Conjugate points on spacelike geodesics or pseudo-self-adjoint Morse-Sturm-Liouville systems

Abstract

This paper develops the basic theory of conjugate points along geodesies in manifolds with indefinite metric; equivalently, that of conjugate points for Morse-Sturm-Liouville systems which are symmetric with respect to an indefinite inner product. The theory is rather different from that for Riemannian manifolds or that for timelike or null geodesies in Lorentzian manifolds. We find that conjugate points may be unstable with respect to perturbation of the geodesic: they may annihilate in pairs. Also the conjugate points need not be isolated: we construct an example where a whole ray is conjugate to a given point. Nevertheless, we give an extension of the Morse Index Theorem to this situation. We also analyze the effects of certain perturbations.