Let x : M → M be the canonical injection of a Null Hypersurface (M, g) in a semi-Riemannian manifold (M ,ḡ). A rigging for M is a vector field L defined on some open set of M containing M such that Lp / ∈ TpM for each p ∈ M . Such a vector field induces a null rigging N . Letη be the 1-form which isḡ-metrically equivalent to N and η = x η its pull back on M . We introduce and study for a given non vanishing function α on M the so-called α-associated (semi-)Riemannian metric gα = g + αη ⊗ η. For a closed rigging N we give a constructive method to find an α-associated metric whose Levi-Civita connection coincides with the connection ∇ induced on M by the Levi-Civita connection ∇ of M and the null rigging N . We relate geometric objects of gα to those of g and g. As application, we show that given a null Monge hypersurface M in R n+1 q , there always exists a rigging and an α-associated metric whose Levi-Civita connection coincides with the induced connection on M .keyword: Perturbation of metric, Monge hypersurface, Null hypersurface, screen distribution, Rigging vector field, Associated Metric MSC[2010] 053C23, 53C25, 53C44, 53C50
In the present paper, we classify all normalized null hypersurfaces $x: (M,g,N)\to\R^{n+2}_1$ endowed with UCC-normalization with vanishing $1-$form $\tau$, satisfying $L_r x =U x +b$ for some (field of) screen constant matrix $U\in \R^{(n+2)\times(n+2)}$ and vector$b\in\R^{n+2}_{1}$, where $L_r$ is the linearized operator of the$(r+1)th-$mean curvature of the normalized null hypersurface for$r=0,...,n$. For $r=0$, $L_0=\Delta^\eta$ is nothing but the (pseudo-)Laplacian operator on $(M, g, N)$. We prove that the lightcone $\Lambda_0^{n+1}$, lightcone cylinders $\Lambda_0^{m+1}\times\R^{n-m}$, $1\leq m\leq n-1$ and $(r+1)-$maximal Monge null hypersurfaces are the only UCC-normalized Monge null hypersurface with vanishing normalization $1-$form $\tau$ satisfying the above equation. In case $U$ is the (field of) scalar matrix $ \lambda I$, $\lambda\in\R$ and hence is constant on the whole $M$, we show that the only normalized Monge null hypersurfaces $x: (M,g,N)\to\R^{n+2}_1$ satisfying $\Delta^\eta x =\lambda x +b$, are open pieces of hyperplanes.
In the present work, we classify normalized null hypersurfaces x : (M, g, N ) → Q n+2 1 (c) immersed into one of the two real standard non-flat Lorentzian space-forms and satisfying the equation Lrx = U x + b for some field of screen constant matrices U and some field of screen constant vectors b ∈ R n+2 , where Lr is the linearized operator of the (r + 1)−mean curvature of the normalized null hypersurface for r = 0, ..., n. We show that if the immersion x is a solution of the equation Lrx = U x + b for 1 ≤ r ≤ n and the normalization N is quasiconformal, then M is either an (r + 1)−maximal null hypersurface, or a totally umbilical (or geodesic) null hypersurface or an almost isoparametric normalized null hypersurface with at most two non-zero principal curvatures. We also show that a null hypersurface M , of a real standard semi-Riemannian non-flat space form Q n+2 t (c), admits a totally umbilical screen distribution (and then M is totally umbilical or totally geodesic) if and only if M is a section of Q n+2 t (c) by a hyperplane of R n+3 . In particular a null hypersurface M → Q n+2 t (c) is totally geodesic if and only if M is a section of Q n+2 t (c) by a hyperplane of R n+3 passing through the origin.
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