Riemannian Connection On A Surface - Principal Connection

Principal Connection

See also: Connection form and Connection (principal bundle)

The theory of connections according to Élie Cartan, and later Charles Ehresmann, revolves around:

• a principal bundle E;
• the exterior differential calculus of differential forms on E.

All "natural" vector bundles associated with the manifold M, such as the tangent bundle, the cotangent bundle or the exterior bundles, can be constructed from the frame bundle using the representation theory of the structure group K = SO(2), a compact matrix group.

Cartan's definition of a connection can be understood as a way of lifting vector fields on M to vector fields on the frame bundle E invariant under the action of the structure group K. Since parallel transport has been defined as a way of lifting piecewise C1 paths from M to E, this automatically induces infinitesimally a way to lift vector fields or tangent vectors from M to E. At a point take a path with given tangent vector and then map it to the tangent vector of the lifted path. (For vector fields the curves can be taken to be the integral curves of a local flow.) In this way any vector field X on M can be lifted to a vector field X* on E satisfying

• X* is a vector field on E;
• the map X ↦ X* is C∞(M)-linear;
• X* is K-invariant and induces the vector field X on C∞(M) C∞(E).

Here K acts as a periodic flow on E, so the canonical generator A of its Lie algebra acts as the corresponding vector field, called the vertical vector field A*. It follows from the above conditions that, in the tangent space of an arbitrary point in E, the lifts X* span a two-dimensional subspace of horizontal vectors, forming a complementary subspace to the vertical vectors. The canonical Riemannian metric on E of Shigeo Sasaki is defined by making the horizontal and vertical subspaces orthogonal, giving each subspace its natural inner product.

Horizontal vector fields admit the following characterisation:

• Every K-invariant horizontal vector field on E has the form X* for a unique vector field X on M.

This "universal lift" then immediately induces lifts to vector bundles associated with E and hence allows the covariant derivative, and its generalisation to forms, to be recovered.

If σ is a representation of K on a finite-dimensional vector space V, then the associated vector bundle E X_K V over M has a C∞(M)-module of sections that can be identified with

the space of all smooth functions ξ : E → V which are K-equivariant in the sense that

for all x ∈ E and g ∈ K.

The identity representation of SO(2) on R2 corresponds to the tangent bundle of M.

The covariant derivative is defined on an invariant section ξ by the formula

The connection on the frame bundle can also be described using K-invariant differential 1-forms on E.

The frame bundle E is a 3-manifold. The space of p-forms on E is denoted Λp(E). It admits a natural action of the structure group K.

Given a connection on the principal bundle E corresponding to a lift X ↦ X* of vector fields on M, there is a unique connection form ω in

,

the space of K-invariant 1-forms on E, such that

for all vector fields X on M and

for the vector field A* on E corresponding to the canonical generator A of .

Conversely the lift X* is uniquely characterised by the following properties:

• X* is K-invariant and induces X on M;
• ω(X*)=0.