Quantum Cohomology and Manifolds of Geodesics

Abstract

This paper is a survey paper to describe the quantum product, the (small) quantum cohomology and their computations. The quantum product and the quantum cohomology of smooth projective quadrics and partial flag manifolds are presented here. The relation between those spaces and manifolds of geodesics are mentioned: the manifold of geodesics on S^n and the smooth projective quadric Q^(n+1) are symplectomoprhic and the manifold of geodesics on P^n and the partial flag manifold F(1, 2, n+1) are diffeomorphic while symplectomoprhism is still unknown.