7. A closed geodesic in is a differentiable closed curve such that its lift to the universal cover is a geodesic.

Theorem. Let be a non-trivial free homotopy class in a compact Riemannian manifold . Then there exists a closed geodesic with .

The set of eigenvalues with multiplicities of the Laplace–Beltrami operator acting on the manifold is called the spectrum of a manifold. Two manifolds are said to be isospectral if they have the same spectrum.

The multiplicity of a given length is the number of distinct free homotopy classes of geodesics that contain a closed geodesic of that length. The length spectrum, denoted by , is the set of lengths of smooth closed geodesics, counted with multiplicities. The marked length spectrum of is a function assigning each free homotopy class the set of lengths of closed geodesics freely homotopic to .

Y. Colin de Verdiere showed that, for a given manifold , there exists a generic subset , in the sense of Baire category, of the set of smooth Riemannian metrics on , such that if , the length spectrum of can be recovered from the Laplace spectrum. Moreover, the set contains all metric with sectional curvature less than zero.

A closed geodesic is said to be a –geodesic if for all . The length spectrum of , denoted by , is the set of lengths of –geodesics in . The length spectrum does not necessarily persist under Gromov–Hausdorff convergence (see here). But the length spectrum does.

6. Let be a measure space and be a measurable map. Then is said to be conservative if for each measurable subset with mutually disjoint .

**Halmos–Sucheston theorem**: If is conservative, so is for each .

*Proof*: let be fixed and be a measurable subset with being mutually disjoint.

Let , and for .

Clearly are measurable (since is), mutually disjoint and .

**Claim**: for all . Therefore and , which completes the argument.

*Proof of Claim*: Suppose on the contrary that there exist and such that . In other words, there exists with . By definition of , there exist with and with . Observe that:

*: for all (since , ).

**: are all different (since ‘s are).

So , contradicting . This completes the proof of Claim.

5. A compact Riemannian manifold is said to be **uniformly secure** if there is a number such that for any two points , the set of geodesics connecting them can be blocked by point obstacles. For example the standard sphere is not secure at all. A nontrivial example is the compact surface of genus of constant curvature -1.

**Conjecture**: uniform security implies flatness.

Wing Kai Ho proved this conjecture for non-simply connected, orientable Riemannian surfaces.

4. Let be a Riemannian manifold, the tangent bundle and the double tangent bundle. The connection map is given by , where . The canonical flip is an involution .

3. Let be a periodic Hamiltonian function and be the induced Hamiltonian flow. Then the –speeder flow is generated by .

2. A diffeo is said to conservative if for any measurable subset with mutually disjoint, .

It is ergodic if for any measurable subset with , . Clearly ergodicity implies conservativity. For Anosov diffeo, conservativity implies transitvie. Is there any special property of the SRB measure of ?

1. A factor is said to be nontrivial if is not a singleton.

Linderstrauss proved that for each nontrivial factor , there exists such that is a factor of . In particular .

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