This paper develops the basic
theory of conjugate points along geodesics 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 geodesics 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.
