A NOTE ON THE GENERAL RELATIONSHIP BETWEEN D-DIFFERENTIATION AND COVARIANT DIFFERENTIATION

2008 ◽  
Vol 05 (04) ◽  
pp. 513-520 ◽  
Author(s):  
D. J. HURLEY ◽  
M. A. VANDYCK
Keyword(s):  
Scalar Curvature ◽  
Tensor Field ◽  
General Relationship ◽  
General Operator ◽  
Covariant Differentiation ◽  
D Operator

The most general operator of D-differentiation is proved to be expressible as a combination of covariant differentiation and a tensor field. The torsion, the curvature and the non-metricity of an arbitrary D-operator are consequently given in terms of those of covariant differentiation. The scalar curvature is also obtained.

TENSORIAL CURVATURE AND D-DIFFERENTIATION PART II: "PRINCIPAL" KIND AND EINSTEIN–MAXWELL THEORY

2007 ◽  
Vol 04 (05) ◽  
pp. 847-860 ◽  
Author(s):  
D. J. HURLEY ◽  
M. A. VANDYCK
Keyword(s):  
Scalar Curvature ◽  
Tensor Field ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Curvature Operator ◽  
Maxwell Theory ◽  
Covariant Differentiation ◽  
Necessary And Sufficient

The class of "commutative" D-operators, which was introduced in the first part of this paper, is generalized to obtain the "principal" class. It is established that principal D-operators are expressible in terms of covariant differentiation and a tensor field. Necessary and sufficient conditions are determined for the curvature operator to be tensorial, and for the scalar curvature to exist. As an application, the Einstein–Maxwell theory is recast in a new geometrical framework.

TENSORIAL CURVATURE AND D-DIFFERENTIATION PART I: "COMMUTATIVE" KIND

2007 ◽  
Vol 04 (05) ◽  
pp. 829-846 ◽  
Author(s):  
D. J. HURLEY ◽  
M. A. VANDYCK
Keyword(s):  
Scalar Curvature ◽  
Tensor Field ◽  
Curvature Operator ◽  
Unified Framework ◽  
Maxwell Theory ◽  
Covariant Differentiation ◽  
The Family ◽  
The Difference ◽  
Non Linear Operator ◽  
Special Case

A special class of operators of D-differentiation is introduced, called the "commutative" kind. It is closely related to the family of D-differentiation operators, the curvature of which is a tensor (as opposed to a non-linear operator), and to that of the D-differentiation operators admitting a scalar curvature. It is found that all commutative D-differentiation operators admit a scalar curvature, but that only a proper subset of them (which is explicitly characterized) has a curvature operator that is a tensor. It is also established that all commutative D-differentiation operators can be expressed in terms of covariant differentiation and a tensor field. This generalizes the well-known result about the difference of two sets of connection coefficients yielding a tensor field. In a companion article, a cognate class of operators is defined, which contains the commutative type as a special case. It enables one to construct a unified framework for the Einstein–Maxwell theory through D-differentiation.

scholarly journals A (CHR)3-flat trans-Sasakian manifold

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.

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

1983 ◽  
Vol 6 (4) ◽  
pp. 727-736
Author(s):  
Liu Shu-Lin ◽  
Xu Sen-Lin
Keyword(s):  
Field Theory ◽  
Scalar Curvature ◽  
Curvature Tensor ◽  
Differential Form ◽  
Ricci Tensor ◽  
Tensor Field ◽  
Second Order ◽  
Dimensional Manifold ◽  
Symmetric Field

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.

Isometry of Riemannian Manifolds to Spheres, II

1976 ◽  
Vol 28 (1) ◽  
pp. 63-72 ◽  
Author(s):  
Neill H. Ackerman ◽  
C. C. Hsiung
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Ricci Tensor ◽  
Symmetric Matrix ◽  
Riemann Tensor ◽  
Positive Definite ◽  
Inverse Matrix ◽  
Covariant Differentiation

Let Mn be a Riemannian manifold of dimension n ≧ 2 and class C3, (gtj) the symmetric matrix of the positive definite metric of Mn, and (gij) the inverse matrix of (gtj), and denote by and R = gijRij the operator of covariant differentiation with respect to gij, the Riemann tensor, the Ricci tensor and the scalar curvature of Mn respectively.

Positive scalar curvature metrics via end-periodic manifolds

2020 ◽  
Vol 5 (3) ◽  
pp. 639-676
Author(s):  
Michael Hallam ◽  
Varghese Mathai
Keyword(s):  
Scalar Curvature ◽  
Positive Scalar Curvature ◽  
Positive Scalar

scholarly journals On the quantization of folded strings in non-critical dimensions

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Jacob Sonnenschein ◽  
Dorin Weissman
Keyword(s):  
Scalar Curvature ◽  
Space Dimension ◽  
Massive Particle ◽  
Open String ◽  
Closed String ◽  
Target Space ◽  
Critical Dimensions ◽  
Complete Set ◽  
Rotating String ◽  
Folding Point

Abstract Classical rotating closed string are folded strings. At the folding points the scalar curvature associated with the induced metric diverges. As a consequence one cannot properly quantize the fluctuations around the classical solution since there is no complete set of normalizable eigenmodes. Furthermore in the non-critical effective string action of Polchinski and Strominger, there is a divergence associated with the folds. We overcome this obstacle by putting a massive particle at each folding point which can be used as a regulator. Using this method we compute the spectrum of quantum fluctuations around the rotating string and the intercept of the leading Regge trajectory. The results we find are that the intercepts are a = 1 and a = 2 for the open and closed string respectively, independent of the target space dimension. We argue that in generic theories with an effective string description, one can expect corrections from finite masses associated with either the endpoints of an open string or the folding points on a closed string. We compute explicitly the corrections in the presence of these masses.

scholarly journals Quaternionic contact 4n + 3-manifolds and their 4n-quotients

2021 ◽  
Author(s):  
Yoshinobu Kamishima
Keyword(s):  
Scalar Curvature ◽  
Lie Group ◽  
Contact Structure ◽  
Einstein Manifolds ◽  
Hermitian Metrics ◽  
Kähler Metric ◽  
Formula One ◽  
Quaternion Space ◽  
Kahler Metric ◽  
Do So

AbstractWe study some types of qc-Einstein manifolds with zero qc-scalar curvature introduced by S. Ivanov and D. Vassilev. Secondly, we shall construct a family of quaternionic Hermitian metrics $$(g_a,\{J_\alpha \}_{\alpha =1}^3)$$ ( g a , { J α } α = 1 3 ) on the domain Y of the standard quaternion space $${\mathbb {H}}^n$$ H n one of which, say $$(g_a,J_1)$$ ( g a , J 1 ) is a Bochner flat Kähler metric. To do so, we deform conformally the standard quaternionic contact structure on the domain X of the quaternionic Heisenberg Lie group$${{\mathcal {M}}}$$ M to obtain quaternionic Hermitian metrics on the quotient Y of X by $${\mathbb {R}}^3$$ R 3 .

scholarly journals Spherically Symmetric Tensor Fields and Their Application in Nonlinear Theory of Dislocations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 830
Author(s):  
Evgeniya V. Goloveshkina ◽  
Leonid M. Zubov
Keyword(s):  
Nonlinear Theory ◽  
Hollow Sphere ◽  
Tensor Field ◽  
Exact Analytical Solution ◽  
Symmetric Tensor ◽  
Symmetric Distribution ◽  
Spherically Symmetric ◽  
Tensor Fields ◽  
Spherically Symmetric Distribution ◽  
Nonlinear Elastic

The concept of a spherically symmetric second-rank tensor field is formulated. A general representation of such a tensor field is derived. Results related to tensor analysis of spherically symmetric fields and their geometric properties are presented. Using these results, a formulation of the spherically symmetric problem of the nonlinear theory of dislocations is given. For an isotropic nonlinear elastic material with an arbitrary spherically symmetric distribution of dislocations, this problem is reduced to a nonlinear boundary value problem for a system of ordinary differential equations. In the case of an incompressible isotropic material and a spherically symmetric distribution of screw dislocations in the radial direction, an exact analytical solution is found for the equilibrium of a hollow sphere loaded from the outside and from the inside by hydrostatic pressures. This solution is suitable for any models of an isotropic incompressible body, i. e., universal in the specified class of materials. Based on the obtained solution, numerical calculations on the effect of dislocations on the stress state of an elastic hollow sphere at large deformations are carried out.

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