Connection (mathematics)

Connections are of central importance in modern geometry in large part because they allow a comparison between the local geometry at one point and the local geometry at another point. Differential geometry embraces several variations on the connection theme, which fall into two major groups: the infinitesimal and the local theory. The local theory concerns itself primarily with notions of parallel transport and holonomy. The infinitesimal theory concerns itself with the differentiation of geometric data. Thus a covariant derivative is a way of specifying a derivative of a vector field along another vector field on a manifold. A Cartan connection is a way of formulating some aspects of connection theory using differential forms and Lie groups. An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field. A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle.

Connections also lead to convenient formulations of geometric invariants, such as the curvature (see also curvature tensor and curvature form), and torsion tensor.

Motivation: the unsuitability of coordinates

Consider the following problem. Suppose that a tangent vector to the sphere S is given at the north pole, and we are to define a manner of consistently moving this vector to other points of the sphere: a means for parallel transport. Naively, this could be done using a particular coordinate system. However, unless proper care is applied, the parallel transport defined in one system of coordinates will not agree with that of another coordinate system. A more appropriate parallel transportation system exploits the symmetry of the sphere under rotation. Given a vector at the north pole, one can transport this vector along a curve by rotating the sphere in such a way that the north pole moves along the curve without axial rolling. This latter means of parallel transport is the Levi-Civita connection on the sphere. If two different curves are given with the same initial and terminal point, and a vector v is rigidly moved along the first curve by a rotation, the resulting vector at the terminal point will be different from the vector resulting from rigidly moving v along the second curve. This phenomenon reflects the curvature of the sphere. A simple mechanical device that can be used to visualize parallel transport is the south-pointing chariot.

For instance, suppose that S is a sphere given coordinates by the stereographic projection. Regard S as consisting of unit vectors in R³. Then S carries a pair of coordinate patches corresponding to the projections from north pole and south pole. The mappings

${\displaystyle \begin{array}{cc}{\phi }_0(x,y)& =\left(\frac{2x}{1+x^2+y^2},\frac{2y}{1+x^2+y^2},\frac{1-x^2-y^2}{1+x^2+y^2}\right)\\ {\phi }_1(x,y)& =\left(\frac{2x}{1+x^2+y^2},\frac{2y}{1+x^2+y^2},\frac{x^2+y^2-1}{1+x^2+y^2}\right)\end{array}}$

cover a neighborhood U₀ of the north pole and U₁ of the south pole, respectively. Let X, Y, Z be the ambient coordinates in R³. Then φ₀ and φ₁ have inverses

${\displaystyle \begin{array}{cc}{\phi }_0^{-1}(X,Y,Z)& =\left(\frac{X}{Z+1},\frac{Y}{Z+1}\right),\\ {\phi }_1^{-1}(X,Y,Z)& =\left(\frac{-X}{Z-1},\frac{-Y}{Z-1}\right),\end{array}}$

so that the coordinate transition function is inversion in the circle:

${\displaystyle {\phi }_{01}(x,y)={\phi }_0^{-1}\circ {\phi }_1(x,y)=\left(\frac{x}{x^2+y^2},\frac{y}{x^2+y^2}\right)}$

Let us now represent a vector field ${\displaystyle v}$ on S (an assignment of a tangent vector to each point in S) in local coordinates. If P is a point of U₀ ⊂ S, then a vector field may be represented by the pushforward of a vector field v₀ on R² by ${\displaystyle {\phi }_0}$:

where ${\displaystyle J_{{\phi }_0}}$ denotes the Jacobian matrix of φ₀ (${\displaystyle d{{\phi }_0}_x(\mathbf{u})=J_{{\phi }_0}(x)\cdot \mathbf{u}}$), and v₀ = v₀(x, y) is a vector field on R² uniquely determined by v (since the pushforward of a local diffeomorphism at any point is invertible). Furthermore, on the overlap between the coordinate charts U₀ ∩ U₁, it is possible to represent the same vector field with respect to the φ₁ coordinates:

To relate the components v₀ and v₁, apply the chain rule to the identity φ₁ = φ₀ o φ₀₁:

${\displaystyle J_{{\phi }_1}\left({\phi }_1^{-1}(P)\right)=J_{{\phi }_0}\left({\phi }_0^{-1}(P)\right)\cdot J_{{\phi }_{01}}\left({\phi }_1^{-1}(P)\right).}$

Applying both sides of this matrix equation to the component vector v₁(φ₁⁻¹(P)) and invoking (1) and (2) yields

We come now to the main question of defining how to transport a vector field parallelly along a curve. Suppose that P(t) is a curve in S. Naïvely, one may consider a vector field parallel if the coordinate components of the vector field are constant along the curve. However, an immediate ambiguity arises: in which coordinate system should these components be constant?

For instance, suppose that v(P(t)) has constant components in the U₁ coordinate system. That is, the functions v₁(φ₁⁻¹(P(t))) are constant. However, applying the product rule to (3) and using the fact that dv₁/dt = 0 gives

${\displaystyle \frac{d}{dt}{\mathbf{v}}_0\left({\phi }_0^{-1}(P(t))\right)=\left(\frac{d}{dt}{J}_{{\phi }_{01}}\left({\phi }_1^{-1}(P(t))\right)\right)\cdot {\mathbf{v}}_1\left({\phi }_1^{-1}\left(P(t)\right)\right).}$

But ${\displaystyle \left(\frac{d}{dt}{J}_{{\phi }_{01}}\left({\phi }_1^{-1}(P(t))\right)\right)}$ is always a non-singular matrix (provided that the curve P(t) is not stationary), so v₁ and v₀ cannot ever be simultaneously constant along the curve.

Resolution

The problem observed above is that the usual directional derivative of vector calculus does not behave well under changes in the coordinate system when applied to the components of vector fields. This makes it quite difficult to describe how to translate vector fields in a parallel manner, if indeed such a notion makes any sense at all. There are two fundamentally different ways of resolving this problem.

The first approach is to examine what is required for a generalization of the directional derivative to "behave well" under coordinate transitions. This is the tactic taken by the covariant derivative approach to connections: good behavior is equated with covariance. Here one considers a modification of the directional derivative by a certain linear operator, whose components are called the Christoffel symbols, which involves no derivatives on the vector field itself. The directional derivative D_uv of the components of a vector v in a coordinate system φ in the direction u are replaced by a covariant derivative:

${\displaystyle {\mathrm{\nabla }}_{\mathbf{u}}\mathbf{v}=D_{\mathbf{u}}\mathbf{v}+\mathrm{\Gamma }(\phi )\{\mathbf{u},\mathbf{v}\}}$

where Γ depends on the coordinate system φ and is bilinear in u and v. In particular, Γ does not involve any derivatives on u or v. In this approach, Γ must transform in a prescribed manner when the coordinate system φ is changed to a different coordinate system. This transformation is not tensorial, since it involves not only the first derivative of the coordinate transition, but also its second derivative. Specifying the transformation law of Γ is not sufficient to determine Γ uniquely. Some other normalization conditions must be imposed, usually depending on the type of geometry under consideration. In Riemannian geometry, the Levi-Civita connection requires compatibility of the Christoffel symbols with the metric (as well as a certain symmetry condition). With these normalizations, the connection is uniquely defined.

The second approach is to use Lie groups to attempt to capture some vestige of symmetry on the space. This is the approach of Cartan connections. The example above using rotations to specify the parallel transport of vectors on the sphere is very much in this vein.