meyer. Periodic geodesics on a compact Riemannian manifold. J. Diff. Geo. 3 (1969), 493-510.

Answer 3

For (4), see the book Manifolds all of whose geodesics are closed by Arthur L. Besse, Ergebnisse der Mathematik und ihrer Grenzgebiete 93. Springer, 1978, ISBN: 3-540-08158-5, MR0496885.

See also Lectures on closed geodesics by Wilhelm Klingenberg, Grundlehren der Mathematischen Wissenschaften 230. Springer, 1978, ISBN: 3-540-08393-6, MR0478069.

Answer 4

You write "I don't get what these closd geodesic tell about the topology of $S^2$", but it might be that some of the interest comes from trying to understand the geometry of $S^2$. Great circles on a sphere (which is what the closed geodesics on $S^2$ are) have been of basic interest in spherical geometry throughout the long history of that subject.

More generally, the study of closed geodesics (and related topics such as Jacobi fields, the cut locus and injectivity radius, and so on) is prima facie a part of geometry, although it can have topological implications and applications.

Answer 5

The question about the general existence of closed geodesics was first studied by Poincaré (https://www.jstor.org/stable/1986219). He was interested in this problem because of the three body problem in celestial mechanics. The reason is that periodic orbits of dynamical systems can be interpreted as closed geodesics.

Poincaré observed that, for closed surfaces, if there is a non-trivial homotopy class of curves, one can always guarantee the existence of such a geodesic by minimizing the length on this class. The question remained open for the sphere until Birkhoff invented the mountain-pass technique to solve it (https://www.ams.org/journals/tran/1917-018-02/S0002-9947-1917-1501070-3/S0002-9947-1917-1501070-3.pdf). His paper is titled "Dynamical systems with two degrees of freedom".

So, back then, the question was seen as a Dynamical Systems problem. Now we see the question as geometrical and it is known that every closed surface admits infinitely many distinct closed geodesics, possibly with self-intersection.