Hypersurfaces Satisfying τ2(φ) = ητ(φ) in Pseudo-Riemannian Space Forms

In this paper, we derived the equations for the hypersurface M-r(n) of a pseudo-Riemannian space form N-q(n+1) (c) to satisfy tau(2)(phi) = eta tau(phi) (eta a constant) with tau(phi) and tau(2)(phi) be the tension and bitension fields of M-r(n). As applications, we prove that a hypersurface M-r(n) satisfying tau(2)(phi) = eta tau(phi) in N-q(n+1) (c) has constant mean curvature, under the assumption that M-r(n) has diagonalizable shape operator with at most three distinct principal curvatures. Then, using this result, we classify partially such hypersurface. We also make a preliminary study of hypersurfaces satisfying tau(2)(phi) = f tau(phi) with f be function.