Foliation Theory Research Paper

The foliation theory is a branch of the geometry which has arisen in the second half of the XX ¬th century on a joint of ordinary differential equations and the differential topology. Now the foliation theory is intensively developed, has wide applications in various areas of mathematics - such, as the optimal control theory, the theory of dynamic polysystems. There are numerous researches on the foliation theory. The review of the last scientific works on the foliation theory and very big bibliography is presented in work of Ph. Tondeur (Tondeur Ph., 1988).

The purpose of first part of this paper is to study the geometry of the geodesics of foliated manifold, namely the question on limit of geodesics of foliated manifold.

The purpose of second part

MAIN PART

I. GEOMETRY OF THE GEODESICS OF FOLIATED MANIFOLD

In modern Riemannian geometry, one of the main objectives is the geometry of the geodesic lines of the Riemannian manifold, i.e. in particular the question on the limit of the geodesic lines. It is known that the limit of the geodesic lines of Riemannian manifold is a geodesic line. In the work (Helgason S., 2001) it is proved a remarkable theorems on the geodesic lines. In the case of a foliated manifold, this question is complicated by the fact that a geodesic line of foliated manifold is not necessary geodesic line of the manifold.

We have the following theorem which is a analog of classic theorem of Riemannian geometry.

Theorem-1. Let be a smooth complete Riemannian manifold of dimension with a smooth foliation of dimension , where . Let be a sequence of geodesics (determined by the induced Riemannian metrics) on leaves . If for for some , then there exists a subsequence of the sequence which pointwise converges to some geodesic of the leaf L(p), passing through the point p at .

Proof of Theorem-1. Let be a - dimensional smooth connected Riemannian manifold with a Riemannian metric , a smooth dimensional foliation on