# Curvature form

From Encyclopedia of Mathematics

A -form on a principal fibre bundle with structure Lie group , taking values in the Lie algebra of the group and defined by the connection form on by the formula

The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition . It satisfies the Bianchi identity

and defines the holonomy algebra (see Holonomy group).

#### Comments

The equation is called the structure equation.

#### References

[a1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. V, VI |

**How to Cite This Entry:**

Curvature form.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Curvature_form&oldid=16422

This article was adapted from an original article by Ãœ. Lumiste (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article