nLab tensor

This is about tensor quantities in the sense of multilinear algebra, differential geometry and physics, as in tensor calculus. For the different notion of a tensor in enriched category theory see under copower.

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Definition

Generally, a tensor is an element of a tensor product.

Traditionally this is considered in differential geometry for the following case:

for $X$ a manifold, $T X$ the tangent bundle, $T^* X$ the cotangent bundle, $\Gamma(T X)$, $\Gamma(T^* X)$ their spaces of sections and $C(X)$ the associative algebra of functions on $X$, a rank-$(p,q)$ tensor or tensor field on $X$ is an element of the tensor product of modules over $C(X)$

$$t \in \Gamma(T X)^{\otimes_{C(X)}^p} \otimes_{C(X)} \Gamma(T^* X)^{\otimes^q_{C(X)}} \,.$$

A rank $(p,0)$-tensor is also called a covariant tensor and a rank $(0,q)$-tensor a contravariant tensor.

Examples

General

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In differential geometry

• A vector field is a rank $(1,0)$-tensor field.

• A Riemannian metric is a symmetric rank $(0,2)$-tensor.

• A differential form of degree $n$ is a skew-symmetric rank $(0,n)$-tensor.

• A Poisson tensor is a skew-symmetric tensor of rank $(2,0)$.

Related concepts

• tensor calculus, tensor network, string diagram

• decomposable tensor

• field (physics)

• hypermatrix

References

Historical discussion:

• Élie Cartan (translated by Vladislav Goldberg from Cartan’s lectures at the Sorbonne in 1926–27): Chapter 8 of: Riemannian Geometry in an Orthogonal Frame, World Scientific (2001) [doi:10.1142/4808, pdf]

Monograph:

• Joseph M. Landsberg?, Tensors: Geometry and Applications (2011) [ISBN:978-0-8218-6907-9, ams:gsm-128]

Discussion with an eye towards application in (particle) physics:

• Theodore Frankel, section 2.4 in: The Geometry of Physics - An Introduction

• Howard Georgi, §10 in: Lie Algebras In Particle Physics, Westview Press (1999), CRC Press (2019) [doi:10.1201/9780429499210]