Infinitesimal Zoll deformations on spheres (Q1079196)

Let \(S^ n\) be the standard sphere, \(n\geq 2\). A Riemannian metric on \(S^ n\) is called Zoll metric when all geodesics are closed and have common length \(2\pi\). Let \(g_ t\) be a one-parameter family of Zoll metrics with \(g_ 0\) being the standard \(SO(n+1)\)-metric. It is known that \(h=dg_ t/dt|_{t=0}\) satisfies \[ (*)\quad \int^{2\pi}_{0}h({\dot \gamma}(s),{\dot \gamma}(s))ds=0 \] for each geodesic \(\gamma\) (s) of \(g_ 0\) parametrized by its arclength s, where \({\dot \gamma}\)(s) is its tangent vector. In general, a symmetric 2-form h on \(S^ n\) is called an infinitesimal Zoll deformation (IZD), if it satisfies (*) for every geodesic of \(g_ 0\). The space of IZD on \(S^ 2\) is known. The aim of the present paper is to give this description of the space IZD on \(S^ n\), \(n\geq 3\).