Tensor bundle

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, and Tensor product bundle

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$$\bigoplus _{(p,q)\in \mathbb {N} \times \mathbb {N} }V^{\otimes p}\otimes (V^{*})^{\otimes q}.$$

References

• Lee, John M. (2012). Introduction to Smooth Manifolds. Graduate Texts in Mathematics. Vol. 218 (Second ed.). New York London: Springer-Verlag. ISBN 978-1-4419-9981-8. OCLC 808682771.
• Saunders, David J. (1989). The Geometry of Jet Bundles. London Mathematical Society Lecture Note Series. Vol. 142. Cambridge New York: Cambridge University Press. ISBN 978-0-521-36948-0. OCLC 839304386.
• Steenrod, Norman (5 April 1999). The Topology of Fibre Bundles. Princeton Mathematical Series. Vol. 14. Princeton, N.J.: Princeton University Press. ISBN 978-0-691-00548-5. OCLC 40734875.

See also

• Fiber bundle – Continuous surjection satisfying a local triviality condition
• Spinor bundle – Geometric structure
• Tensor field – Assignment of a tensor continuously varying across a region of space