Generalized Wintgen inequality for slant submanifolds in metallic Riemannian space forms (Q2044468)

This paper is devoted to the study of the generalized Wintgen inequality on certain submanifolds in metallic Riemannian space forms. Recall that the Wintgen inequality is a geometric inequality that involves intrinsic and extrinsic invariants of a surface in the \(4\)-dimensional Euclidean space given by \[ \|\mathcal{H}\|^2 \geq \mathcal{K}+ \vert \mathcal{K}^\perp \vert, \] where \(\mathcal{K}\) and \(\mathcal{K}^\perp\) denote the Gaussian and the normal curvatures, respectively, and \(\mathcal{H}\) denotes the mean curvature. The generalized Wintgen inequality was conjectured as a natural generalization of this inequality for submanifolds of arbitrary codimension in a real space form with constant sectional curvature \(c\) as \[ \rho \leq \|\mathcal{H}\|^2-\rho^\perp +c, \] being \(\rho\) the Ricci scalar curvature and \(\rho^\perp\) the normalized normal scalar curvature of the submanifold. Also recall that a metallic pseudo-Riemannian structure \(J\) on a pseudo-Riemannian manifold \((M,g)\) is a \(g\)-symmetric \((1,1)\)-tensor field on \(M\) such that \(J^2=pJ + qI\) for some \(p,q\in\mathbb{R}\), being \(I\) the identity matrix. In Theorem 3.1. the authors derive the generalized Wintgen inequality for a slant submanifold of a locally metallic product space form. The cases where the equality holds are also studied. In Section 4 they discuss some applications of Theorem 3.1.