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How can we differentiate a section in a vector bundle over a manifold ? It takes values in different vector spaces for each , but differentiating involves comparing values at different points . This needs an extra structure on which connects the different vector spaces , , among each other; it is therefore called connection. It can be defined in two different ways. Either it is viewed as a linear map which assigns to each vector field on a so called covariant derivative, a differential operator on the space of sections on . Or else it is given as a parallel displacement along curves in : Given two points and a curve connecting these points, a connection allows to move any element of to along . Infinitesimally it is given by a distribution on or on its frame bundle . Let us repeat: In ordinary analysis we differentiate functions with values in a constant vector space; differentiating functions with values in a variable vector space (a vector bundle) needs an extra structure called connection. The greater generality leads to a new notion: curvature. In the covariant derivative model, curvature is the commutator of covariant derivatives in two coordinate directions. For ordinary derivatives this quantity vanishes, and for covariant derivatives it is an algebraic quantity (a tensor) rather than a differential operator. In the parallel displacement model, curvature is just the non-integrability of the distribution, the tensor . A case of particular importance is the tangent bundle, . A connection on yields a second tensor quantity beneath curvature, the so called torsion. This is explained best by passing to the affine frame bundle; therefore a connection on is sometimes called affine connection.
2 Covariant derivatives
A covariant derivative on a vector bundle over a smooth manifold is a directional derivative for sections of . It can be viewed as a bilinear map , which is a tensor (linear over ) in the first argument and a derivation in the second argument:
where is a smooth function and a vector field on and a section of , and where is the ordinary derivative of the function in the direction of . By these properties, is defined locally and even pointwise regarding the first argument: For any we may define where is any (local) vector field with .
A covariant derivative has all properties of the ordinary directional derivative for functions with exception of the commutativity: For functions we have with , where is a local diffeomorphism (local parametrization of ) and its -th partial derivative. Instead, for covariant derivatives of a section on a vector bundle , the quantity
is in general nonzero but just a tensor (rather than a differential operator): . For arbitrary vector fields with and we put
where denotes the Lie bracket of vector fields.
4 Moving frames
A moving frame on is an -tuple of sections on some open subset such that for any the vectors form a basis of . Denoting by the linear map , we have
where we have used and for arbitrary 1-forms . On the other hand we let
for some 2-forms called (local) curvature forms, and thus we obtain the following relation between connection and curvature forms:
called Cartan Structure equations. Using matrix notation , , we may write the latter equation in the short form
5 Parallelity and connection
A section defined along an injective smooth curve is called parallel if . This is a linear ODE for which is solvable along with arbitrary initial values. If is a basis of parallel sections and is an arbitrary section along , then . Thus a covariant derivative is determined by its notion of parallelity and vice versa. Parallelity is given by a distribution on , i.e.\ a subbundle , where for consists of the initial vectors of parallel sections with along curves starting at . Stated differently, where is a local section defined near such that and for all . Vice versa, a section along any curve (a smooth map with for all ) is parallel iff for all . Since this distribution "connects" the distinct fibres of among each other, it is called a "connection".
For any piecewise smooth curve from to and any initial value we have a parallel section along with . The mapping is an invertible linear map called parallel transport along the curve . In general, parallel transport depends on the curve itself, not only on the end points , but it is independent of the parametrization of . This dependence is measured by the holonomy group at which is the set of parallel transports along all loops at , i.e.piecewise smooth curves with . It is known by the Ambrose-Singer theorem [Kobayashi&Nomizu1963, Theorem 8.1] that the connected component of is a Lie subgroup and its Lie algebra is spanned by the linear maps for all curves starting from and all where .
6 Connection on the frame bundle
Since it is useful to work with frames instead of single sections, we may replace with the linear frame bundle whose fibre over is the set of all frames (bases) of the vector space . Then parallelity of a frame , i.e. parallelity of all sections in this frame, is expressed by a distribution on which is also called . Together with the the "vertical space" , the tangent space of the fibre through , it yields a direct decomposition and hence we have projections and of onto the two subbundles. is a principal fibre bundle for the group , i.e.the fibres are the orbits of a free action of from the right given by where is a frame and a matrix and where is the line with -th component . Fixing , the action , is a diffeomorphism which is equivariant with respect to right translation, and the left invariant vector fields on are turned by into vector fields on tangent to the fibres, so called fundamental vector fields. In particular, the vertical space is canonically isomorphic to the Lie algebra of , via the infinitesimal action . Using this identification, the projection is a linear form on with values in ; it will be called (global) connection form . The form in Section 4 will be better called since it depends on a moving frame . We have
7 Curvature on the frame bundle
We get the same Cartan structure equations as in Section 4
where the (global) curvature form is given by
for all vector fields on . To prove (10) we work with two special kinds of vector fields on , the fundamental vertical fields etc. which are of type for some , and the horizontal lifts etc. which are horizontal vector fields projecting to a constant tangent vector on along every fibre; they are -related to a vector field on . Choosing and similarly , we have and and hence since are constant elements of . What remains is
8 Connections on general principal bundles
More generally, let be a -principal bundle over : A manifold with a smooth submersion and a free action of a Lie group on from the right such that the orbits are precisely the fibres, the preimages , . A connection on is a -invariant distribution on (also called the "horizontal distribution") which is complementary to the tangent spaces of the fibres forming the "vertical distribution" . As before, each vertical space can be identified with the Lie algebra of , and thus the vertical projection can be viewed as a -valued 1-form , and the equations (10) and (11) hold accordingly. If , then splits geometrically as at least locally, and (10) becomes the Maurer-Cartan equation of the Lie group . For any smooth action of on a smooth manifold we consider the associated bundle where the action on is given by . This is a bundle over with fibre , and since the distribution is -invariant, it can be transferred to via , defining a connection on . In the case for a vector bundle and with its linear -action we have , using the map , . This map is obviously invariant under the diagonal -action on since ; it is the usual identification of with the vector space by means of the basis .
9 Connections on the tangent bundle
The tangent bundle is somewhat special since it carries another 1-form besides . In the moving frame language where a local frame of is given on an open subset , any vector field can be written as . The coefficients depend linearly on , and we may write where the 1-forms om form the dual basis of , i.e. . Thus
If we have a covariant derivative on and another vector field , we obtain
from which we derive (interchanging the roles of and in the second term)
This tensor is called Torsion tensor; letting
for some and putting (called torsion form) and (sometimes called soldering form), we end up with the second Cartan structure equation
The following section explains why beneath (11) a second equation occurs for .
10 Affine connections
An affine frame on is a pair where is a frame of and . This is acted on from the right by the affine group which consists of the inhomogeneous linear transformations on with and : we let
where the frame is considered as the isomorphism mapping the standard basis vector onto . This action turns the set of affine frames on into a -principal bundle. A connection on the -principal bundle will be called generalized affine connection. Its connection and curvature forms , are -valued where is the Lie algebra of . Since , the forms split accordingly into a matrix and a vector component. Now we consider the embedding with . For the pull back forms on we have the same splitting:
where the first term on the right takes values in , the second on in . Moreover, the Cartan structure equations for the affine group are
using agian the splitting . If has the special property (12), we call the connection affine, and equals the torsion form as introduced in the last section.
- [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