Editing Tensor field (section)

== Tensor bundles ==

A tensor bundle is a [[fiber bundle]] where the fiber is a tensor product of any number of copies of the [[tangent space]] and/or [[cotangent space]] of the base space, which is a manifold. As such, the fiber is a [[vector space]] and the tensor bundle is a special kind of [[vector bundle]].

The vector bundle is a natural idea of "vector space depending continuously (or smoothly) on parameters" – the parameters being the points of a manifold ''M''. For example, a ''vector space of one dimension depending on an angle'' could look like a [[Möbius strip]] or alternatively like a [[cylinder (geometry)|cylinder]]. Given a vector bundle ''V'' over ''M'', the corresponding field concept is called a ''section'' of the bundle: for ''m'' varying over ''M'', a choice of vector
: ''v<sub>m</sub>'' in ''V<sub>m</sub>'',

where ''V<sub>m</sub>'' is the vector space "at" ''m''.

Since the [[tensor product]] concept is independent of any choice of basis, taking the tensor product of two vector bundles on ''M'' is routine. Starting with the [[tangent bundle]] (the bundle of [[tangent space]]s) the whole apparatus explained at [[component-free treatment of tensors]] carries over in a routine way – again independently of coordinates, as mentioned in the introduction.

We therefore can give a definition of '''tensor field''', namely as a [[section (fiber bundle)|section]] of some [[tensor bundle]].  (There are vector bundles that are not tensor bundles: the Möbius band for instance.) This is then guaranteed geometric content, since everything has been done in an intrinsic way. More precisely, a tensor field assigns to any given point of the manifold a tensor in the space
: <math>V \otimes \cdots \otimes V \otimes V^* \otimes  \cdots  \otimes V^* ,</math>
where ''V'' is the [[tangent space]] at that point and ''V''<sup>∗</sup> is the [[cotangent space]]. See also [[tangent bundle]] and [[cotangent bundle]].

Given two tensor bundles ''E'' → ''M'' and ''F'' → ''M'', a linear map ''A'': Γ(''E'') → Γ(''F'') from the space of sections of ''E'' to sections of ''F'' can be considered itself as a tensor section of <math>\scriptstyle E^*\otimes F</math> if and only if it satisfies ''A''(''fs'') = ''fA''(''s''), for each section ''s'' in Γ(''E'') and each smooth function ''f'' on ''M''.  Thus a tensor section is not only a linear map on the vector space of sections, but a ''C''<sup>∞</sup>(''M'')-linear map on the [[module (mathematics)|module]] of sections.  This property is used to check, for example, that even though the [[Lie derivative]] and [[covariant derivative]] are not tensors, the [[torsion tensor|torsion]] and [[Affine connection|curvature tensors]] built from them are.