Connection (affine bundle)
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Let Y → X be an affine bundle modelled over a vector bundle Y → X. A connection Γ on Y → X is called the affine connection if it as a section Γ : Y → J1Y of the jet bundle J1Y → Y of Y is an affine bundle morphism over X. In particular, this is the case of an affine connection on the tangent bundle TX of a smooth manifold X.
With respect to affine bundle coordinates (xλ, yi) on Y, an affine connection Γ on Y → X is given by the tangent-valued connection form
An affine bundle is a fiber bundle with a general affine structure group GA(m, ℝ) of affine transformations of its typical fiber V of dimension m. Therefore, an affine connection is associated to a principal connection. It always exists.
For any affine connection Γ : Y → J1Y, the corresponding linear derivative Γ : Y → J1Y of an affine morphism Γ defines a unique linear connection on a vector bundle Y → X. With respect to linear bundle coordinates (xλ, yi) on Y, this connection reads
Since every vector bundle is an affine bundle, any linear connection on a vector bundle also is an affine connection.
If Y → X is a vector bundle, both an affine connection Γ and an associated linear connection Γ are connections on the same vector bundle Y → X, and their difference is a basic soldering form on
Thus, every affine connection on a vector bundle Y → X is a sum of a linear connection and a basic soldering form on Y → X.
It should be noted that, due to the canonical vertical splitting VY = Y × Y, this soldering form is brought into a vector-valued form
where ei is a fiber basis for Y.
Given an affine connection Γ on a vector bundle Y → X, let R and R be the curvatures of a connection Γ and the associated linear connection Γ, respectively. It is readily observed that R = R + T, where
is the torsion of Γ with respect to the basic soldering form σ.
In particular, let us consider the tangent bundle TX of a manifold X coordinated by (xμ, ẋμ). There is the canonical soldering form
on TX which coincides with the tautological one-form
on X due to the canonical vertical splitting VTX = TX × TX. Given an arbitrary linear connection Γ on TX, the corresponding affine connection
on TX is the Cartan connection. The torsion of the Cartan connection A with respect to the soldering form θ coincides with the torsion of a linear connection Γ, and its curvature is a sum R + T of the curvature and the torsion of Γ.
- Connection (fibred manifold)
- Affine connection
- Connection (vector bundle)
- Connection (mathematics)
- Affine gauge theory
- Kobayashi, S.; Nomizu, K. (1996). Foundations of Differential Geometry. 1–2. Wiley-Interscience. ISBN 0-471-15733-3.
- Sardanashvily, G. (2013). Advanced Differential Geometry for Theoreticians. Fiber bundles, jet manifolds and Lagrangian theory. Lambert Academic Publishing. arXiv: . ISBN 978-3-659-37815-7.
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