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arXiv · 2109.08911

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

Abdulqader Mustafa·Cenap Ozel·Alexander Pigazzini·Ramandeep Kaur·Gauree Shanker
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Abstract

In this paper, the first Chen inequality is proved for general warped product submanifolds in Riemannian space forms, this inequality involves intrinsic invariants ($\delta$-invariant and sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer for Problem 1. As a geometric application, 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. For further research directions, we address a couple of open problems; namely Problem 3 and Problem 4.

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Abdulqader Mustafa, Cenap Ozel, Alexander Pigazzini, Ramandeep Kaur, Gauree Shanker. 2021-09-18. First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications. https://arxiv.org/abs/2109.08911

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