First Chen Inequality for CR-Warped Product Submanifolds of a Complex Space Form and Applications
Abstract
In this paper, the first Chen inequality is proved for CR-warped product submanifolds in complex space forms. This inequality involves intrinsic invariants (a leaf-wise -invariant and the sectional curvature) controlled by an extrinsic one (the mean curvature vector), which provides an answer to Problem [1]. We carefully distinguish the leaf-wise -invariant of a factor (used in the bound) from the intrinsic Chen invariant of the same factor, the two being related, on the totally real factor, by the Bishop--O'Neill formula. The bound is sharp and is uniform in the sign of the holomorphic sectional curvature . As a geometric application, we derive necessary conditions for the immersed CR-warped product submanifold to be minimal in a complex space form, providing a partial answer to a well-known problem proposed by S.S. Chern (Problem [2]). For further research directions, we address a couple of open problems (Problem [3]} and Problem [4]).
Keywords
Cite
@article{arxiv.2605.19601,
title = {First Chen Inequality for CR-Warped Product Submanifolds of a Complex Space Form and Applications},
author = {Abdulqader Mustafa and Monika Sati and Uday Chand De and Cenap Ozel and Alexander Pigazzini},
journal= {arXiv preprint arXiv:2605.19601},
year = {2026}
}