On infinitesimal \(C_{2\pi}\)-deformations of standard metrics on spheres
Let \(g_ t\) for any \(t\) be a Riemannian metric on \(S^ n\) whose geodesics are all closed and of length \(2\pi\) and where \(g_ 0\) is the standard metric. Then the symmetric 2-form \(h\) which is the derivative of \(g_ t\) with respect to \(t\) at \(t=0\) is called an infinitesimal \(C_{2\pi}\)-deformation of \(g_ 0\). \textit{V. Guillemin} [Adv. Math. 22, 85--119 (1976; Zbl 0353.53027)] proved that any symmetric 2-form \(h\) on \(S^ 2\) such that the integral of \(h({\dot \gamma},{\dot \gamma})\) along a great circle \(\gamma\) vanishes is a \(C_{2\pi}\)-deformation of \(g_ 0\). In the paper under review a necessary condition is introduced on 2-forms of the type \(fg_ 0\) to be an infinitesimal \(C_{2\pi}\)-deformation of \(g_ 0\), where \(f\) is a restriction of an odd polynomial on \({\mathbb R}^{n+1}\) to \(S^ n\). As a consequence it follows that Guillemin's result cannot be generalized to \(S^ n\) for \(n\geq 3\) without introducing new conditions on \(h\).
scientific article; zbMATH DE number 3857904