A quadric representation of pseudo-Riemannian product immersions

Abstract

In this paper we introduce a quadric representation $\varphi$ of the product of two pseudo-Riemannian isometric immersions. We characterize the product of submanifolds whose quadric representation satisfies $\Delta H_{\varphi}=\lambda H_{\varphi}$, for a real constant $\lambda$, where $H_{\varphi}$ is the mean curvature vector field of $\varphi$. As for hypersurfaces, we prove that the only ones satisfying that equation are minimal products as well as products of a minimal hypersurface and another one which has constant mean and constant scalar curvatures with an appropriate relation between them. In particular, the family of these surfaces consists of $H^{2}(-1)$ and $S^{1}(2/3)\times H^{1}(-2)$ in $S_{1}^{3}(1)$ and $S_{1}^{2}(1),H_{1}^{1}(-2/3)\times S^{1}(2)$, $S_{1}^{1}(2)\times H^{1}(-2/3)$ and a B-scroll over a null Frenet curve with torsion $\pm\sqrt{2}$ in $H_{1}^{3}(-1)$.