Connections
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1 Introduction
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:
![\begin{matrix} \nabla_{(fX)}s &=& f\nabla_Xs\,\cr \nabla_{X}(fs) &=& (Xf)s + f\nabla_Xs \,, \end{matrix}](/images/math/0/3/9/03926605e39b127d570124ddb84c209c.png)
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
.
3 Curvature
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
![R_{ij}s := [\nabla_i,\nabla_j]s = \nabla_i\nabla_js - \nabla_j\nabla_is](/images/math/e/0/7/e07385894231745e1b0b4fbb9cb41373.png)
is in general nonzero but just a tensor (rather than a differential operator):
. For arbitrary vector fields
with
and
we put
![R(X,Y)s = \sum_{ij} \xi^i\eta^j R_{ij}s = [\nabla_X,\nabla_Y]s - \nabla_{[X,Y]}s](/images/math/a/d/2/ad2e427e6cc5cd8d760f3ac414e3e33e.png)
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
![\nabla s_j = \sum_i\omega_{ij} s_i](/images/math/4/c/1/4c1bcac3fd68004d7eef0210da8eb648.png)
![\omega_{ij}\in \Omega^1(M_o)](/images/math/b/a/b/bab2a24e9a2af2e726fadc57009b9e5f.png)
![\nabla_X\nabla_Ys_j = \sum_i \left\{X\omega_{ij}(Y)s_i + \omega_{ij}(Y)\sum_{k}\omega_{ki}(X)s_k\right\}](/images/math/5/9/f/59f5595c66b9b379f617285a15d43f09.png)
![k](/images/math/a/0/9/a09fe38af36f6839f4a75051dc7cea25.png)
![i](/images/math/a/1/6/a16d2280393ce6a2a5428a4a8d09e354.png)
![R(.,.)s_j = \sum_i \left(d\omega_{ij} + \sum_k \omega_{ik}\wedge \omega_{kj}\right)s_i](/images/math/1/7/6/17619bc1ea861eaf784696d48035564e.png)
where we have used and
for arbitrary 1-forms
. On the other hand we let
![R(.,.)s_j = \Omega_{ij}s_i](/images/math/9/f/5/9f58df9ba1c8c6164df01eab7dcf2985.png)
for some 2-forms called (local) curvature forms, and thus we obtain the following relation between connection and curvature forms:
![\Omega_{ij} = d\omega_{ij} + \sum_k \omega_{ik}\wedge \omega_{kj}](/images/math/2/a/c/2ac81a9896438497fc8ac60851b44dba.png)
![\Omega = (\Omega_{ij})](/images/math/3/8/7/387a18e1f924fc00d228cd76750fa1d7.png)
![\omega = (\omega_{ij})](/images/math/e/a/c/eac30358702fc8fd38a40770e0c98ec2.png)
![\Omega = d\omega + \omega\wedge\omega.](/images/math/9/3/5/93558f4e455685477bd36fe7d10fdbfb.png)
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|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
![\omega_{\textup{f}} = \textup{f}^*\omega.](/images/math/b/0/a/b0adaa4952d528e468481e63ba858739.png)
7 Curvature on the frame bundle
We get the same Cartan structure equations as in Section 4
![d\omega + \omega\wedge\omega = \Omega](/images/math/a/2/0/a209fa2bb3ac42882054462a267df775.png)
where the (global) curvature form is given by
![\Omega(U,V) = -\omega([\pi_{\mathcal{H}}U,\pi_{\mathcal{H}}V])](/images/math/9/8/2/98258f9b264db5b0a57b13da382d861b.png)
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
, using 11. Here we see the curvature form in a new r\^ole: it measures the non-integrability of the horizontal distribution:
is integrable
is horizontal
.
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 ``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
![Y = \sum_i \theta_i(Y)f_i](/images/math/c/a/b/cab297b43d8498deb1aa711d4db76113.png)
If we have a covariant derivative on
and another vector field
, we obtain
![\displaystyle \nabla_X Y = \sum_i \left\{X(\theta_i(Y))f_i + \theta_i(Y)\nabla_Xf_i\right\} = \sum_{ij} \left\{X(\theta_i(Y))f_i + \theta_i(Y)\omega_{ji}(X)f_j\right\}](/images/math/5/5/5/555416566f627fcc5e73c99532db0c71.png)
from which we derive (interchanging the roles of and
in the second term)
![\displaystyle T(X,Y): = \nabla_XY-\nabla_YX - [X,Y] = \sum_{ij}\left(d\theta_i + \omega_{ij}\wedge\theta_j\right)(X,Y)f_i.](/images/math/9/e/8/9e8fd6b52be12df9fc8a2d3af87c04b1.png)
This tensor is called Torsion tensor; letting
![T(.,,) = \sum\Theta_if_i](/images/math/2/7/8/278138d5cd626344ee66a3b9e159fa8b.png)
for some and putting
(called torsion form) and
(sometimes called soldering form), we end up with the second Cartan structure equation
![\Theta = d\theta + \omega \wedge \theta.](/images/math/e/7/6/e76d32eb9ae9354302ddf10eaabf7440.png)
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
![(f,v)(A,a) = (fA,v+f^{-1}a)](/images/math/0/3/d/03dec4d14933af5d1567d365d7bdf0ac.png)
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:
![\gamma^*\hat\omega = \omega \oplus \theta](/images/math/0/5/c/05c3af3ef9ca0d845b83b8dd5e3ca4ad.png)
![\gamma^*\hat\Omega = \Omega \oplus \Theta](/images/math/8/0/b/80b33ae4977034ed850b4e85b1a4e1f4.png)
where the first term on the right takes values in , the second on in
. Moreover, the Cartan structure equations for the affine group are
![\begin{matrix} \Omega &=& d\omega + \omega\wedge \omega,\\ \Theta &=& d\theta + \omega\wedge\theta, \end{matrix}](/images/math/e/3/b/e3b93eebd06cffcb444f877f75516ddc.png)
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.
11 References
- [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
12 External links
- The Encylopedia of Mathematics article on connections
- The Wikipedia page about connections