{{Short description|Concept in mathematics}} {{no footnotes|date=September 2025}} In mathematics, the '''tensor bundle''' of a manifold is the 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 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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