  Let M be a possibly noncompact manifold. We prove, generically in the
C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on
M do not possess any degenerate geodesics satisfying suitable boundary
conditions. This extends a result of Biliotti, Javaloyes and Piccione for
geodesics with fixed endpoints to the case where endpoints lie on a compact
submanifold P of the product MxM that satisfies an admissibility condition.
Such condition holds, for example, when P is transversal to the diagonal of
MxM. Further aspects of these boundary conditions are discussed and general
conditions under which metrics without degenerate geodesics are C^k-generic are
given.
