Tsukuba Journal of Mathematics

A quadric representation of pseudo-Riemannian product immersions

Angel Ferrandez, Pascual Lucas, and Miguel A. Merono

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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 $H_{1}^{3}(-1)$.

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Tsukuba J. Math., Volume 20, Number 2 (1996), 435-456.

First available in Project Euclid: 30 May 2017

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Ferrandez, Angel; Lucas, Pascual; Merono, Miguel A. A quadric representation of pseudo-Riemannian product immersions. Tsukuba J. Math. 20 (1996), no. 2, 435--456. doi:10.21099/tkbjm/1496163093. https://projecteuclid.org/euclid.tkbjm/1496163093

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