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Natural vector fields and 2-vector fields on the tangent bundle of a pseudo-Riemannian manifold

Josef Janyška (2001)

Archivum Mathematicum

Let M be a differentiable manifold with a pseudo-Riemannian metric g and a linear symmetric connection K . We classify all natural (in the sense of [KMS]) 0-order vector fields and 2-vector fields on T M generated by g and K . We get that all natural vector fields are of the form E ( u ) = α ( h ( u ) ) u H + β ( h ( u ) ) u V , where u V is the vertical lift of u T x M , u H is the horizontal lift of u with respect to K , h ( u ) = 1 / 2 g ( u , u ) and α , β are smooth real functions defined on R . All natural 2-vector fields are of the form Λ ( u ) = γ 1 ( h ( u ) ) Λ ( g , K ) + γ 2 ( h ( u ) ) u H u V , where γ 1 , γ 2 are smooth real functions defined...

New characterizations of linear Weingarten hypersurfaces immersed in the hyperbolic space

Cícero P. Aquino, Henrique F. de Lima (2015)

Archivum Mathematicum

In this paper, we deal with complete linear Weingarten hypersurfaces immersed in the hyperbolic space n + 1 , that is, complete hypersurfaces of n + 1 whose mean curvature H and normalized scalar curvature R satisfy R = a H + b for some a , b . In this setting, under appropriate restrictions on the mean curvature and on the norm of the traceless part of the second fundamental form, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder of n + 1 . Furthermore, a rigidity result...

New estimates for the first eigenvalue of the Jacobi operator on closed hypersurfaces in Riemannian space forms

Jiancheng Liu, Rong Mi (2020)

Czechoslovak Mathematical Journal

We study the first eigenvalue of the Jacobi operator on closed hypersurfaces with constant mean curvature in non-flat Riemannian space forms. Under an appropriate constraint on the totally umbilical tensor of the hypersurfaces and following Meléndez's ideas in J. Meléndez (2014) we obtain a new sharp upper bound of the first eigenvalue of the Jacobi operator.

Non-degenerate hypersurfaces of a semi-Riemannian manifold with a semi-symmetric metric connection

Ahmet Yücesan, Nihat Ayyildiz (2008)

Archivum Mathematicum

We derive the equations of Gauss and Weingarten for a non-degenerate hypersurface of a semi-Riemannian manifold admitting a semi-symmetric metric connection, and give some corollaries of these equations. In addition, we obtain the equations of Gauss curvature and Codazzi-Mainardi for this non-degenerate hypersurface and give a relation between the Ricci and the scalar curvatures of a semi-Riemannian manifold and of its a non-degenerate hypersurface with respect to a semi-symmetric metric connection....

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