Connection on a principal bundle

Latest revision as of 12:26, 22 May 2013

1 Definition

Let ${{Authors|Jost Eschenburg}}{{Stub}} == Definition == <wikitex include>; Let $G$ be a Lie group with Lie algebra $\frak{g}$ and $F \to M$ a [[Principal bundle|principal bundle]] for $G$ over a smooth manifold $M$. A ''connection'' on $F$ is a ''distribution'' (a subbundle of the tangent bundle) $\mathcal{H} \subset TF$ on $F$, called the "horizontal distribution", which is $G$-invariant and complementary to the vertical distribution $\mathcal{V}$ on $F$. The decomposition $TF = \mathcal{V} \oplus \mathcal{H}$ can be given by the projection $\pi_\mathcal{V} : TF \to \mathcal{V}$ onto the vertical distribution. Since each vertical space $\mathcal{V}_f$ can be identified with $\frak{g}$ (see [[Principal bundle]]), this map $\pi_V$ can be viewed as a $\frak{g}$-valued 1-form on $F$, a linear map $\omega : TF \to \frak{g}$; this is called the ''connection form''. The $\frak{g}$-valued 2-form $\Omega := d\omega + [\omega,\omega]$ is called ''curvature form'' and measures the non-integrability of the distribution $\mathcal{H}$, see the theory page [[Connections]] for details. A connection $\mathcal{H}$ on a $G$-principal bundles $F$ induces a distribution on any associated bundle $E = (F \times E_o)/G$ (see [[Principal bundle]]) since $\mathcal{H}$ passes trivially to $F \times E_o$ and by $G$-invariance to $E$. The induced distribution is called a ''connection on $E$''. If $E_o$ is a vector bundle (the action of $G$ on $E_o$ is linear), the connection on $E$ is closely related to a [[covariant derivative]] (see [[Connections]]). </wikitex> == References == {{#RefList:}} [[Category:Definitions]]G$ be a Lie group with Lie algebra $\frak{g}$ and $F \to M$ a principal bundle for $G$ over a smooth manifold $M$. A connection on $F$ is a distribution (a subbundle of the tangent bundle) $\mathcal{H} \subset TF$ on $F$, called the "horizontal distribution", which is $G$-invariant and complementary to the vertical distribution $\mathcal{V}$ on $F$.

The decomposition $TF = \mathcal{V} \oplus \mathcal{H}$ can be given by the projection $\pi_\mathcal{V} : TF \to \mathcal{V}$ onto the vertical distribution. Since each vertical space $\mathcal{V}_f$ can be identified with $\frak{g}$ (see Principal bundle), this map $\pi_V$ can be viewed as a $\frak{g}$-valued 1-form on $F$, a linear map $\omega : TF \to \frak{g}$; this is called the connection form.

The $\frak{g}$-valued 2-form $\Omega := d\omega + [\omega,\omega]$ is called curvature form and measures the non-integrability of the distribution $\mathcal{H}$, see the theory page Connections for details.

A connection $\mathcal{H}$ on a $G$-principal bundles $F$ induces a distribution on any associated bundle $E = (F \times E_o)/G$ (see Principal bundle) since $\mathcal{H}$ passes trivially to $F \times E_o$ and by $G$-invariance to $E$. The induced distribution is called a connection on $E$. If $E_o$ is a vector bundle (the action of $G$ on $E_o$ is linear), the connection on $E$ is closely related to a covariant derivative (see Connections).

2 Examples

A (semi-)Riemannian metric on $M$ defines a connection on the orthonormal frame bundle $F$, the Levi-Civita connection: The horizontal space $\mathcal{H}_f$ at some orthonormal basis $f$ of $T_pM$ consists of the derivatives of all curves $f(t)$ in $F$ which are parallel along their base point curve $p(t)$ in $M$.

Another type of example is the canonical connection on the principal bundle $G \to G/H$ of a reductive homogeneous space $M = G/H$.

For further information see [Kobayashi&Nomizu1963].

References

• [Kobayashi&Nomizu1963] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002