First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

Abstract

This article has three recurrent goals. Firstly, we prove the existence of a wide class of warped product submanifolds possessing a geometrical property; namely, D i -minimal warped product submanifolds. Secondly, the first Chen inequality is derived for this class of warped products in Riemannian space forms, this inequality involves intrinsic invariants (δ-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. Thirdly, this inequality is applied to derive a necessary condition for the immersed submanifold to be minimal in Riemannian space forms, which presents a partial answer for the well-known problem proposed by S.S. Chern, Problem 2. Also, a concrete example is constructed to insure the existence of D i -minimal warped product submanifolds for warped products other than CR and contact CR-warped product submanifolds. For further research directions, we address a couple of open problems; namely Problem 3 and Problem 4.

Chen inequality for general warped product submanifold of Riemannian warped products I×fSm(c)

“…If f = 1, then theorem 7.1 is a generalized result of [9]. Therefore, our result is a generalization of the work [9,18].…”

“…Remark 7.1. It should be noticed that theorem 7.1 coincides with theorem 4.1 from [18]. If f = 1, then theorem 7.1 is a generalized result of [9].…”

“…It should be noted that there is not so much study on δ-invariant for warped product structure other than Chen derived an optimal inequality for CR-warped product in complex space form [17]. Recently, Mustafa et al [18] constructed the first Chen invariant for warped product submanifolds in real space forms and discussed the minimality conditions on submanifolds. Our primary goal in this study is to introduce a novel method for establishing inequalities for the δ-invariant curvature inequalities for warped product submanifolds isometrically immersed in a Riemannian warped product manifolds of types ( ) ´ I c f m which has been discussed [19].…”

An Invariant of Riemannian Type for Legendrian Warped Product Submanifolds of Sasakian Space Forms

“…As the main objective of our study, we present a novel method for establishing inequalities for δ-invariant curvature inequalities for warped product Legendrian submanifolds isometrically immersed in Sasakian space. This has been discussed in [20,21,35]. As a consequence of the main results discussed in this paper, we generalize a number of inequalities for areas on Euclidean spheres and Euclidean spaces.…”

“…It should be noted that there are few studies on the δ-invariant for warped product structures other than the Chen-derived optimal inequality for CR-warped products in complex space form [19]. Recently, Mustafa et al [20] constructed the first Chen invariant for warped product submanifolds in real space forms and discussed the minimality conditions on submanifolds. From this point of view, by using the Gauss equation instead of the Codazzi equation in the sense of [13], in the first part of this paper, we provide a sharp estimate of the squared norm of the mean curvature in terms of a warping function and the constant holomorphic sectional curvature in the spirit of [21][22][23][24][25][26][27][28][29][30][31][32][33], motivated by the historical development on the study of a warping function of a warped product submanifold [34].…”