# Connection on a principal bundle

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## 1 Definition


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

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

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

## 2 Examples

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

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

For further information see [Kobayashi&Nomizu1963].