Geometric aspects of riemannian submanifolds and their warped products / Akram Ali - Akram, Ali (2017)
The objective of this thesis is to devote a self-contained study of Riemannian submanifolds and their warped products. This is completed in five major footprints: constructing basic lemmas, proving existence, deriving characterizations, and applying geometric inequalities to obtain geometric applications in physical sciences. The whole thesis is divided into seven chapters. The first two chapters are an excursion from the origins of this area of research to the current study. It includes the definitions, basic formulae, and research open problems. It is known that the existence problem is central in the field of Riemannian geometry, particularly in the warped product submanifolds. Moreover, a lot of important results such as preparatory lemmas for subsequent chapters are stated in these two chapters. In this thesis, we have hypothesized Sahin (2009b) open problems in more general settings, and new inequalities are established by means of a new method such as mixed totally geodesic submanifolds with equality cases are discussed in details. In a hope to provide new methods by means of Gauss equation; instead of Codazzi equation, deriving the Chen (2003) type inequalities with slant immersions, and equalities are considered. An advantage has been taken from Nash’s embedding theorem to deliberate geometric situations in which the immersion may pass such minimality, totally geodesic, totally umbilical and totally mixed geodesic submanifolds. As applications, we present the non-existence conditions of a warped product submanifold in a different ambient space forms using Green theory on a compact Riemannian manifold with or without boundary in various mathematical and physical terms such as Hessian, Hamiltonian, kinetic energy and Euler-Lagrange equation as well. The rest of this work is devoted to establish a relationship between intrinsic invariant and extrinsic invariants in terms of slant angle and pointwise slant functions. As a consequence, a wide scope of research was presented.