Chen inequalities on warped product Legendrian submanifolds in Kenmotsu space forms and applications

In the current work, we study the geometry and topology of warped product Legendrian submanifolds in Kenmotsu space forms F²ⁿ⁺¹(ϵ) and derive the first Chen inequality, including extrinsic invariants such as the mean curvature and the length of the warping functions. Additionally, sectional curvature and the δ-invariant are intrinsic invariants related to this inequality. An integral bound is also given in terms of the gradient Ricci curvature for the Bochner operator formula of compact warped product submanifolds. Our primary technique is applying geometry to number structures and solving problems such as problems with Dirichlet eigenvalues.