### Notes on Ehresmann’s connections -Part 1

Ehresmann’s connection

Consider a fibre bundle with a fibred chart where are local coordinates on the base manifold and are coordinates along the fibres. If we have two fibred charts, then on their intersection domain we have

The chart induces vector fields on the tangent space

with the transformation laws:

Therefore

and the inverse

Vector fields span what is called the vertical subspace in This subspace, consisting of vectors tangent to the fibers, is independent of the chart. There are no a priori distinguished `horizontal’ subspaces. The subspaces determined by vector fields change when the charts change. A preferred distribution of horizontal subspaces is determined by what is called an Ehresmann connection. Once such a connection is given, vector fields tangent to can be lifted to horizontal vector fields tangent to

Remark: In principle we should use a different notation for vector fields tangent to and vector fields induced on by a chart Using the same notation for both can lead to misunderstandings when is not paying attention to the context.

Let us denote by the horizontal lifts induced by an Ehresmann connection on They can be written uniquely in the form:

Remark: In Finsler geometry instead of one usually uses so that the formula for the horizontal lift reads

The functions may be called the coefficient functions of the Ehresmann connection with respect to the chart

If is a vector field on we denote its horizontal lift by thus

Suppose now we have two charts, and Corresponding to these two charts we will have two sets of horizontal lifts of the same connection and Since both are supposed to describe the same horizontal distribution, they must be related by an invertible matrix :

Substituting now the definitions we have

or

Comparing the terms we find that

and therefore

We can distinguish two special classes of transformations. First class consists of changes of coordinates on without reparametrizing of the fibers. For these transformations transforms like one–forms:

The second class consists of transformations of the form For these transformations we have

{\bf Remark}: If is a vector bundle, then the reparametrizations of the fibers compatible with the vector bundle structure are linear and of the form:

In this case the last formula reads:

Curvature

Given an Ehresmann connection represented by horizontal vector fields we can calculate their commutator. As it happens their commutator is a vertical vector field that can be written as

where

More generally, for any two vector fields on one defines

If are functions on then

Under coordinate transformations transforms like a tensor:

Under fiber reparametrization we have:

Connection form

It is convenient to code the connection given by a horizontal distribution in one geometrical object. This can be done by introducing the curvature form – a one-form on with values in the vertical distribution. Given a vector tangent to at we decompose it into a horizontal and vertical part and define the connection form on this vector as the vertical part. It follows that the connection form is the identity map on vertical vectors. In coordinates we can uniquely write the connection form as

The curvature form vanishes automatically on the horizontal vectors therefore it easily follows that