Title:

Total pth curvature and foliations and connections

This thesis is in two parts. In Part I we consider integrals of the pth power of the total curvature of a manifold immersed in R(^n) and thus introduce the notions of total pth curvature and pconvex. This generalises the ideas of total curvature(which corresponds to total 1st curvature)and tight(which corresponds to 1convex)introduced by Chern, Lashof , and Kuiper. We find lower bounds for the total pth curvature in terms of the betti numbers of the immersed manifold and describe pconvex spheres. We also give some properties of 2convex surfaces. Finally, through a discussion of volume preserving transformations of R(^n) we are able to characterise those transformations which preserve the total pth 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.
