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.
