Tensor bundle

Short description: Concept in mathematics

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

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

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 $(p,q)\in \mathbb{\mathbb{N}}\times \mathbb{\mathbb{N}}$.

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

References

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

See also

• Spinor bundle – Geometric structure
• Tensor field – Assignment of a tensor continuously varying across a region of space