  We use techniques based on the splitting tensor to explicitly integrate the
Codazzi equation along the relative nullity distribution and express the second
fundamental form in terms of the Jacobi tensor of the ambient space. This
approach allows us to easily recover several important results in the
literature on complete submanifolds with relative nullity of the sphere as well
as derive new strong consequences in hyperbolic and Euclidean spaces. Among the
consequences of our main theorem are results on submanifolds with sufficiently
high index of relative nullity, submanifolds with nonpositive extrinsic
curvature and submanifolds with integrable relative conullity. We show that no
complete submanifold of hyperbolic space with sufficiently high index of
relative nullity has extrinsic geometry bounded away from zero. As an
application of these results, we derive an interesting corollary for complete
submanifolds of hyperbolic space with nonpositive extrinsic curvature and
discourse on their relation to Milnor's conjecture about complete surfaces with
second fundamental form bounded away from zero. Finally, we also prove that
every complete Euclidean submanifold with integrable relative conullity is a
cylinder over the relative conullity.
