Immersions into manifolds without conjugate points

Abstract

Many differential geometric concepts such as (isometric) immersion, stability, etc., realized in Euclidean spaces proved to be also realized in manifolds without conjugate points while other concepts are found to be strictly associated with Euclidean spaces. In fact, this thesis may be considered as a trial for finding out to what extent geometric phenomena in Euclidean spaces are still l valid in manifolds without conjugate points. In the introduction, we have quoted the necessary background material for the following chapters. Specially, we have concentrated on the geometry of submanifolds. The interesting problem of rigidity of submanifolds lies in three different categories : finite rigidity, continuous rigidity and infinitesimal rigidity. These three types of rigidity have been studied in hyperbolic spaces in chapter I, sections 1 and 2. K. Nomizu, B. Snmyth (1969) and S. Braidi, C.C. Hsuing (1970) studied some geometric properties of immersed submanifolds in Euclidean sphere essentially the behaviour of the second fundamental form and the Gauss map. In chapter II (sections 1, 2) we have carried out similar study for immersed submanifolds in hyperbolic spaces which shows some deviations from the corresponding one in Euclidean sphere. Since B.Y. Chen's paper (1973) which established the geometric concept of stability of submanifolds in Euclidean spaces, other geometers tried to extend this concept to non-Euclidean spaces. In chapter II (section 3) we share this development through studying stability of surfaces in hyperbolic 3-dimensional space. The most interesting part of our thesis is the last chapter which deals with tight and taut (convex-minimal) immersions in manifolds without conjugate points. Some geometric concepts such as (spherical) two-piece property, h-two-piece property, total (absolute) curvature,... e t c . , have been introduced. Relations between the above concepts have been adopted. We expect for this part to receive more attention in the future to discover more results and to generalize other Euclidean concepts which we did not touch.