부산대학교 도서관

Let $(M,g)$ be an $(n+1)$-dimensional Lorentzian manifold and let $H\subset M$ be a hypersurface. This paper concerns isometricimmersions of $(M,g)$ or hypersurfaces of arbitrary signature $H$ inthe Minkowski space $\Bbb{L}^{n+1}$ under metrics having Sobolevregularity and curvature defined in the distributional sense. Thefirst result is as follows: Assume that $(M,g)$ is simplyconnected 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 inthe distributional sense, vanishes.\parAs for hypersurfaces, they are assumed to have a transverse field (calledrigging) $l$ along $H$, i.e., a vector field $l\in TM$ which istransversal to $H$. Then, the Levi-Civita connection on $M$ can bedecomposed into tangent and transversal components. The authors areable to characterize immersions $f\colon H\to\Bbb{L}^{n+1}$ in termsof generalized Gauss and Codazzi equations.\parFinally, applications are made to spacelike, timelike and lightlikehypersurfacesof any signature which possibly changes from point to point.