An index theory for paths that are solutions of a class of strongly indefinite variational problems

We generalize the Morse index theorem of 12,15] and we apply the new result to obtain lower estimates on the number of geodesics joining two fixed non conjugate points in certain classes of semi-Riemannian manifolds. More specifically, we consider semi-Riemannian manifolds admitting a smooth distribution spanned by commuting Killing vector fields and containing a maximal negative distribution for . In particular we obtain Morse relations for stationary semi-Riemannian manifolds (see 7]) and for the G?del-type manifolds (see 3]). Received: 4 April 2001 / Accepted: 27 September 2001 / Published online: 23 May 2002 The authors are partially sponsored by CNPq (Brazil) Proc. N. 301410/95 and N. 300254/01-6. Parts of this work were done during the visit of the two authors to the IMPA, Instituto de Matemática Pura e Aplicada, Rio de Janeiro, Brazil, in January and February 2001. The authors wish to express their gratitude to all Faculty and Staff of the IMPA for their kind hospitality.