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

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

Vectors	Distribution Lie bracket Pushforward Tangent space bundle Torsion Vector field Vector flow
Covectors	Closed/Exact Covariant derivative Cotangent space bundle De Rham cohomology Differential form Vector-valued Exterior derivative Interior product Pullback Ricci curvature flow Riemann curvature tensor Tensor field density Volume form Wedge product
Bundles	Adjoint Affine Associated Cotangent Dual Fiber (Co) Fibration Jet Lie algebra (Stable) Normal Principal Spinor Subbundle Tangent Tensor Vector
Connections	Affine Cartan Ehresmann Form Generalized Koszul Levi-Civita Principal Vector Parallel transport

Tensors

Glossary of tensor theory

Scope

Mathematics	Coordinate system Differential geometry Dyadic algebra Euclidean geometry Exterior calculus Multilinear algebra Tensor algebra Tensor calculus
Physics Engineering	Computer vision Continuum mechanics Electromagnetism General relativity Transport phenomena

Notation

Tensor
definitions

Operations

Related
abstractions

Notable tensors

Mathematics	Kronecker delta Levi-Civita symbol Metric tensor Nonmetricity tensor Ricci curvature Riemann curvature tensor Torsion tensor Weyl tensor
Physics	Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein tensor EM tensor Gluon field strength tensor Metric tensor (GR)

Mathematicians