# Tensor bundle

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{{Short description|Concept in mathematics}}
{{no footnotes|date=September 2025}}
In [mathematics](/source/mathematics), the '''tensor bundle''' of a [manifold](/source/manifold) is the [direct sum](/source/Direct_sum_of_vector_bundles) of all tensor products of the [tangent bundle](/source/tangent_bundle) and the [cotangent bundle](/source/cotangent_bundle) of that manifold. To do [calculus](/source/calculus) on the tensor bundle a [connection](/source/Connection_(mathematics)) is needed, except for the special case of the [exterior derivative](/source/exterior_derivative) of antisymmetric tensors.

==Definition==

{{See also|Tensor field#Tensor bundles}}

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

{{differential-geometry-stub}}

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