Some invariant theorems on geometry of Einstein non-symmetric field theory

This paper generalizes Einstein's theorem. It is shown that under the transformationTΛ:Uikℓ→U¯ikℓ≡Uikℓ+δiℓΛk−δkℓΛi, curvature tensorSkℓmi(U), Ricci tensorSik(U), and scalar curvatureS(U)are all invariant, whereΛ=Λjdxjis a closed1-differential form on ann-dimensional manifoldM.It is still shown that for arbitraryU, the transformation that makes curvature tensorSkℓmi(U)(or Ricci tensorSik(U)) invariantTV:Uikℓ→U¯ikℓ≡Uikℓ+Vikℓmust beTΛtransformation, whereV(its components areVikℓ) is a second order differentiable covariant tensor field with vector value.

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A (CHR)3-flat trans-Sasakian manifold

Proceedings of the International Geometry Center ◽

10.15673/tmgc.v12i2.1438 ◽

2019 ◽

Vol 12 (2) ◽

Author(s):

Koji Matsumoto

Keyword(s):

Riemannian Manifold ◽

Scalar Curvature ◽

Curvature Tensor ◽

Ricci Tensor ◽

Tensor Field ◽

Einstein Manifold ◽

Sasakian Manifold ◽

Hermitian Manifolds ◽

Riemannian Curvature Tensor ◽

Contact Riemannian Manifold

In [4] M. Prvanovic considered several curvaturelike tensors defined for Hermitian manifolds. Developing her ideas in [3], we defined in an almost contact Riemannian manifold another new curvaturelike tensor field, which is called a contact holomorphic Riemannian curvature tensor or briefly (CHR)3-curvature tensor. Then, we mainly researched (CHR)3-curvature tensor in a Sasakian manifold. Also we proved, that a conformally (CHR)3-flat Sasakian manifold does not exist. In the present paper, we consider this tensor field in a trans-Sasakian manifold. We calculate the (CHR)3-curvature tensor in a trans-Sasakian manifold. Also, the (CHR)3-Ricci tensor ρ3 and the (CHR)3-scalar curvature τ3 in a trans-Sasakian manifold have been obtained. Moreover, we define the notion of the (CHR)3-flatness in an almost contact Riemannian manifold. Then, we consider this notion in a trans-Sasakian manifold and determine the curvature tensor, the Ricci tensor and the scalar curvature. We proved that a (CHR)3-flat trans-Sasakian manifold is a generalized ɳ-Einstein manifold. Finally, we obtain the expression of the curvature tensor with respect to the Riemannian metric g of a trans-Sasakian manifold, if the latter is (CHR)3-flat.

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Some characterizations of three-dimensional trans-Sasakian manifolds admitting η- Ricci solitons and trans-Sasakian manifolds as Kagan subprojective space

Ukrains’kyi Matematychnyi Zhurnal ◽

10.37863/umzh.v72i3.6047 ◽

2020 ◽

Vol 72 (3) ◽

pp. 427-432

Author(s):

A. Sarkar ◽

A. Sil ◽

A. K. Paul

Keyword(s):

Curvature Tensor ◽

Ricci Tensor ◽

Three Dimensional ◽

Ricci Soliton ◽

Second Order ◽

Ricci Solitons ◽

Riemann Curvature ◽

Sasakian Manifolds ◽

Riemann Curvature Tensor ◽

Order Parallel

UDC 514.7 The object of the present paper is to study three-dimensional trans-Sasakian manifolds admitting η -Ricci soliton. Actually, we study such manifolds whose Ricci tensor satisfy some special conditions like cyclic parallelity, Ricci semisymmetry, ϕ -Ricci semisymmetry, after reviewing the properties of second order parallel tensors on such manifolds. We determine the form of Riemann curvature tensor of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces. We also give some classification results of trans-Sasakian manifolds of dimension greater than three as Kagan subprojective spaces.

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Connections Satisfying a Generalized Ricci Lemma

Canadian Journal of Mathematics ◽

10.4153/cjm-1964-056-2 ◽

1964 ◽

Vol 16 ◽

pp. 549-560 ◽

Cited By ~ 2

Author(s):

J. R. Vanstone

Keyword(s):

Differential Geometry ◽

Covariant Derivative ◽

Tensor Field ◽

Positive Definite ◽

Second Order ◽

Dimensional Manifold ◽

Very Old ◽

Christoffel Symbols ◽

Image Position

In this paper we shall consider a generalization of a very old problem in differential geometry; namely, given a second-order covariant tensor field aij(x) on an n-dimensional manifold, when does there exist a connection such that the covariant derivative, defined byvanishes?The earliest question of this type arose in the case when is symmetric and positive definite. A solution connection of the problem is then given by the Christoffel symbols

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A quarter-symmetric metric connection on almost contact B-metric manifolds

Filomat ◽

10.2298/fil1916181b ◽

2019 ◽

Vol 33 (16) ◽

pp. 5181-5190

Author(s):

Şenay Bulut

Keyword(s):

Scalar Curvature ◽

Curvature Tensor ◽

Ricci Tensor ◽

Civita Connection ◽

Metric Connection ◽

Levi Civita Connection

The aim of this paper is to study the notion of a quarter-symmetric metric connection on an almost contact B-metric manifold (M,?,?,?,g). We obtain the relation between the Levi-Civita connection and the quarter-symmetric metric connection on (M,?,?,?,g).We investigate the curvature tensor, Ricci tensor and scalar curvature tensor with respect to the quarter-symmetric metric connection. In case the manifold (M,?,?,?,g) is a Sasaki-like almost contact B-metric manifold, we get some formulas. Finally, we give some examples of a quarter-symmetric metric connection.

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Erratum to: Hints of unitarity at large N in the O(N)3 tensor field theory

Journal of High Energy Physics ◽

10.1007/jhep08(2020)167 ◽

2020 ◽

Vol 2020 (8) ◽

Author(s):

Dario Benedetti ◽

Razvan Gurau ◽

Sabine Harribey ◽

Kenta Suzuki

Keyword(s):

Field Theory ◽

Tensor Field ◽

Normalization Factor ◽

Large N

The measure in equation (2.11) contains a wrong normalization factor, and it should be multiplied by 21−dΓ(d − 1)/Γ(d/2)2.

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Singularity Theorems in the Effective Field Theory for Quantum Gravity at Second Order in Curvature

Universe ◽

10.3390/universe6100171 ◽

2020 ◽

Vol 6 (10) ◽

pp. 171

Author(s):

Folkert Kuipers ◽

Xavier Calmet

Keyword(s):

Field Theory ◽

Quantum Gravity ◽

Effective Field Theory ◽

Energy Condition ◽

Null Energy Condition ◽

Effective Field ◽

Second Order ◽

Jordan Frame ◽

Singularity Theorems ◽

Massive Spin

In this paper, we discuss singularity theorems in quantum gravity using effective field theory methods. To second order in curvature, the effective field theory contains two new degrees of freedom which have important implications for the derivation of these theorems: a massive spin-2 field and a massive spin-0 field. Using an explicit mapping of this theory from the Jordan frame to the Einstein frame, we show that the massive spin-2 field violates the null energy condition, while the massive spin-0 field satisfies the null energy condition, but may violate the strong energy condition. Due to this violation, classical singularity theorems are no longer applicable, indicating that singularities can be avoided, if the leading quantum corrections are taken into account.

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A variational approach to second-order multisymplectic field theory

Journal of Geometry and Physics ◽

10.1016/s0393-0440(00)00012-7 ◽

2000 ◽

Vol 35 (4) ◽

pp. 333-366 ◽

Cited By ~ 27

Author(s):

Shinar Kouranbaeva ◽

Steve Shkoller

Keyword(s):

Field Theory ◽

Variational Approach ◽

Second Order

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On a Set of Conform-Invariant Equations of the Gravitational Field

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s001309150001419x ◽

1953 ◽

Vol 10 (1) ◽

pp. 16-20 ◽

Cited By ~ 8

Author(s):

H. A. Buchdahl

Keyword(s):

Gravitational Field ◽

Linear Combination ◽

Curvature Tensor ◽

Wide Class ◽

Ricci Tensor ◽

Empty Space ◽

Image Position ◽

Derivatives Of

Eddington has considered equations of the gravitational field in empty space which are of the fourth differential order, viz. the sets of equations which express the vanishing of the Hamiltonian derivatives of certain fundamental invariants. The author has shown that a wide class of such equations are satisfied by any solution of the equationswhere Gμν and gμν are the components of the Ricci tensor and the metrical tensor respectively, whilst λ is an arbitrary constant. For a V4 this applies in particular when the invariant referred to above is chosen from the setwhere Bμνσρ is the covariant curvature tensor. K3 has been included since, according to a result due to Lanczos3, its Hamiltonian derivative is a linear combination of and , i.e. of the Hamiltonian derivatives of K1 and K2. In fact

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