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# [FORTHCOMING] Première inégalité de Chen pour des produits tordus de sous-variétés d’un espace forme riemannien et applications

## [FORTHCOMING] First Chen Inequality for General Warped Product Submanifolds of a Riemannian Space Form and Applications

Palestine Technical University
Palestine

Cenap OZEL
King Abdulaziz University
Saudi Arabia

Alexander PIGAZZINI
Mathematical and Physical Science Foundation
Denmark

Ramandeep KAUR
Central University of Punjab
India

Gauree SHANKER
Central University of Punjab
India

Publié le 24 mai 2023   DOI :

### Keywords

In this paper, the first Chen inequality is proved for general warped product submanifolds 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 Chen’s Problem 1 relating to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. 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.

In this paper, the first Chen inequality is proved for general warped product submanifolds 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 Chen’s Problem 1 relating to establish simple relationships between the main extrinsic invariants and the main intrinsic invariants of a submanifold. 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.