부산대학교 도서관

Let $(M,g)$ be an $(n+1)$-dimensional Lorentzian manifold and let $H \subset M$ be a hypersurface. This paper concerns isometric immersions of $(M,g)$ or hypersurfaces of arbitrary signature $H$ in the Minkowski space $\Bbb{L}^{n+1}$ under metrics having Sobolev regularity and curvature defined in the distributional sense. The first result is as follows: Assume that $(M,g)$ is simply connected and $g$ is of class $W_{\rm loc}^{1,p}$; then $f\colon M \to \Bbb{L}^{n+1}$ is an isometric immersion of class $W_{\rm loc}^{2,p}$ if and only if the Riemann curvature tensor ${\rm Riem}_g$, defined in the distributional sense, vanishes. \par As for hypersurfaces, they are assumed to have a transverse field (called rigging) $l$ along $H$, i.e., a vector field $l\in TM$ which is transversal to $H$. Then, the Levi-Civita connection on $M$ can be decomposed into tangent and transversal components. The authors are able to characterize immersions $f\colon H\to\Bbb{L}^{n+1}$ in terms of generalized Gauss and Codazzi equations. \par Finally, applications are made to spacelike, timelike and lightlike hypersurfaces of any signature which possibly changes from point to point.