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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.

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4 Answers 4

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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.

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Thanks for mentioning systolic geometry: it's a nice interplay between metric and topological properties. But let me focus on the rest of your answer, as I'd like to understand what's going on here: 1) Cartan's theorem implies that if I have no closed geodesic on a compact manifold, then it's simply connected. But Lusternik-Fet's theorem says I always have a closed geodesic on a compact manifold. So the point is: is that closed geodesic homotopically trivial? Am I right? 2) Why doesn't your cylinder have closed geodesics? I can see lots of them (though not minimizing)..what am I missing? –  Lor May 23 '12 at 14:05
    
In 2) I'm referring to the various circles at fixed radius. They are all freely homotopical and non homotopically trivial geodesics, arent't they? –  Lor May 23 '12 at 14:06
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1) The sphere is a good example to have in mind: it is simply connected, but it has lots of closed geodesics. Note that you can introduce nullhomotopic closed geodesics to any manifold by attaching a spherical handle to it, so I guess I would interpret Cartan's theorem as saying that if a manifold doesn't have very many closed geodesics then its fundamental group is small. –  Paul Siegel May 24 '12 at 2:30
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2) Consider the intersection of the standard unit sphere in $\mathbb{R}$ with the plane $x = 1/2$. The intersection is a circle and it might feel like it's a geodesic, but it's not: given two nearby points on the circle it's a little bit cheaper to move toward the point $(1,0,0)$ than it is to move along the circle. The same is true of your circles on the sloped cylinder. –  Paul Siegel May 24 '12 at 2:36
    
In the case of the cylinder, do you mean something like this? postimage.org/image/ucw687gzf What does the geodesic do away from this? will the two branches go away for ever never meeting again? or will the geodesic selfintersect infinitely many times, but never close (like on an elliptic paraboloid)? On the other hand, if I let the radius increase (i.e. take a one-sheeted hyperboloid), the "obvious shortest geodesic" is indeed a shortest geodesi, right? Has this something to do with the sign of curvature? Thanks a lot! –  Lor May 24 '12 at 13:16

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.

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Oh huh, I didn't know that Besse "wrote" more than one book! Neat reference. –  Willie Wong May 24 '12 at 8:38

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.

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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.

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