Riemannian invariants for warped product submanifolds in Q ε m × R {{Q}}εᵐ× {R} and their applications

This article investigates the geometric and topologic of warped product submanifolds in Riemannian warped product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msubsup> <m:mrow> <m:mi mathvariant="double-struck">Q</m:mi> </m:mrow> <m:mrow> <m:mi>ε</m:mi> </m:mrow> <m:mrow> <m:mi>m</m:mi> </m:mrow> </m:msubsup> <m:mo>×</m:mo> <m:mi mathvariant="double-struck">R</m:mi> </m:math> {{Q}}εᵐ× {R} . In this respect, we obtain the first Chen inequality that involves extrinsic invariants like the length of the warping functions and the mean curvature. This inequality involves two intrinsic invariants (sectional curvature and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>δ</m:mi> </m:math> δ -invariant). In addition, an integral bound is provided for the Bochner operator formula of compact warped product submanifolds in terms of the Ricci curvature gradient. We aim to apply this theory to many structures and obtain Dirichlet eigenvalues for problem applications. Some new results regarding the vanishing mean curvature are presented as a partial solution, and this can be considered for the well-known problem given by Chern.