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
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
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