Covariant exterior derivative

In differential geometry, the connection form describes connection on principal bundles (or vector bundles). It can be considered as an generalization/alternative for Christoffel symbols.

Principal bundles

For a principal G-bundle <math>E\to B <math>, for each <math>x\in E<math> let <math>T_x(E)<math> denotes the tangent space at x and <math>V_x<math> the veritiacl subspace tangent to the fiber . Then connection is an assignment of a horizontal subspace <math>H_x<math> of <math>T_x(E)<math> such that

1. <math>T_x(E)<math> is direct sum of <math>V_x<math> and <math>H_x<math>,
2. The distribution of <math>H_x<math> is invariant with respect to the G-action on E, i.e. <math>H_{ax}=D_x(R_a)H_{x}<math> for any <math>x\in E<math> and <math>a\in G<math>, here <math>D_x(R_a)<math> denotes the differential of the group action by a at x.
3. The distribution <math>H_x<math> depends smoothly on x.

The subspace <math>V_x<math> can be naturally identifyed with the Lie algebra g of group G, say by map <math>\iota:V_x\to g<math>. Then the connection form is a form <math>\omega<math> on <math>E<math> whith values in g defined by <math>\omega(X)=\iota\circ v(X)<math> where <math>v<math> denotes projection of <math>x\in E<math> to <math>V_x<math> with kernel <math>H_x<math>.

Given a local trivialization one can reduce <math>\omega<math> to the horizontal vector fields (in this trivialization). It defined form say <math>\omega'<math> on B. The form <math>\omega'<math> defines <math>\omega<math> cmpletely, but it depends on the choice of trivialization. (This form often also called connection form and denoted also by <math>\omega<math>)

Related definitions

Exterior covariant derivative

The exterior covariant derivative is a very useful notion which makes possible to simplify formulas in using connection. Given a tensor-valued differential k-form <math>\phi <math> its exterior covariant derivative defined by

<math>D\phi(X_0,X_1,...,X_k)=d\phi(h(X_0),h(X_1),...,h(X_k))<math>

where h denotes the projection to the horizontal subspace, <math>H_x<math> with kernel <math>V_x<math> and <math>X_i<math> are arbitrary vector fields on E.

Curvature form

The curvature form <math>\Omega<math>, a g-valued 2-form, can be defined by

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

where <math>[*,*]<math> denotes the Lie bracket. This equasion also called the second structure equasion

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 <math>\theta=\theta^i<math> on E defined by identity <math> X=\sum_i\theta^i(X)e_i.<math>. Then the torsion form, an Rⁿ-valued 2-form can be defined by

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

This equasion also called the first structure equasion.

Vector bundles

The connection form for the vector bundle is the form on the total space of associated principle bundle, but it can be completely described by the following form (on the base on a NOT invariant way). This subsection can be considered as a smoother but bit wrong introduction to connection form.

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

(i) <math>\nabla_u(v_1+v_2)=\nabla_uv_1+\nabla_uv_2<math> and <math>\nabla_{u_1+u_2}v=\nabla_{u_1}v+\nabla_{u_2}v<math> (Linearity)

(ii) <math>\nabla_u(fv)=df(u) v +f\nabla_uv <math> and <math>\nabla_{f u}v=f\nabla_{u}v<math> for any smooth function <math>f<math>

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

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

The choice of trivialisation is equivalent to choice of frames in each fiber, that expaignes the reason for the name Method of moving frames. Let us choose (a local smooth section of) basis frames <math>e_i<math> in fibers. Then the matrix of 1-forms <math>\omega=\omega_i^j<math> is defined by the following identity:

<math>\nabla_u e_i=\sum_j\omega^j_i(u)e_j<math>

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

Related definitions

Curvature

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

The following 2-form with values in <math>Hom(F,F)<math> is called curvature form

<math>\Omega=d\omega +\omega\wedge\omega,<math>

where <math>d<math> stands for exterior derivative and <math>\wedge<math> is the wedge product. This equasion also called the second structure equasion.

Torsion

For the connection on tangent 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 <math>\theta=\theta^i<math> on B defined by identity <math> X=\sum_i\theta^i(X)e_i.<math>

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

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

This equation also called the first structure equation.

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