Riemannian curvature

Let be a -manifold equipped with an affine connexion . The Riemannian curvature is a tensor field defined so that

and thus

where is the torsion tensor. A manifold with null Riemannian curvature is said to be flat.

Proof of equality and tensoriality

Let be vector fields. Then

and thus using the first equation

for any , so indeed

as claimed.

To show tensoriality it suffices to show that the map is -linear. To this end let be a scalar field. Then

and

Thus

where the final terms cancel since .

Conflicting conventions

The convention used here is that used by Evgeny Buchbinder and (for the most part) Wikipedia. Wald's General relativity defines the torsion-free case acting on a 1-form so that

meaning the action on a vector field is

meaning .

Given local coördinates , the components of can be computed explicitly in terms of the connexion coëfficients as¹

Derivation

We have

as required.

Relation to parallel transport

The curvature of an (intrinsic) space can be thought of as the failure a vector to return to its initial value after parallel transport along a loop, explaining why the structure of an affine connexion is a prerequisite.

#to/complete

Properties

, i.e. .

For a torsion free connexion

Assume is torsion-free.

Bianchi Identity I. .

Proof

#missing/proof

Computing curvature

Besides the above expression using the connexion coëfficients, see the Vielbein method for computing curvature.