Tensor bundle

Template:Short description Template:No footnotes 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. Explicitly, for fixed non-negative integers $p$ and $q$, a tensor bundle is the fiber bundle

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

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

$V_x\otimes \cdots \otimes V_x\otimes V_x^{\ast }\otimes \cdots \otimes V_x^{\ast }=V_x^{\otimes p}\otimes (V_x^{\ast }{)}^{\otimes q}$

where $V_x$ is the tangent space of x and $V_x^{\ast }$ 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$$\underset{(p,q)\in \mathbb{N}\times \mathbb{N}}{⨁}{V}^{\otimes p}\otimes (V^{\ast }{)}^{\otimes q}.$$

References

• Template:Lee Introduction to Smooth Manifolds
• Template:Saunders The Geometry of Jet Bundles
• Template:Steenrod The Topology of Fibre Bundles 1999

See also

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