CONVEXITY IN SPACE FORMS

Abstract

The convexity of subsets in general Riemannian manifolds is a very interesting research area in both local and global differential geometry. Accordingly, a lot of mathematicians such as (Karcher [17], 1968), (do Carmo [9], 1969) and (Alexander [l], 1978) have paied a great deal of attention towards such a topic. The convexity in differential geometry has several types such as weak convexity, strong convexity, strict convexity, ...etc [l]. One of the interesting types of convexity, called k-convexity, has been introduced by D. Mejia and D. Minda [20] in (1990). They actually established the concept of k-convex region Q with boundary 09 in Euclidean plane E? In (1996), M. Beltagy and I. sakr [5] defined and studied-as a continuation-the concept of k-convexity of regions in the Euclidean 3-space E'using the idea of revolution surfaces. They established some results relating the k-convexity property to other geometric characteristics such as sectional curvatures as well as focal points of the boundary of the considered region. In this thesis we are going to define and study the concept of k-convex region Q with boundary ~Q in the hyperbolic plane. It is worth mentioning that the k-convexity of subsets in general Riemannian manifolds with arbitrary dimension may be considered as an open subject which deserves a lot of attention.