### A certain complete space-like hypersurface in Lorentz manifolds.

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If $(M,\nabla )$ is a manifold with a symmetric linear connection, then ${T}^{*}M$ can be endowed with the natural Riemann extension $\overline{g}$ (O. Kowalski and M. Sekizawa (2011), M. Sekizawa (1987)). Here we continue to study the harmonicity with respect to $\overline{g}$ initiated by C. L. Bejan and O. Kowalski (2015). More precisely, we first construct a canonical almost para-complex structure $\mathcal{P}$ on $({T}^{*}M,\overline{g})$ and prove that $\mathcal{P}$ is harmonic (in the sense of E. García-Río, L. Vanhecke and M. E. Vázquez-Abal (1997)) if and only if $\overline{g}$ reduces to the...

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...

We consider paraKähler Lie algebras, that is, even-dimensional Lie algebras g equipped with a pair (J, g), where J is a paracomplex structure and g a pseudo-Riemannian metric, such that the fundamental 2-form Ω(X, Y) = g(X, JY) is symplectic. A complete classification is obtained in dimension four.

We prove that a locally symmetric and a null-complete Lorentz manifold is geodetically complete.

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.

We obtain nonexistence results concerning complete noncompact spacelike hypersurfaces with polynomial volume growth immersed in a Lorentzian space form, under the assumption that the support functions with respect to a fixed nonzero vector are linearly related. Our approach is based on a suitable maximum principle recently established by Alías, Caminha and do Nascimento [3].

We consider a certain pseudo-Riemannian metric G on the tangent bundle TM of a Riemannian manifold (M,g) and obtain necessary and sufficient conditions for the pseudo-Riemannian manifold (TM,G) to be Ricci flat (see Theorem 2).