## Loop products and closed geodesics.(English)Zbl 1181.53036

The closed geodesics on a compact Riemannian manifold $$M$$ are the critical points of the length function on the free loop space $$\Lambda(M)$$. The length function provides a filtration of the homology of $$\Lambda(M)$$. The authors prove that the Chas-Sullivan product
$H_i(\Lambda)\times H_j(\Lambda)@>*>> H_{i+j-n}(\Lambda)$
is compatible with this filtration. Furthermore, they also interprete Sullivan’s coproduct $$\vee$$ on $$C_*(\Lambda)$$ as a product in cohomology
$H^i(\Lambda,\Lambda_0)\times H^j(\Lambda,\Lambda_0)@>\circledast>> H^{i+j+n-1}(\Lambda, \Lambda_0),$
where $$\Lambda_0= M)$$ is the constant loop. They show that $$\circledast$$ is also compatible with the length filtration and provide a similar expression for the ring $$\text{Gr\,}H^*(\Lambda, \Lambda_0)$$. Among other results they also determine the full ring structure $$(H^*(\Lambda, \Lambda_0),\circledast)$$ for spheres $$M= S^n$$, where $$n\geq 3$$. They obtain a significant contribution on loop products and closed geodesics.

### MSC:

 53C22 Geodesics in global differential geometry 58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces 58E10 Variational problems in applications to the theory of geodesics (problems in one independent variable) 55N45 Products and intersections in homology and cohomology
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