Tensor
Fields| Tensor
Fields |
In General > s.a. [tensors]; types
of fiber bundles [tensor bundles].
$ Def: An element of
the space
p T*x X q
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 x 
X.
* And other structure:
The set of all tensor fields on X forms an
algebra, 
(X).
Tensor Density
$ Def: A tensor density 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 
abcd .
> 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 def can be extended to contravariant tensor fields.
@ Generalizations: Akhmedov TMP(05)
[non-abelian], TMP(06) [non-abelian, gauge transformations and curvature].
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 – 
cab kc
, a kb = a kb
+
bac 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: Spivak 65; Synge & Schild 69; Dodson & Poston 91 [geometry].
@ 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.
@ References: Mars & Zalaletdinov JMP(97)dg [averaging].
Main page – Abbreviations – Journals – Comments – Other
sites – Acknowledgements
Send feedback and suggestions to bombelli at olemiss.edu – Modified
20 jun 2008