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Description of  Geometry of Submanifolds and Homogeneous Spaces

In the submanifolds theory, creating a relationship between extrinsic and intrinsic invariants is considered to be one of the most basic problems. Most of these relations play a notable role in submanifolds geometry. The role of immersibility and non-immersibility in studying the submanifolds geometry of a Riemannian manifold was affected by the pioneering work of the Nash embedding theorem , where every Riemannian manifold realizes an isometric immersion into a Euclidean space of sufficiently high codimension. This becomes a very useful object for the submanifolds theory, and was taken up by several authors. Its main purpose was considered to be how Riemannian manifolds could always be treated as Riemannian submanifolds of Euclidean spaces. Inspired by this fact, Nolker classified the isometric immersions of a warped product decomposition of standard spaces. Motivated by these approaches, Chen started one of his programs of research in order to study the impressibility and non-immersibility of Riemannian warped products into Riemannian manifolds, especially in Riemannian space forms. Recently, a lot of solutions have been provided to his problems by many geometers (see [18] and references therein). The field of study which includes the inequalities for warped products in contact metric manifolds and the Hermitian manifold is gaining importance. In particular, in, Chen observed the strong isometrically immersed relationship between the warping function f of a warped product M 1 f M 2 and the norm of the mean curvature, which isometrically immersed into a real space form. i will be very grateful when you support me and buy Or Renew Your Premium from my i appreciate your support Too much as it will help me to post more and more