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A class of metrics on tangent bundles of pseudo-Riemannian manifolds

H. M. Dida, A. Ikemakhen (2011)

Archivum Mathematicum

We provide the tangent bundle T M 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 ( T M , g s ) contains the one of ( M , g ) . What allows us to show that if ( T M , g s ) is indecomposable reducible, then the basis manifold ( M , g ) is also indecomposable-reducible. We determine completely the holonomy group of ( T M , g n ) according to the one of ( M , g ) . Secondly we found conditions on the base manifold under which ( T M , g s ) ( respectively ( T M , g n ) ) is Kählerian, locally symmetric or Einstein...

A new characterization of r -stable hypersurfaces in space forms

H. F. de Lima, M. A. Velásquez (2011)

Archivum Mathematicum

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.

Algebraic restrictions on geometric realizations of curvature models

Corey Dunn, Zoë Smith (2021)

Archivum Mathematicum

We generalize a previous result concerning the geometric realizability of model spaces as curvature homogeneous spaces, and investigate applications of this approach. We find algebraic restrictions to realize a model space as a curvature homogeneous space up to any order, and study the implications of geometrically realizing a model space as a locally symmetric space. We also present algebraic restrictions to realize a curvature model as a homothety curvature homogeneous space up to even orders,...

An Osserman-type condition on g.f.f-manifolds with Lorentz metric

Letizia Brunetti (2014)

Annales Polonici Mathematici

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....

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