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Vector Bundle Connection


A connection on a vector bundle pi:E->M is a way to "differentiate" bundle sections, in a way that is analogous to the exterior derivative df of a function f. In particular, a connection del is a function from smooth sections Gamma(M,E) to smooth sections of E with one-forms Gamma(M,E tensor T^*M) that satisfies the following conditions.

1. del fs=s tensor df+fdel s (Leibniz rule), and

2. del s_1+s_2=del s_1+del s_2.

Alternatively, a connection can be considered as a linear map from bundle sections of E tensor TM, i.e., a section of E with a vector field X, to sections of E, in analogy to the directional derivative. The directional derivative of a function f, in the direction of a vector field X, is given by df(X). The connection, along with a vector field X, may be applied to a section s of E to get the section del _Xs. From this perspective, connections must also satisfy

 del _(fX)s=fdel _Xs
(1)

for any smooth function f. This property follows from the first definition.

For example, the trivial bundle E=M×R^k admits a flat connection since any bundle section s corresponds to a function s^~:M->R^k. Then setting del s=ds gives the connection. Any connection on the trivial bundle is of the form del s=ds+s tensor alpha, where alpha is any one-form with values in Hom(E,E)=E^* tensor E, i.e., alpha is a matrix of one-forms.

The matrix of one-forms

 alpha=[dx 2xdy 0; 0 dx-3dy 0; xydx 0 y^2dx+dy]
(2)

determines a connection del on the rank-3 bundle over R^2. It acts on a section s=(s_1,s_2,s_3) by the following.

del _(partial/partialx)s=s_x+alpha(partial/partialx)s
(3)
=s_x+[1 0 0; 0 1 0; xy 0 y^2]s
(4)
=((partials_1)/(partialx)+s_1,(partials_2)/(partialx)+s_2,(partials_3)/(partialx)+xys_1+y^2s_3)
(5)
del _(partial/partialy)s=s_y+alpha(partial/partialy)s
(6)
=s_y+[0 2x 0; 0 -3 0; 0 0 1]s
(7)
=((partials_1)/(partialx)+2xs_2,(partials_2)/(partialx)-3s_2,(partials_3)/(partialx)+s_3).
(8)

In any trivialization, a connection can be described just as in the case of a trivial bundle. However, if the bundle E is not trivial, then the exterior derivative ds is not well-defined (globally) for a bundle section s. Still, the difference between any two connections must be one-forms with values in endomorphisms of E,i.e., matrices of one forms. So the space of connections forms an affine space.

The bundle curvature of the bundle is given by the formula Omega=del  degreesdel . In coordinates, Omega=alpha ^ alpha is matrix of two-forms. For instance, in the example above,

 Omega=[0 2xdx ^ dy 0; 0 -3dx ^ dy 0; 0 2x^2ydx ^ dy y^2dx ^ dy]
(9)

is the curvature.

Another way of describing a connection is as a splitting of the tangent bundle TE of E as TM direct sum E. The vertical part of TE corresponds to tangent vectors along the fibers, and is the kernel of dpi:TE->TM. The horizontal part is not well-defined a priori. A connection defines a subspace of TE_((x,v)) which is isomorphic to TM_x. It defines k flat sections s_i such that del s_i=0, which are a vector basis for the fiber bundles of E, at least nearby x. These flat sections determine the horizontal part of TE near x. Also, a connection on a vector bundle can be defined by a principal bundle connection on the associated principal bundle.

In some settings there is a canonical connection. For example, a Riemannian manifold has the Levi-Civita connection, given by the Christoffel symbols of the first and second kinds, which is the unique torsion-free connection compatible with the metric. A holomorphic vector bundle with a Hermitian metric has a unique connection which is compatible with both metric and the complex structure.


See also

Bundle Curvature, Bundle Section, Bundle Torsion, Curvature, Hermitian Metric, Levi-Civita Connection, Parallel Transport, Principal Bundle, Principal Bundle Connection, Second Fundamental Form

This entry contributed by Todd Rowland

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Cite this as:

Rowland, Todd. "Vector Bundle Connection." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/VectorBundleConnection.html

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