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.


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

tangent cohesion

differential cohesion

graded differential 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& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\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

For instance section 2.4 of

Revised on April 20, 2016 05:00:18 by Urs Schreiber (131.220.184.222)