Topology and Geometry

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Sur un groupe remarquable de difféomorphismes du cercle

1. Lie Groups and Homogeneous Spaces 2. Minimal Models 3. Manifolds 4. Complex and Symplectic Manifolds 5. Geodesics 6. Curvature 7. G-Spaces 8. Blow-ups and Intersection Products 9. A Florilege of Geometric Applications APPENDICES A. De Rham Forms B. Spectral Sequences C. Basic Homotopy Recollections

/pdf/algebraic-models-in-geometry-4m4p1db1xw.pdf

Algebraic models in geometry

Manifolds all of whose geodesics are closed

In this paper we introduce symplectic invariants for convex Hamiltonian energy surfaces and their periodic trajectories and show that these quentities satisfy several nontrivial relations. In particular we show that they can be used to prove multiplicity results for the number of periodic trajectories.

https://projecteuclid.org/download/pdf_1/euclid.cmp/1104160288

Convex Hamiltonian energy surfaces and their periodic trajectories

/pdf/equivariant-morse-theory-and-closed-geodesics-4rcc9x1dwp.pdf

Equivariant Morse theory and closed geodesics

On the growth of the number of closed geodesics on the two-sphere

Let the torus T2n be equipped with the standard symplectic structure and a periodic Hamiltonian Dt e C3(SX x T2n, R). We look for periodic orbits of the Hamiltonian flow u(t) = J VDt(t, (/)). A subharmonic solution is a periodic orbit with minimal period an integral multiple m of the period of Dt, with m > 1. We prove that if the Hamiltonian flow has only finitely many orbits with the same period as Dt, then there are subharmonic solutions with arbitrarily high minimal period. Thus there are always infinitely many distinct periodic orbits. The results proved here were proved in the nondegenerate case by Conley and Zehnder and in the case n = 1 by Le Calvez.

/pdf/subharmonic-solutions-of-hamiltonian-equations-on-tori-3khhw3dwdp.pdf

Subharmonic solutions of Hamiltonian equations on tori

The critical points of the length function on the free loop space Λ(M) of a compact Riemannian manifold M are the closed geodesics on M. The length function gives a filtration of the homology of Λ(M), and we show that the Chas-Sullivan product Hi(Λ)×Hj(Λ)→*Hi+j−n(Λ) is compatible with this filtration. We obtain a very simple expression for the associated graded homology ring GrH*(Λ(M)) when all geodesics are closed, or when all geodesics are nondegenerate. We also interpret Sullivan's coproduct ∨ (see [Su1], [Su2])) on C*(Λ) as a product in cohomology Hi(Λ,Λ0)×Hj(Λ,Λ0)→⊛Hi+j+n−1(Λ,Λ0) (where Λ0=M is the constant loop). We show that ⊛ is also compatible with the length filtration, and we obtain a similar expression for the ring GrH*(Λ,Λ0). The nonvanishing of products σ*n and τ⊛n is shown to be determined by the rate at which the Morse index grows when a geodesic is iterated. We determine the full ring structure (H*(Λ,Λ0),⊛) for spheres M=Sn, n≥3

/pdf/loop-products-and-closed-geodesics-2y8fuiz435.pdf

Loop products and closed geodesics

We show the Chas-Sullivan product (on the homology of the free loop space of a Riemannian manifold) is related to the Morse index of its closed geodesics. We construct related products in the cohomology of the free loop space and of the based loop space, and show they are nontrivial.

Nancy Hingston

Papers

Equivariant Morse theory and closed geodesics

On the growth of the number of closed geodesics on the two-sphere

Subharmonic solutions of Hamiltonian equations on tori

Loop products and closed geodesics

Loop Products and Closed Geodesics