nLab tensor

Redirected from "tensor field".
Contents

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.


Context

Category theory

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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

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

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

tΓ(TX) C(X) p C(X)Γ(T *X) C(X) q. t \in \Gamma(T X)^{\otimes_{C(X)}^p} \otimes_{C(X)} \Gamma(T^* X)^{\otimes^q_{C(X)}} \,.

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

Examples

General

(…)

In differential geometry

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:

Last revised on May 24, 2024 at 10:19:41. See the history of this page for a list of all contributions to it.