Some properties of the spectral flow in semiriemannian geometry

Let be the action integral on a semiriemannian manifold ( , g) defined on the space of the curves z : 0, 1] → joining two given points z₀ and z₁. The critical points of ? are the geodesics joining z₀ and z₁. Let s ε 0, 1]. We study the behavior, in dependence of s, of the eigenvalues of the Hessian form of ? evaluated at z, restricted to the interval 0, s]. A formula for the derivative of the eigenvalues is given and some applications are shown.