View source for Tensor bundle

In [[mathematics]], the '''tensor bundle''' of a [[manifold]] is the [[Direct sum of vector bundles|direct sum]] of all tensor products of the [[tangent bundle]] and the [[cotangent bundle]] of that manifold. To do [[calculus]] on the tensor bundle a [[Connection (mathematics)|connection]] is needed, except for the special case of the [[exterior derivative]] of antisymmetric tensors.

==Definition==

{{See also|Tensor field#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]].

==References==

{{reflist}}

* {{Lee Introduction to Smooth Manifolds|edition=2}} <!--{{sfn|Lee|2012|p=}}-->
* {{Saunders The Geometry of Jet Bundles}} <!--{{sfn|Saunders|1989|p=}}-->
* {{Steenrod The Topology of Fibre Bundles 1999}} <!--{{sfn|Steenrod|1999|p=}}-->

==See also==

* {{annotated link|Fiber bundle}}
* {{annotated link|Spinor bundle}}
* {{annotated link|Tensor field}}

{{Manifolds}}
{{Tensors}}

[[Category:Vector bundles]]

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