Some properties of the spectral flow in semiriemannian geometry

Let f(z) = integral(0)(1) g(Z)[(z) over dot, (z) over dot] ds be the action integral on a semiriemannian manifold (M, g) defined on the space of the curves z : [0, 1] --> M joining two given points z(0) and z(1). The critical points of f are the geodesics joining z(0) and z(1). Let s is an element of [0, 1]. We study the behavior, in dependence of s, of the eigenvalues of the Hessian form of f evaluated at z, restricted to the interval [0, s]. A formula for the derivative of the eigenvalues is given and some applications are shown.