# nLab tensor

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

category theory

## Applications

#### Differential geometry

differential geometry

synthetic differential geometry

# 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

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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 $(2,0)$.

## References

For instance section 2.4 of

Revised on October 19, 2015 04:47:27 by Urs Schreiber (89.204.139.236)