7. Covariant Differentiation and Geodesic Curves

2018 ◽  
pp. 40-46
Keyword(s):  
Covariant Differentiation

WEYL SPACES OF COMPOSITIONS OF SIX BASIC MANIFOLDS

2021 ◽  
Vol 10 (10) ◽  
pp. 3337-3347
Author(s):  
M. Ajeti ◽  
M. Teofilova ◽  
G. Zlatanov
Keyword(s):  
Covariant Differentiation ◽  
Weyl Spaces ◽  
Riemannian Spaces

By help of prolonged covariant differentiation, Cartesian compositions of six basic manifolds are studied. Weyl spaces of such compositions are characterized. Eleven-dimensional Riemannian spaces containing compositions of six basic manifolds are also considered.

Connections and Covariant Differentiation

1991 ◽  
pp. 205-245
Author(s):  
Christopher Terence John Dodson ◽  
Timothy Poston
Keyword(s):  
Covariant Differentiation

scholarly journals On projective-symmetric spaces

1964 ◽  
Vol 4 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Bandana Gupta
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Image Position ◽  
Riemannian Spaces

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

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.

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