On bi-conservative hypersurfaces in the Lorentz-Minkowski ۴-space E_۱^۴

In the ۱۹۲۰s, D. Hilbert has showed that the tensor of stress-energy, related to a given functional \Lambda, is a conservative symmetric bicovariant tensor \Theta at the critical points of \Lambda, which means that div\Theta =۰. As a routine extension, the bi-conservative condition (i.e. div\Theta_۲=۰) on  the tensor of stress-bienergy  \Theta_۲ is introduced by G. Y. Jiang (in ۱۹۸۷). This subject has been followed by many mathematicians. In this paper, we study an extended version of bi-conservativity condition on the Lorentz hypersurfaces of the Einstein space. A Lorentz hypersurface M_۱^۳ isometrically immersed into the Einstein space is called \mathcal{C}-bi-conservative if it satisfies the condition n_۲(\nabla H_۲)=\frac{۹}{۲} H_۲\nabla H_۲, where n_۲ is the second Newton transformation, H_۲ is the ۲nd mean curvature function on M_۱^۳ and \nabla is the gradient tensor. We show that the C-bi-conservative Lorentz hypersurfaces of Einstein space have constant second mean curvature.

Keywords:

Lorentz hypersurface , bi-conservative , bi-harmonic , isoparametric