Geometry of warped product submanifolds of Riemannian manifolds / Abdalqader Mahmoud Abdalqader Mustafa - Abdalqader Mahmoud , Abdalqader Mustafa (2016)
The purpose of this thesis is to present a self-contained study of Riemannian warped product submanifolds. This is accomplished in four major steps; proving existence, deriving basic lemmas, constructing geometric inequalities and applying them to obtain some geometric applications. The whole thesis is divided into nine chapters. The first two chapters are a journey from the origins of this field to the recent results. Here, definitions, basic formulas and open problems are included. It is well known that the existence problem is central in the field of differential geometry, especially in warped product submanifolds. This problem is investigated in the third and the fourth chapters. Moreover, a lot of key results as preparatory lemmas for subsequent chapters can be found in these two chapters. In the second section of chapter five, a benefit has been taken from Nash’s embedding theorem to discuss geometrical situations the immersion may possess such as minimality, total geodesic and total umbilical submanifolds. The rest of this work is devoted to establish basic simple relationships between intrinsic and extrinsic invariants. In a hope to provide new solutions to the question asked by Chern (1968), about whether we can find other necessary conditions for an isometric immersion to be minimal or not, Chen (1993, 2002) has considered this problem in his research programs. In this thesis, and following Chen (2002) and Chern (1968), we have hypothesized their open problems in a more general way in the first chapter. As a result, a wider scope of research becomes available. Therefore, new inequalities are constructed by means of new methods, where equality cases are discussed in details.