On the nullity distribution of a submanifold of a space form
This paper analyzes the nullity distribution \({\mathcal N}\) of a submanifold \(M\) of the Euclidean space or of the sphere. The nullity distribution is the common kernel of the shape operators of the submanifold. The subset \({\mathcal C}\) of \(M\) where the dimension of the nullity distribution is locally constant is open and dense, and the nullity distribution is autoparallel on \({\mathcal C}\). The paper presents both a local and a global result. The local theorem asserts that, around any point \(p \in {\mathcal C}\) one and only one of the following two things may happen: (i) for any open neighbourhood \(U\) of \(p\), the set of points that can be reached from \(p\) by curves within \(U\) with tangent vector everywhere orthogonal to the nullity distribution is, itself, a neighbourhood of \(p\), or (ii) there exists an open neighbourhood \(U\) of \(p\) and a proper submanifold \(S\) of \(U\) containing \(p\) such that \(U\) is the union of manifolds parallel to \(S\) over a flat sub-bundle of the normal bundle of \(S\). This sub-bundle is contained in the nullity distribution of \(M\) and, moreover, the leaves of this parallel foliation are such that, locally, any two points in a leaf can be joined by a curve within \(U\) with tangent vector everywhere orthogonal to the nullity distribution. The global theorem states that if \(M\) is an immersed, complete and irreducible submanifold of the Euclidean space or the sphere and the nullity distribution has constant dimension, then any two points in \(M\) can be joined by a curve everywhere orthogonal to the nullity distribution (i.e. the orthogonal complement to the nullity distribution is a completely non-integrable distribution). Irreducible means, in the Euclidean case, that \(M\) cannot be written, locally, as the product of submanifolds of the Euclidean space. In the spherical case, \(M\) is irreducible if it is irreducible as a submanifold of Euclidean space. The paper also shows, with a simple example, that the global statement is false for submanifolds of the hyperbolic space.
complete non-integrability of perpendicular distribution
scientific article; zbMATH DE number 6097137
scientific article