Tensor bundle: Difference between revisions
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{{Short description|Concept in mathematics}} | |||
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In [[Mathematics|mathematics]], the '''tensor bundle''' of a [[Manifold|manifold]] is the direct sum of all tensor products of the [[Tangent bundle|tangent bundle]] and the [[Cotangent bundle|cotangent bundle]] of that manifold. To do [[Calculus|calculus]] on the tensor bundle a [[Connection (mathematics)|connection]] is needed, except for the special case of the [[Exterior derivative|exterior derivative]] of antisymmetric tensors. | In [[Mathematics|mathematics]], the '''tensor bundle''' of a [[Manifold|manifold]] is the direct sum of all tensor products of the [[Tangent bundle|tangent bundle]] and the [[Cotangent bundle|cotangent bundle]] of that manifold. To do [[Calculus|calculus]] on the tensor bundle a [[Connection (mathematics)|connection]] is needed, except for the special case of the [[Exterior derivative|exterior derivative]] of antisymmetric tensors. | ||
==Definition== | ==Definition== | ||
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* {{annotated link|Tensor field}} | * {{annotated link|Tensor field}} | ||
{{Manifolds}} | |||
{{Tensors}} | {{Tensors}} | ||
Latest revision as of 05:14, 15 April 2026
Short description: Concept in mathematics
 | This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations. (September 2025) (Learn how and when to remove this template message) |
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
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
- Template:Lee Introduction to Smooth Manifolds
- Template:Saunders The Geometry of Jet Bundles
- Template:Steenrod The Topology of Fibre Bundles 1999
See also
- Spinor bundle – Geometric structure
- Tensor field – Assignment of a tensor continuously varying across a region of space
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