On the nullity distribution of a submanifold of a space form

Abstract

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 its perpendicular complementary distribution 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.