Séance Séminaire

Séminaire Gaston Darboux

Cosphere Bundle Reduction

The copshere bundle of a differentiable manifold $M$ (denoted by $S^*(M)$) is the quotient of its cotangent bundle without the zero section with respect to the action by multiplications of $\RR^+$ which covers the identity on $M$. It is a contact manifold which has the same privileged position in contact geometry that cotangent bundles have in symplectic geometry. Using a Riemannian metric on $M$, we can identify $S^*(M)$ with its unitary tangent bundle and its Reeb vector field with the geodesic field on $M$. If $M$ is endowed with the proper action of a Lie group $G$, the lift of this action on $S^*(M)$ respects the contact structure and admits an equivariant momentum map $J$. We will study the topological and geometrical properties of the reduced space of $S^*(M)$ at zero momentum, i.e. $\left(S^*(M)\right)_0 :=J^{-1}(0)/G$. In particular, we will annalyse the behaviour of the geodesic flow with respect to the Lie group symmetries of $M$.