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.

Scalar curvature of Lagrangian Riemannian submersions and their harmonicity

2017 ◽  
Vol 14 (12) ◽  
pp. 1750171 ◽  
Author(s):  
Şemsi Eken Meri̇ç ◽  
Erol Kiliç ◽  
Yasemi̇n Sağiroğlu
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Sufficient Conditions ◽  
Riemannian Submersion ◽  
Hermitian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Riemannian Submersions ◽  
Totally Geodesic ◽  
Totally Umbilical ◽  
Necessary And Sufficient

In this paper, we consider a Lagrangian Riemannian submersion from a Hermitian manifold to a Riemannian manifold and establish some basic inequalities to obtain relationships between the intrinsic and extrinsic invariants for such a submersion. Indeed, using these inequalities, we provide necessary and sufficient conditions for which a Lagrangian Riemannian submersion [Formula: see text] has totally geodesic or totally umbilical fibers. Moreover, we study the harmonicity of Lagrangian Riemannian submersions and obtain a characterization for such submersions to be harmonic.

scholarly journals The geometry of contact pseudo-slant submanifolds of a Sasakian manifold

Filomat ◽  
2019 ◽  
Vol 33 (17) ◽  
pp. 5551-5559
Author(s):  
Ümit Yıldırım
Keyword(s):  
Tensor Field ◽  
Sufficient Conditions ◽  
Sasakian Manifold ◽  
Necessary And Sufficient Conditions ◽  
Slant Submanifold ◽  
Slant Submanifolds ◽  
Necessary And Sufficient

In this paper, we study the geometry of the contact pseudo-slant submanifolds of a Sasakian manifold. We verify some properties of the components of the tensor field acting on that kind of submanifold and find out the necessary and sufficient conditions for them to be parallel. Also, necessary and sufficient conditions are given for a submanifold to be a pseudo-slant submanifold, contact pseudo-slant product, D?, D? and mixed-geodesic in Sasakian manifold.

scholarly journals The scalar curvature of the tangent bundle of a Finsler manifold

2011 ◽  
Vol 89 (103) ◽  
pp. 57-68
Author(s):  
Aurel Bejancu ◽  
Reda Farran
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Riemannian Manifolds ◽  
Tangent Bundle ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Finsler Manifold ◽  
Finsler Metric ◽  
Degree Zero ◽  
Necessary And Sufficient

Let Fm = (M, F) be a Finsler manifold and G be the Sasaki-Finsler metric on the slit tangent bundle TM0 = TM \{0} of M. We express the scalar curvature ?~ of the Riemannian manifold (TM0,G) in terms of some geometrical objects of the Finsler manifold Fm. Then, we find necessary and sufficient conditions for ?~ to be a positively homogenenous function of degree zero with respect to the fiber coordinates of TM0. Finally, we obtain characterizations of Landsberg manifolds, Berwald manifolds and Riemannian manifolds whose ?~ satisfies the above condition.

The extinction time of a general birth and death process with catastrophes

1986 ◽  
Vol 23 (04) ◽  
pp. 851-858 ◽  
Author(s):  
P. J. Brockwell
Keyword(s):  
Laplace Transform ◽  
Size Distribution ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Death Process ◽  
Extinction Time ◽  
Birth And Death Process ◽  
Different Types ◽  
The Laplace Transform ◽  
Necessary And Sufficient

The Laplace transform of the extinction time is determined for a general birth and death process with arbitrary catastrophe rate and catastrophe size distribution. It is assumed only that the birth rates satisfyλ0= 0,λj> 0 for eachj> 0, and. Necessary and sufficient conditions for certain extinction of the population are derived. The results are applied to the linear birth and death process (λj=jλ, µj=jμ) with catastrophes of several different types.

Nonlinear boundary-value problems for degenerate differential-algebraic systems

2020 ◽  
Vol 17 (3) ◽  
pp. 313-324
Author(s):  
Sergii Chuiko ◽  
Ol'ga Nesmelova
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Algebraic System ◽  
Boundary Value ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Weakly Nonlinear ◽  
Differential Algebraic System ◽  
Linear Differential ◽  
Necessary And Sufficient

The study of the differential-algebraic boundary value problems, traditional for the Kiev school of nonlinear oscillations, founded by academicians M.M. Krylov, M.M. Bogolyubov, Yu.A. Mitropolsky and A.M. Samoilenko. It was founded in the 19th century in the works of G. Kirchhoff and K. Weierstrass and developed in the 20th century by M.M. Luzin, F.R. Gantmacher, A.M. Tikhonov, A. Rutkas, Yu.D. Shlapac, S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko, O.A. Boichuk, V.P. Yacovets, C.W. Gear and others. In the works of S.L. Campbell, L.R. Petzold, Yu.E. Boyarintsev, V.F. Chistyakov, A.M. Samoilenko and V.P. Yakovets were obtained sufficient conditions for the reducibility of the linear differential-algebraic system to the central canonical form and the structure of the general solution of the degenerate linear system was obtained. Assuming that the conditions for the reducibility of the linear differential-algebraic system to the central canonical form were satisfied, O.A.~Boichuk obtained the necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and constructed a generalized Green operator of this problem. Based on this, later O.A. Boichuk and O.O. Pokutnyi obtained the necessary and sufficient conditions for the solvability of the weakly nonlinear differential algebraic boundary value problem, the linear part of which is a Noetherian differential algebraic boundary value problem. Thus, out of the scope of the research, the cases of dependence of the desired solution on an arbitrary continuous function were left, which are typical for the linear differential-algebraic system. Our article is devoted to the study of just such a case. The article uses the original necessary and sufficient conditions for the solvability of the linear Noetherian differential-algebraic boundary value problem and the construction of the generalized Green operator of this problem, constructed by S.M. Chuiko. Based on this, necessary and sufficient conditions for the solvability of the weakly nonlinear differential-algebraic boundary value problem were obtained. A typical feature of the obtained necessary and sufficient conditions for the solvability of the linear and weakly nonlinear differential-algebraic boundary-value problem is its dependence on the means of fixing of the arbitrary continuous function. An improved classification and a convergent iterative scheme for finding approximations to the solutions of weakly nonlinear differential algebraic boundary value problems was constructed in the article.

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