Stability of the conjugate index, degenerate conjugate points and the Maslov index in semi-Riemannian geometry

Abstract

We investigate the problem of the stability of the number of conjugate or focal points (counted with multiplicity) along a semi-Riemannian geodesic γ. For a Riemannian or a nonspacelike Lorentzian geodesic, such number is equal to the intersection number (Maslov index) of a continuous curve with a subvariety of codimension one of the Lagrangian Grassmannian of a symplectic space. In the general semi-Riemannian case, under a certain nondegeneracy assumption on the conjugate points, this number is equal to an algebraic count of their multiplicities. In this paper we reprove some results that were incorrectly stated by Helfer in 1994, where the occurrence of degeneracies was overlooked; in particular, a counterexample to one of Helfer's results, which is essential for the theory, is given. In the last part of the paper we discuss a general technique for the construction of examples and counterexamples in the index theory for semi-Riemannian geodesics, in which some new phenomena appear.

On the distribution of conjugate points along semi-Riemannian geodesies

“…Clearly, the subspace LQ is Lagrangian in IR n ffi lR n * and therefore (2.6) defines a smooth curve £ in A(2n, JR); such curve is used in [8] to study the conjugate points along a semi-Riemannian geodesic. We now introduce the smooth curve £ : It is clear that isomorphic abstract symplectic systems have the same conjugate instants.…”

“…It is easy to check that a = ipia)' 1 is an isomorphism iromZ{X) to (V,a;,f)-□ We now want to characterize which abstract symplectic systems correspond to nondegenerate symplectic differential systems. To this aim, we recall a couple of simple facts about the geometry of the Lagrangian Grassmannian (see for instance [3,8] Let now X be a symplectic differential system and define £ as in (2.10); obviously £ = f3 Lo o $" 1 . By (2.11) and (2.5) we have: since a;(-X"(t)-, OUoxLo = -B(*)J we see that ^C*) is the push-forward of -B(t) by the isomorphism ^(t)" 1 : LQ -►£(*).…”

“…To each symplectic differential system is naturally associated the notion of Maslov index; this formalism is used in [11] to prove a Morse index theorem for non convex Hamiltonian systems and for semi-Riemannian geometry (see also [8,10]). In [11] it is also defined the notions of multiplicity and of signature of a conjugate instant of a symplectic differential system; these notions, as well as that of Maslov index, can be defined directly in the context of abstract symplectic systems.…”

“…Conjugate points along 7 correspond therefore to intersections of this curve with the subvariety of A consisting of Lagrangians that are not transverse to {0} ffi iR n *. Details of this construction can be found in [6,8,11,13]. The problem of determining precisely which curves of Lagrangians arise from a semi-Riemannian geodesic is a rather difficult task.…”

“…The problem of determining precisely which curves of Lagrangians arise from a semi-Riemannian geodesic is a rather difficult task. A partial result in this direction can be found in the last section of [8], where it is proven that a necessary condition for a smooth curve in the Lagrangian Grassmannian A to arise from a semiRiemannian geodesic is that it be tangent to a singular distribution of affine planes in A. However, this condition alone is not sufficient, and attempts to produce interesting examples of conjugate points along geodesies using this characterization lead quickly to rather involved computations.…”