Curvature properties of some class of warped product manifolds

We prove that warped product manifolds with [Formula: see text]-dimensional base, [Formula: see text] satisfy some pseudosymmetry type curvature conditions. These conditions are formed from the metric tensor [Formula: see text], the Riemann–Christoffel curvature tensor [Formula: see text], the Ricci tensor [Formula: see text] and the Weyl conformal curvature [Formula: see text] of the considered manifolds. The main result of the paper states that if [Formula: see text] and the fiber is a semi-Riemannian space of constant curvature (when [Formula: see text] is greater or equal to 5) then the [Formula: see text]-tensors [Formula: see text] and [Formula: see text] of such warped products are proportional to the [Formula: see text]-tensor [Formula: see text] and the tensor [Formula: see text] is a linear combination of some Kulkarni–Nomizu products formed from the tensors [Formula: see text] and [Formula: see text]. We also obtain curvature properties of this kind of quasi-Einstein and 2-quasi-Einstein manifolds, and in particular, of the Goedel metric, generalized spherically symmetric metrics and generalized Vaidya metrics.

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On some classes of mixed-super quasi-Einstein manifolds

Acta Universitatis Sapientiae Mathematica ◽

10.1515/ausm-2016-0003 ◽

2016 ◽

Vol 8 (1) ◽

pp. 32-52

Author(s):

Santu Dey ◽

Buddhadev Pal ◽

Arindam Bhattacharyya

Keyword(s):

Warped Product ◽

Einstein Manifold ◽

Warped Products ◽

Einstein Manifolds

Abstract Quasi-Einstein manifold and generalized quasi-Einstein manifold are the generalizations of Einstein manifold. The purpose of this paper is to study the mixed super quasi-Einstein manifold which is also the generalizations of Einstein manifold satisfying some curvature conditions. We define both Riemannian and Lorentzian doubly warped product on this manifold. Finally, we study the completeness properties of doubly warped products on MS(QE)4 for both the Riemannian and Lorentzian cases.

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On some classes of generalized quasi Einstein manifolds

Filomat ◽

10.2298/fil1503443g ◽

2015 ◽

Vol 29 (3) ◽

pp. 443-456 ◽

Cited By ~ 1

Author(s):

Sinem Güler ◽

Sezgin Demirbağ

Keyword(s):

Curvature Tensor ◽

Ricci Tensor ◽

Einstein Manifold ◽

Sufficient Conditions ◽

Riemannian Curvature ◽

Einstein Manifolds ◽

Conformal Curvature ◽

Conformal Curvature Tensor ◽

Riemannian Curvature Tensor ◽

Concircular Curvature Tensor

In the present paper, we investigate generalized quasi Einstein manifolds satisfying some special curvature conditions R?S = 0,R?S = LSQ(g,S), C?S = 0,?C?S = 0,?W?S = 0 and W2?S = 0 where R, S, C,?C,?W and W2 respectively denote the Riemannian curvature tensor, Ricci tensor, conformal curvature tensor, concircular curvature tensor, quasi conformal curvature tensor and W2-curvature tensor. Later, we find some sufficient conditions for a generalized quasi Einstein manifold to be a quasi Einstein manifold and we show the existence of a nearly quasi Einstein manifolds, by constructing a non trivial example.

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On warped product manifolds satisfying Ricci-Hessian class type equations

Publications de l Institut Mathematique ◽

10.2298/pim1817069g ◽

2018 ◽

Vol 103 (117) ◽

pp. 69-75 ◽

Cited By ~ 1

Author(s):

Sinem Güler ◽

Altay Demirbağ

Keyword(s):

Smooth Function ◽

Ricci Tensor ◽

Warped Product ◽

Einstein Manifold ◽

Warped Products ◽

Product Manifold ◽

Warped Product Manifold

We deal with a study of warped product manifold which is also a generalized quasi Einstein manifold. Then, we investigate the relationships between such warped products and certain manifolds that provide some Ricci-Hessian type equations, such as Ricmf = ?g for some smooth function ?, where Ricmf denotes the m-Bakery-Emery Ricci tensor. Finally, we obtain some rigidity conditions for such manifolds.

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A study on killing vector fields in four-dimensional spaces

Filomat ◽

10.2298/fil1904249k ◽

2019 ◽

Vol 33 (4) ◽

pp. 1249-1257

Author(s):

Bahar Kırık

Keyword(s):

Ricci Tensor ◽

Vector Fields ◽

Killing Vector ◽

Positive Definite ◽

Killing Vector Field ◽

Killing Vector Fields ◽

Metric Tensor ◽

Conformal Curvature ◽

Conformal Curvature Tensor ◽

Weyl Conformal Curvature Tensor

In the present study, some properties of Killing vector fields are investigated on 4-dimensional manifolds in case of the signature of the metric tensor 1 is either Lorentz or positive definite or neutral. First of all, the notation and the main object of the study are introduced on these manifolds. Later on, some special subalgebras are examined for the members of the Killing algebra when the Killing vector field vanishes at a point of the manifold admitting any of these metric signatures. The constraints of this examination to the Weyl conformal curvature tensor and the Ricci tensor are then studied and some results are obtained. Finally, some examples related to these results are given for all metric signatures.

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The Fundamental Groups of m-Quasi-Einstein Manifolds

ISRN Geometry ◽

10.5402/2011/812541 ◽

2011 ◽

Vol 2011 ◽

pp. 1-4 ◽

Cited By ~ 1

Author(s):

Hee Kwon Lee

Keyword(s):

Fundamental Group ◽

Positive Constant ◽

Ricci Flow ◽

Ricci Tensor ◽

Flow Theory ◽

Ricci Soliton ◽

Fundamental Groups ◽

Einstein Manifolds ◽

Constant Multiple ◽

Metric Tensor

In Ricci flow theory, the topology of Ricci soliton is important. We call a metric quasi-Einstein if the m-Bakry-Emery Ricci tensor is a constant multiple of the metric tensor. This is a generalization of gradient shrinking Ricci soliton. In this paper, we will prove the finiteness of the fundamental group of m-quasi-Einstein with a positive constant multiple.

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Curvature inequalities for C-totally real doubly warped products of locally conformal almost cosymplectic manifolds

Filomat ◽

10.2298/fil1720449a ◽

2017 ◽

Vol 31 (20) ◽

pp. 6449-6459 ◽

Cited By ~ 2

Author(s):

Akram Ali ◽

Siraj Uddin ◽

Wan Othman ◽

Cenap Ozel

Keyword(s):

Sectional Curvature ◽

Mean Curvature ◽

Warped Product ◽

Warped Products ◽

Equality Case ◽

Cosymplectic Manifolds ◽

Almost Cosymplectic Manifold ◽

Locally Conformal ◽

Totally Real ◽

Optimal Inequalities

In this paper, we establish some optimal inequalities for the squared mean curvature in terms warping functions of a C-totally real doubly warped product submanifold of a locally conformal almost cosymplectic manifold with a pointwise ?-sectional curvature c. The equality case in the statement of inequalities is also considered. Moreover, some applications of obtained results are derived.

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Pointwise pseudo-slant submanifolds of a Kenmotsu manifold

Filomat ◽

10.2298/fil1718833k ◽

2017 ◽

Vol 31 (18) ◽

pp. 5833-5853 ◽

Cited By ~ 1

Author(s):

Viqar Khan ◽

Mohammad Shuaib

Keyword(s):

Present Article ◽

Warped Product ◽

Shape Operator ◽

Warped Products ◽

Canonical Structure ◽

Slant Submanifold ◽

Kenmotsu Manifold ◽

Slant Submanifolds ◽

Kenmotsu Manifolds

In the present article, we have investigated pointwise pseudo-slant submanifolds of Kenmotsu manifolds and have sought conditions under which these submanifolds are warped products. To this end first, it is shown that these submanifolds can not be expressed as non-trivial doubly warped product submanifolds. However, as there exist non-trivial (single) warped product submanifolds of a Kenmotsu manifold, we have worked out characterizations in terms of a canonical structure T and the shape operator under which a pointwise pseudo slant submanifold of a Kenmotsu manifold reduces to a warped product submanifold.

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A Note on the Transformability of Spherically Symmetric Metrics

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091500013973 ◽

1953 ◽

Vol 9 (1) ◽

pp. 13-16 ◽

Cited By ~ 1

Author(s):

Paul Kustaanheimo

Keyword(s):

Spherically Symmetric ◽

Symmetric Metrics ◽

Spherically Symmetric Metric

SummaryIt is shown that every spherically symmetric metric can be transformed into the isotropic form. As illustration an example is given.

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Black hole in balance with dark matter

International Journal of Modern Physics A ◽

10.1142/s0217751x20400503 ◽

2020 ◽

Vol 35 (02n03) ◽

pp. 2040050

Author(s):

Boris E. Meierovich

Keyword(s):

Black Hole ◽

Dark Matter ◽

Phase Equilibrium ◽

Vector Field ◽

Scalar Field ◽

Total Mass ◽

Metric Tensor ◽

Static Solutions ◽

Tensor Formula ◽

Galaxy Rotation Curves

Equilibrium of a gravitating scalar field inside a black hole compressed to the state of a boson matter, in balance with a longitudinal vector field (dark matter) from outside is considered. Analytical consideration, confirmed numerically, shows that there exist static solutions of Einstein’s equations with arbitrary high total mass of a black hole, where the component of the metric tensor [Formula: see text] changes its sign twice. The balance of the energy-momentum tensors of the scalar field and the longitudinal vector field at the interface ensures the equilibrium of these phases. Considering a gravitating scalar field as an example, the internal structure of a black hole is revealed. Its phase equilibrium with the longitudinal vector field, describing dark matter on the periphery of a galaxy, determines the dependence of the velocity on the plateau of galaxy rotation curves on the mass of a black hole, located in the center of a galaxy.

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RETRACTED Warped Product Pointwise Semi-slant Submanifolds of Sasakian Manifol Retracted

Kragujevac Journal of Mathematics ◽

10.46793/kgjmat2105.721m ◽

2021 ◽

Vol 45 (5) ◽

pp. 721-738

Author(s):

ION MIHAI ◽

◽

SIRAJ UDDIN ◽

АДЕЛА MIHAI

Keyword(s):

Warped Product ◽

Warped Products ◽

Sasakian Manifolds ◽

Slant Submanifolds ◽

Hermitian Manifolds ◽

Almost Hermitian Manifolds

Recently, B.-Y. Chen and O. J. Garay studied pointwise slant submanifolds of almost Hermitian manifolds. By using the notion of pointwise slant submanifolds, we investigate the geometry of pointwise semi-slant submanifolds and their warped products in Sasakian manifolds. We give non-trivial examples of such submanifolds and obtain several fundamental results, including a characterization for warped product pointwise semi-slant submanifolds of Sasakian manifolds.

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