Let ((M,g,N)) be a normalized null hypersurface with rigged vector field ξ.
Let ((M,g,N)) be a normalized null hypersurface of a pseudo-Riemannian manifold ((overline{M}^{n+2},overline{g})), ∇ the induced connection on M. In order to relate the main geometric objects of both null and associated non-degenerate geometry on the null hypersurface, we first need to relate the covariant derivatives (nabla^{eta}) and ∇.
Let ((M, g, N )) be a closed normalized null hypersurface with rigged vector field ξ.
Let ((M^{n+1},g,N)) be a closed normalized null hypersurface in a Lorentzian ((n+2))-manifold ((overline{M},overline{g})), (N-xi) be a proper conformal Killing field.
Let ((M,g,N) ) be a closed normalized null hypersurface with rigged ξ and (tau^{N}(xi) = 0 ) in a Lorentzian manifold.
Let ((M, g, N )) be a closed normalized null hypersurface with rigged vector field ξ and (tau^{N} (xi) = 0 ) in a pseudo-Riemannian manifold.
Let ((M^{n+1},g,N)) be a closed normalized null hypersurface in Lorentzian manifold ((overline{M},overline{g})), ((N-xi)) be a proper conformal Killing with conformal factor 2λ and (lambda= frac{1}{n+2}operatorname{div}^{overline{g}} N-xi)) and (tau^{N} N-xi0).
Let ((M,g,N )) be a closed normalized null hypersurface in a ((n+2))-dimensional Lorentzian manifold ((overline{M},overline{g})), (N-xi) be a proper conformally Killing field with conformal factor 2λ and (lambda=frac {1}{n+2}operatorname{div}^{overline{g}} N-xi)).
Let ((M, g,N )) be a closed normalized null hypersurface in ((n+2))-dimensional Lorentzian manifold ((overline{M},overline{g})), (N-xi) be a proper conformal Killing field with conformal factor 2λ and (lambda= frac{1}{n+2}operatorname{div}^{overline{g}} N-xi)).
Let ((M,g,N )) be a closed normalized null hypersurface in a ((n+2))-dimensional Lorentzian manifold ((overline{M},overline{g})) with constant sectional curvature k, ((N-xi)) be a proper conformal Killing field with conformal factor 2λ and (lambda= frac{1}{n+2}operatorname{div}^{overline{g}} N-xi)).
We shall give the following result, involving the extrinsic Ricci curvature Ric to the associated Ricci curvature (operatorname{Ric}^{eta}). Let ((M,g,N)) be a closed normalized null hypersurface with rigged vector field ξ and (tau^{N} (xi =0 ) in a ((n + 2 -pseudo-Riemannian manifold.
