Tensor Fields  

In General > s.a. tensors; types of fiber bundles [tensor bundles].
$ Def: An element of the space ⊗p T*x Xq Tx X of multilinear forms on Tx X ⊗ ... ⊗ Tx X ⊗ T*x X ⊗ ... ⊗ T*x X (p copies of Tx X, and q copies of T*x X), for all xX.
* And other structure: The set of all tensor fields on X forms an algebra, \(\cal D\)(X).
* Tensors and physical theories: A commonplace attributed to Kretschmann states that any local physical theory can be represented in arbitrary coordinates using tensor calculus, but this may not be true for theories with spinors.
@ General references: Wasserman 09 [and physics applications].
@ Related topics: Brännlund et al IJMPD(10)-a1003 [covariant averaging procedure, and Weitzenböck connection].
> Online resources: see Wikipedia page.

Tensor Density > s.a. projective structures.
$ Def: A tensor density (sometimes called a relative tensor) of weight w on a manifold is an object that transforms as

T' a... bc... d = |∂x/∂x'|w {∂x'a/∂xm} ··· {∂x'b/∂xn} {∂xp/∂x'c} ··· {∂xq/∂x'd} T' m... np... q .

* With a metric: A tensor density of weight w on a manifold can be expressed as

T a... bc... d = |g|w/2 T a... bc... d ,

where Ta... bc... d is a tensor and does not depend on the choice of a volume element εabcd .
* Special cases: The ones with w = 1 are the tensor densities proper; The ones with w = –1 are sometimes called tensor capacities.
@ References: in Dalarsson & Dalarsson 05 ["relative tensors"].
> In physics: see ADM, connection, other formulations, and actions for general relativity; canonical quantum theory.

Types of Tensor Fields > s.a. 3D geometries [transverse traceless]; decomposition; forms; vector field [vertical].
$ Horizontal: Given a fibration of a manifold, a covariant tensor field is horizontal if any contraction of it with a vector tangent to a fiber vanishes; With a metric, the definition can be extended to contravariant tensor fields.
@ Generalizations: Akhmedov TMP(05) [non-abelian], TMP(06) [non-abelian, gauge transformations and curvature]; Gallego a1207 [higher-order, and applications to electrodynamics]; Navarro JMP(14)-a1306 [second-order, divergence-free tensors].

Derivatives, Tensor Calculus > s.a. analysis; Calculus; connections; lie derivative.
* Covariant derivative: For a covariant/contravariant vector field, it is given respectively by

a kb = ∂a kb – Γcab kc ,   ∇a kb = ∂a kb + Γbac kc .

* Weak derivative: A locally integrable tensor field T has a weak derivative if there exists a tensor field X such that their associated distributions are related by Xma... cb... d = ∇m Ta... cb... d.
@ General references: Frederiks & Friedmann 24; Spivak 65; Synge & Schild 69; Dodson & Poston 91 [geometry]; Akivis & Goldberg 03; Hackbusch 12 [numerical].
@ Related topics: Ashtekar et al GRG(82) [generalization]; Geroch & Traschen PRD(87) [weak]; Hall JMP(91) [covariantly constant, and holonomy groups]; Thiffeault JPA(01)n.CD [time derivatives]; Tapia gq/04 [differential invariants]; Boulanger JMP(05)ht/04 [Weyl-covariant].

Spacetime Tensors > s.a. Potential for a Field.
@ Averaging: Mars & Zalaletdinov JMP(97)dg; Boero & Moreschi a1610.
> As dynamical fields: see gravity theories; types of field theories.

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