This paper investigates the relative nullity distribution of an indefi- nite Riemannian manifold isometrically immersed into an indefinite space form. Introduction. In this paper we investigate the relative nullity distribu- tion of an indefinite Riemannian manifold isometrically immersed into an indefinite space form. This distribution is totally geodesic and gives rise to a Ricatti-type differential equation along a geodesic in a leaf of the distribution. This differential equation is applied in several ways to estimate the index of relative nullity v for geodesically complete, connected, Lorentzian submanifolds M of Af+1(c), the Lorentzian sphere. These applications extend the work of Abe (1), (2), (3), Ferus (7), (8), and others to the setting of indefinite manifolds. Some of the work in §2 was obtained previously by Graves (10) in the codimension one case and by M. Dajczer. In particular Theorem 2 was conjectured by Dajczer (5). Sections 1 and 2 lay the groundwork and derive the Ricatti-type differential equation. In §3 an integer vn is defined and it is shown that if M is as above and if v > vn then M is totally geodesic. This integer is used to formulate a geometric condition which guarantees that a complete connected hypersurface of S(c) is totally geodesic. We also estimate v given a natural condition on the space-like Ricci curvature of the sub- manifold. In (6) other conditions on Ricci curvature are given. The general scheme of our investigation is very similar to that of the Riemannian case as formulated in the papers mentioned above. However, there are a few basic and non-trivial differences from the Riemannian case. These differences are due to the indefinite metric and are to be overcome. Therefore, we think it worthwhile to include the details of the proofs for most of our results.