The notion of quasi-Einstein manifolds arose during the study of exact solutions of the Einstein field equations as well as during considerations of quasi-umbilical hypersurfaces. For instance, the Robertson-Walker spacetimes are quasi-Einstein manifolds. The object of the present paper is to study Lorentzian quasi-Einstein manifolds. Some basic geometric properties of such a manifold are obtained. The applications of Lorentzian quasi-Einstein manifolds to the general relativity and cosmology are investigated. Theories of gravitational collapse and models of Supernova explosions [5] are based on a relativistic fluid model for the star. In the theories of galaxy formation, relativistic fluid models have been used in order to describe the evolution of perturbations of the baryon and radiation components of the cosmic medium [32]. Theories of the structure and stability of neutron stars assume that the medium can be treated as a relativistic perfectly conducting magneto fluid. Theories of relativistic stars (which would be models for supermassive stars) are also based on relativistic fluid models. The problem of accretion onto a neutron star or a black hole is usually set in the framework of relativistic fluid models. Among others it is shown that a quasi-Einstein spacetime represents perfect fluid spacetime model in cosmology and consequently such a spacetime determines the final phase in the evolution of the universe. Finally the existence of such manifolds is ensured by several examples constructed from various well known geometric structures.

background

/pdf/on-lorentzian-quasi-einstein-manifolds-42x03qjvqy.pdf

Sasakian Manifolds with Lorentzian Metric

On lorentzian quasi-einstein manifolds

. The object of the present paper is to study certain curva-ture restriction on an LP-Sasakian manifold with a coeﬃcient α . Amongothers it is shown that if an LP-Sasakian manifold with a coeﬃcient α is a manifold of constant curvature, then the manifold is the productmanifold. Also it is proved that a 3-dimensional Ricci semisymmetricLP-Sasakian manifold with a constant coeﬃcient α is a spaceform. 1. IntroductionIn 1989, Matsumoto [6] introduced the notion of LP-Sasakian manifolds.Then Mihai and Rosca [7] introduced the same notion independently and theyobtained several results in this manifold. In a recent paper, De, Shaikh, andSengupta [3] introduced the notion of LP-Sasakian manifolds with a coeﬃcient α which generalizes the notion of LP-Sasakian manifolds. Recently, T. Ikawaand his coauthors [4], [5] studied Sasakian manifolds with Lorentzian metricand obtained several results in this manifold. The object of the present paperis to study certain curvature restriction on an LP-Sasakian manifold with acoeﬃcient

result

/pdf/certain-curvature-conditions-on-an-lp-sasakian-manifold-with-4uz2f6tlee.pdf

CERTAIN CURVATURE CONDITIONS ON AN LP-SASAKIAN MANIFOLD WITH A COEFFICIENT α

If the scalar normal curvature of a spacelike maximal surface in a 5-dimensional normal contact Lorentzian manifold with constant φ-sectional curvature is constant, then the surface is totally geodesic or nonpositively curved.

/pdf/spacelike-maximal-surfaces-with-constant-scalar-normal-429vjoqvv4.pdf

Spacelike maximal surfaces with constant scalar normal curvature in a normal contact Lorentzian manifold.

/pdf/on-lorentzian-trans-sasakian-manifolds-l7gjn2d68z.pdf

On lorentzian trans-sasakian manifolds

We study biharmonic curves in contact geometry whose mean curvature vector field is in the kernel of Laplacian. We give some results for biharmonic curves in Sasakian 3-space. We also give some characterizations for Legendre curves in the same space.

methods

/pdf/biharmonic-curves-in-contact-geometry-4c9nhsgd7j.pdf

Sasakian Manifolds with Lorentzian Metric

Citations

On lorentzian quasi-einstein manifolds

CERTAIN CURVATURE CONDITIONS ON AN LP-SASAKIAN MANIFOLD WITH A COEFFICIENT α

Spacelike maximal surfaces with constant scalar normal curvature in a normal contact Lorentzian manifold.

On lorentzian trans-sasakian manifolds

Biharmonic curves in contact geometry

Related Papers (5)

On a type of lorentzian Para-Sasakian manifolds

Some Results on Lorentzian Para-Sasakian Manifolds

Semi-invariant submanifolds of lorentzian sasakian manifolds

On lorentzian {sasakian manifolds admitting a type of semi-symmetric metric connection

On Lorentzian α -sasakian Manifolds