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Tensor bundle

Concept in mathematics

Wikipedia contributorsAdapted for Fathomly · View authors
1 min read
Source updated August 16, 2026

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.

01Definition

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. Explicitly, for fixed non-negative integers p and q, a tensor bundle is the fiber bundle

V\otimes \cdots \otimes V\otimes V^{*}\otimes \cdots \otimes V^{*}=V^{\otimes p}\otimes (V^{*})^{\otimes q}

where V is the tangent bundle of M and V∗ is the cotangent bundle of M, and its fibers are

V_{x}\otimes \cdots \otimes V_{x}\otimes V_{x}^{*}\otimes \cdots \otimes V_{x}^{*}=V_{x}^{\otimes p}\otimes (V_{x}^{*})^{\otimes q}

where V_{x} is the tangent space of x and V_{x}^{*} is the cotangent space of x, for every x in M. The elements of the fibers are tensors of type (p,q)\in \mathbb {N} \times \mathbb {N}.

The direct sum of vector bundles allows to consider all these posibilities of tensor bundle for fixed (p,q) as a single mathematical object\bigoplus _{(p,q)\in \mathbb {N} \times \mathbb {N} }V^{\otimes p}\otimes (V^{*})^{\otimes q}.

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This article is adapted from the Wikipedia article “Tensor bundle”, written by its contributors and licensed under CC BY-SA 4.0. Fathomly has changed the layout, removed citation markers, navigation and maintenance notices, and adjusted punctuation. This adapted version is shared under the same license. For references, see the original article.

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