Total p-th curvature and foliations and connections

Abstract

This thesis is in two parts. In Part I we consider integrals of the p-th power of the total curvature of a manifold immersed in R(^n) and thus introduce the notions of total p-th curvature and p-convex. This generalises the ideas of total curvature(which corresponds to total 1st curvature)and tight(which corresponds to 1-convex)introduced by Chern, Lashof , and Kuiper. We find lower bounds for the total p-th curvature in terms of the betti numbers of the immersed manifold and describe p-convex spheres. We also give some properties of 2-convex surfaces. Finally, through a discussion of volume preserving transformations of R(^n) we are able to characterise those transformations which preserve the total p-th curvature (when p>1)as the isometries of R(^n). Part II is concerned with the theory of foliations. Three groups associated with a leaf of a foliation are described. They are all factor groups of the fundamental group of the leaf: the Ehresmann group, the holonomy group of A.G.Walker, and the "Jet group". This Jet group is introduced as the group of transformations of the fibres of a suitable bundle induced by lifting closed loops on the leaf, and also by a geometric method which gives a means of calculating them. The relationship between these groups is discussed in a series of examples and the holonomy groups and Jet groups of each leaf are shown to be isomorphic. The holonomy group of a leaf is shown to be not a Lie group and, v/hen the foliation is of codimension 1, it is proved that the holonomy group is a factor group of the first homology group with integer coefficients and has a torsion subgroup which is either trivial or of order 2.