  If M is a submanifold of a space form, the nullity distribution N of its
second fundamental form is (when defined) the common kernel of its shape
operators. In this paper we will give a local description of any submanifold of
the Euclidean space by means of its nullity distribution. We will also show the
following global result: if M is a complete, irreducible submanifold of the
Euclidean space or the sphere then N is completely non integrable. This means
that any two points in M can be joined by a curve everywhere perpendicular to
N. We will finally show that this statement is false for a submanifold of the
hyperbolic space.

Differential Geometry