# Difference between revisions of "Curvature form"

From Encyclopedia of Mathematics

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+ | A $2$-form $\Omega$ on a principal fibre bundle $P$ with structure Lie group $G$, taking values in the Lie algebra $\mathfrak g$ of the group $G$ and defined by the [[Connection form|connection form]] $\theta$ on $P$ by the formula | ||

− | + | $$\Omega=d\theta+\frac12[\theta,\theta].$$ | |

− | The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition | + | The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition $\Omega\equiv0$. It satisfies the Bianchi identity |

− | + | $$d\Omega=[\Omega,\theta]$$ | |

and defines the holonomy algebra (see [[Holonomy group|Holonomy group]]). | and defines the holonomy algebra (see [[Holonomy group|Holonomy group]]). | ||

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====Comments==== | ====Comments==== | ||

− | The equation | + | The equation $\Omega=d\theta+[\theta,\theta]/2$ is called the structure equation. |

====References==== | ====References==== | ||

<table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1''' , Interscience (1963) pp. Chapt. V, VI</TD></TR></table> | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , '''1''' , Interscience (1963) pp. Chapt. V, VI</TD></TR></table> |

## Latest revision as of 12:45, 31 July 2014

A $2$-form $\Omega$ on a principal fibre bundle $P$ with structure Lie group $G$, taking values in the Lie algebra $\mathfrak g$ of the group $G$ and defined by the connection form $\theta$ on $P$ by the formula

$$\Omega=d\theta+\frac12[\theta,\theta].$$

The curvature form is a measure of the deviation of the given connection from the locally flat connection characterized by the condition $\Omega\equiv0$. It satisfies the Bianchi identity

$$d\Omega=[\Omega,\theta]$$

and defines the holonomy algebra (see Holonomy group).

#### Comments

The equation $\Omega=d\theta+[\theta,\theta]/2$ 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=32609

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