On some optimal inequalities for bi-slant submanifolds in metallic Riemannian space forms

Abstract: In this paper, we derive some important optimal relationships for bi-slant submanifolds in metallic Riemannian product space forms enriching the understanding of their geometric properties and deepening the connection between intrinsic and extrinsic curvature invariants. We establish generalized Wintgen inequality for bi-slant submanifolds in metallic Riemannian product space forms and discussed the equality case. Next we derive optimal inequalities involving $\delta$-invariants, also known as Chen-invariants and discuss the conditions for Chen ideal submanifolds. Further, we derive optimal relationships involving Ricci curvature and shape operator invariants along with the discussion about the equality cases. In the last section, we establish optimal inequalities involving generalized normalized $\delta$-Casorati curvatures for bi-slant submanifolds of metallic Riemannian product space form and discuss the conditions under which the equality holds. Furthermore, we examine how the main findings specialize to slant, semi-slant, hemi-slant, and semi-invariant submanifolds in metallic Riemannian product space forms, offering a better understanding of their geometric characteristics.