K-umblilcity Of Spacelike Hypersurfaces In Lorentzian Space Forms

In this article, we mainly investigate hypersurfaces with higher order umbilic-ity in Lorentzian space forms. Under the assumption conditions, we give some k-umbility characterizations and we give some applications of the result obtained about r-Newton operators.The article divided into three sections:In section one, we mainly introduce some basic definitions and formulas of elementary k-symmetric polynomial in Lorentzian space forms.In section two, we mainly introduce some basic definitions and formulas of r-Newton operators in Lorentzian space forms and give some applications of the result about r-Newton operators..In section three, we extend the concept of umbilicity to higher order umbilicity in Lorentzian space forms saying that an spacelike hypersurfaces is k-umbilical when APk-1(A) is a multiple of the identity, where Pk(A) is the kth Newton polynomial in the second fundamental form A with P0(A) being the identity. Thus, for k=1, one-umbilical coincides with umbilical. We also introduce an operator φk which measures how an isometric immersion fails to be k-umbilical, giving in particular that φ1=0if and only if the immersion is totally umbilical. We characterize the two-umbilical hypersurfaces of a space form as images of isometric immersions of Einstein manifolds.