Connection on a principal bundle

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1 Definition

Let $G$${{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 of smooth manifolds|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 of smooth manifolds|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 of smooth manifolds|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> == Examples == <wikitex>; A (semi-)Riemannian metric on $M$ defines a connection on the orthonormal frame bundle $F$, the [[Levi-Civita connection|''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|''canonical connection'']] on the principal bundle $G \to G/H$ of a reductive homogeneous space $M = G/H$. For further information see \cite{Kobayashi&Nomizu1963}. == References == {{#RefList:}} [[Category:Definitions]] [[Category:Connections and curvature]]G$ be a Lie group with Lie algebra $\frak{g}$$\frak{g}$ and $F \to M$$F \to M$ a principal bundle for $G$$G$ over a

smooth manifold $M$$M$. A connection on

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$F$ is a distribution

(a subbundle of the tangent bundle) $\mathcal{H} \subset TF$$\mathcal{H} \subset TF$

on

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$F$, called the "horizontal distribution", which is $G$$G$-invariant and complementary to the vertical distribution $\mathcal{V}$$\mathcal{V}$ on

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$F$.

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

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$F$,

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

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$F$ 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

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$E$. The induced distribution is called a connection on

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$E$. If $E_o$$E_o$ is a vector bundle (the action of $G$$G$ on $E_o$$E_o$ is linear), the connection on

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$E$ 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

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$F$, 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

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$F$ 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].

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