Pseudo Focal Points Along Lorentzian Geodesics and Morse Index

Given a Lorentzian manifold (M,g), a geodesic gamma in M and a timelike Jacobi field Y along gamma, we introduce a special class of instants along gamma that we call Y-pseudo conjugate (or focal relatively to some initial orthogonal submanifold). We prove that the Y-pseudo conjugate instants form a finite set, and their number equals the Morse index of (a suitable restriction of) the index form. This gives a Riemannian-like Morse index theorem. As special cases of the theory, we will consider geodesics in stationary and static Lorentzian manifolds, where the Jacobi field Y is obtained as the restriction of a globally defined timelike Killing vector field.

Regional Junta Andalucia

Regional Junta Andalucia[P06-FQM-01951]

Fundacion Seneca[04540/GERM/06]

Fundacion Seneca

Spanish MEC

Spanish MEC[MTM2007-64504]

M.I.U.R.

M.I.U.R.[PRIN07]

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Capes, Brazil[BEX 1509/08-0]