Connection on a principal bundle
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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]]). | 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> | </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 == | == References == | ||
{{#RefList:}} | {{#RefList:}} |
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1 Definition
Let be a Lie group with Lie algebra and a principal bundle for over a smooth manifold . A connection on is a distribution (a subbundle of the tangent bundle) on , called the "horizontal distribution", which is -invariant and complementary to the vertical distribution on .
The decomposition can be given by the projection onto the vertical distribution. Since each vertical space can be identified with (see Principal bundle), this map can be viewed as a -valued 1-form on , a linear map ; this is called the connection form.
The -valued 2-form is called curvature form and measures the non-integrability of the distribution , see the theory page Connections for details.
A connection on a -principal bundles induces a distribution on any associated bundle (see Principal bundle) since passes trivially to and by -invariance to . The induced distribution is called a connection on . If is a vector bundle (the action of on is linear), the connection on is closely related to a covariant derivative (see Connections).
2 Examples
A (semi-)Riemannian metric on defines a connection on the orthonormal frame bundle , the Levi-Civita connection: The horizontal space at some orthonormal basis of consists of the derivatives of all curves in which are parallel along their base point curve in .
Another type of example is the canonical connection on the principal bundle of a reductive homogeneous space .
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