Connection form

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In mathematics, and specifically differential geometry, the connection form captures the invariant aspects of the connection on principal bundles, vector bundles and line bundles. In a certain sense, it captures the idea of Christoffel symbols on a Riemannian manifold and re-expresses this idea in a more general way, so that it is applicable on a principal bundle.

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[edit] Principal bundles

For a principal G-bundle E\to B, for each x\in E let Tx(E) denote the tangent space at x and Vx the vertical subspace tangent to the fiber . Then connection is an assignment of a horizontal subspace Hx of Tx(E) such that

  1. Tx(E) is direct sum of Vx and Hx,
  2. The distribution of Hx is invariant with respect to the G-action on E, i.e. Hax = Dx(Ra)Hx for any x\in E and a\in G, here Dx(Ra) denotes the differential of the group action by a at x.
  3. The distribution Hx depends smoothly on x.

This can be recast more elegantly using the jet bundle JE \rightarrow E. The assignment of a horizontal subspace at each point is none other than a smooth section of this jet bundle.

The one-parameter subgroups of G act vertically on E. The differential of this action allows one to identify the subspace Vx with the Lie algebra g of group G, say by map \iota:V_x\to g. Then the connection form is a form ω on E with values in g defined by \omega(X)=\iota\circ v(X) where v denotes projection at x \in E of X \in T_x to Vx with kernel Hx.

The connection form satisfies the following two properties:

  • The connection transforms equivariantly under the G action: R_h^*\omega=\hbox{Ad}(h^{-1})\omega for all h∈G.
  • The connection maps vertical vector fields to their associated elements of the Lie algebra: ω(X) = ι(X) for all X∈V.

Conversely, it can be shown that such a g-valued 1-form on a principal bundle generates a horizontal distribution satisfying the aforementioned properties.

Given a local trivialization one can reduce ω to the horizontal vector fields (in this trivialization). It defines form say ω' on B via pullback. The form ω' defines ω completely, but it depends on the choice of trivialization. (This form is often also called a connection form and denoted also by ω.)

[edit] Related definitions

[edit] Exterior covariant derivative

The exterior covariant derivative is a very useful notion which makes it possible to simplify formulas using a connection. Given a tensor-valued differential k-form φ its exterior covariant derivative Dφ is defined by

Dφ(X0,X1,...,Xk): = dφ(h(X0),h(X1),...,h(Xk)),

where h denotes the projection to the horizontal subspace, Hx with kernel Vx and Xi are arbitrary vector fields on E.

[edit] Curvature form

The curvature form Ω, a g-valued 2-form, can be defined by

\Omega=d\omega +{1\over 2}[\omega,\omega]=D\omega,

where [ * , * ] denotes the Lie bracket. This equation is also called the second structure equation.

[edit] Torsion

For the connection on a frame bundle, the curvature is not the only invariant of connection since the additional structure should be taken into account. Namely one has an extra canonical Rn-valued form θ = θi on E defined by identity

X=\sum_i\theta^i(X)e_i.\,

Then the torsion form, an Rn-valued 2-form can be defined by

\Theta=d\theta+{1\over 2}[\omega, \theta]=D\theta.

This equation is also called the first structure equation.

[edit] Vector bundles

The connection form for the vector bundle is the form on the total space of the associated principal bundle, but it can also be completely described by the following form (on the base in a not invariant way). This subsection can be considered as a smoother but somewhat inaccurate introduction to connection forms.

A covariant derivative on a vector bundle is a way to "differentiate" bundle sections along tangent vectors; it is also sometimes called a connection. Let \zeta:E\to B be a vector bundle over a smooth manifold B with an n-dimensional vector space F as a fiber. Let us denote by \nabla_uv a section of the vector bundle, the result of differentiation of the section of vector bundle v along the tangent vector field u. In order to be a covariant derivative, \nabla must satisfy the following identities:

(i) \nabla_u(v_1+v_2)=\nabla_uv_1+\nabla_uv_2 and \nabla_{u_1+u_2}v=\nabla_{u_1}v+\nabla_{u_2}v (linearity)
(ii) \nabla_u(fv)=df(u) v +f\nabla_uv and \nabla_{f u}v=f\nabla_{u}v for any smooth function f.

The simplest example: if \zeta:E=F\times B \to B is the projection, i.e. ζ is a trivial vector bundle, then any section can be described by a smooth map v:B\to F. Therefore, one can consider the trivial covariant derivative defined by partial derivatives: \nabla_u v=\partial v/\partial u.

If one has two connections \nabla and \nabla' on the same vector bundle then the difference \omega(u)v=\nabla_uv-\nabla'_uv depends only on values of u and v at a point. ω is a 1-form on B with values in Hom(F,F); i.e. \omega(u)\in Hom(F,F) and ω can be described as an n\times n-matrix of one-forms. In particular, if one chooses a local trivialization of the vector bundle and takes \nabla' to be the corresponding trivial connection, then ω gives a complete local description of \nabla.

The choice of trivialization is equivalent to choosing frames in each fiber; this explains the reason for the name method of moving frames. Let us choose (a local smooth section of) basis frames ei in fibers. Then the matrix of 1-forms \omega=\omega_i^j is defined by the following identity:

\nabla_u e_i=\sum_j\omega^j_i(u)e_j.

If G\subset GL(F) is the structure group of the vector bundle and the connection \nabla respects the group structure then the form ω is a 1-form with values in g, the Lie algebra of G. In particular, for the tangent bundle of a Riemannian manifold we have O(n) as the structure group and the form ω for the Levi-Civita connection takes values in so(n), the Lie algebra of O(n) (which can be thought of as antisymmetric matrices in an orthonormal basis).

[edit] Related definitions

[edit] Curvature

The connection form (ω) describes a connection (\nabla) in a non-invariant way; it depends on the choice of local trivialization. The following construction extracts invariant information out of ω:

A 2-form with values in Hom(F,F) is called curvature form if it can be written as

\Omega=d\omega +\omega\wedge\omega,

where d stands for exterior derivative and \wedge is the wedge product. This equation also called the second structure equation.

[edit] Torsion

For the connection on tangent bundle, the curvature is not the only invariant of the connection since the additional structure should be taken into account. Namely, one has an extra canonical Rn-valued form θ = θi on B defined by identity

X = ∑ θi(X)ei.
i

Then the torsion, an Rn-valued 2-form, can be defined by

\Theta=d\theta+\omega\wedge \theta\ \ \mbox{or} \ \ \Theta^i=d\theta^i+\sum_j\omega^i_j\wedge \theta^j.

This equation is also called the first structure equation.

[edit] See also

[edit] References

  • Kobayashi, Shoshichi; Nomizu, Katsumi; Foundations of differential geometry Vol. I. Reprint of the 1963 original. Wiley Classics Library. A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996. xii+329 pp. ISBN 0-471-15733-3
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