# nLab tensor

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.

category theory

## Applications

#### Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

• (shape modality $\dashv$ flat modality $\dashv$ sharp modality)

$(\esh \dashv \flat \dashv \sharp )$

• dR-shape modality$\dashv$ dR-flat modality

$\esh_{dR} \dashv \flat_{dR}$

tangent cohesion

differential cohesion

singular cohesion

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

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 $X$ 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

(…)

### 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 $(2,0)$.

## References

For instance section 2.4 of

Last revised on February 3, 2019 at 05:41:30. See the history of this page for a list of all contributions to it.