We study connected orientable spacelike hypersurfaces $x:M^{n}rightarrowM_q^{n+1}(c)$, isometrically immersed into the Riemannian or Lorentzian space form of curvature $c=-1,0,1$, and index $q=0,1$, satisfying the condition $~L_kx=Ax+b$,~ where $L_k$ is the $textit{linearized operator}$ of the $(k+1)$-th mean curvature $H_{k+1}$ of the hypersurface for a fixed integer $0leq k

We consider the Dirac operator D of a Lorentzian spin manifold of even dimension n 4. We prove, that the square D 2 of the Dirac operator on plane wave manifolds and the shifted operator D 2 ? K on Lorentzian space forms of constant sectional curvature K are of Huygens type. Furthermore, we study the Huygens property for coupled Dirac operators on 4-dimensional Lorentzian spin manifolds.

Sasaki manifolds admit a nowhere vanishing vector field and it is always possible to consider a Lorentz metric on them. Then we are able to obtain a classification result for compact Lorentz–Sasaki space forms.

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in S, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint surfaces. These new surfaces are again conformal Willmore surfaces. For them holds...