### $(m+1)$-dimensional spacelike parallel ${p}_{i}$-equidistant ruled surfaces in the Minkowski space ${R}_{1}^{n}$.

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On a 4-dimensional anti-Kähler manifold, its zero scalar curvature implies that its Weyl curvature vanishes and vice versa. In particular any 4-dimensional anti-Kähler manifold with zero scalar curvature is flat.

We provide the tangent bundle $TM$ of pseudo-Riemannian manifold $(M,g)$ with the Sasaki metric ${g}^{s}$ and the neutral metric ${g}^{n}$. First we show that the holonomy group ${H}^{s}$ of $(TM,{g}^{s})$ contains the one of $(M,g)$. What allows us to show that if $(TM,{g}^{s})$ is indecomposable reducible, then the basis manifold $(M,g)$ is also indecomposable-reducible. We determine completely the holonomy group of $(TM,{g}^{n})$ according to the one of $(M,g)$. Secondly we found conditions on the base manifold under which $(TM,{g}^{s})$ ( respectively $(TM,{g}^{n})$ ) is Kählerian, locally symmetric or Einstein...

In this paper we study the $r$-stability of closed hypersurfaces with constant $r$-th mean curvature in Riemannian manifolds of constant sectional curvature. In this setting, we obtain a characterization of the $r$-stable ones through of the analysis of the first eigenvalue of an operator naturally attached to the $r$-th mean curvature.

A condition of Osserman type, called the φ-null Osserman condition, is introduced and studied in the context of Lorentz globally framed f-manifolds. An explicit example shows the naturality of this condition in the setting of Lorentz 𝓢-manifolds. We prove that a Lorentz 𝓢-manifold with constant φ-sectional curvature is φ-null Osserman, extending a well-known result in the case of Lorentz Sasaki space forms. Then we state a characterization of a particular class of φ-null Osserman 𝓢-manifolds....

Our purpose is to apply suitable maximum principles in order to obtain Bernstein type properties for two-sided hypersurfaces immersed with constant mean curvature in a Killing warped product $M\u207f{\times}_{\rho}\mathbb{R}$, whose curvature of the base Mⁿ satisfies certain constraints and whose warping function ρ is concave on Mⁿ. For this, we study situations in which these hypersurfaces are supposed to be either parabolic, stochastically complete or, in a more general setting, L¹-Liouville. Rigidity results related to entire...