**Principal Connection**

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 **R**2 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 Λ*p*(*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.

Read more about this topic: Riemannian Connection On A Surface

### Other articles related to "connection":

... An Ehresmann

**connection**on a fibre bundle (endowed with a structure group) sometimes gives rise to an Ehresmann

**connection**on an associated bundle ... For instance, a (linear)

**connection**in a vector bundle E, thought of giving a parallelism of E as above, induces a

**connection**on the associated bundle of ... Conversely, a

**connection**in PE gives rise to a (linear)

**connection**in E provided that the

**connection**in PE is equivariant with respect to the action of the general linear group ...

### Famous quotes containing the words connection and/or principal:

“Children of the same family, the same blood, with the same first associations and habits, have some means of enjoyment in their power, which no subsequent connections can supply; and it must be by a long and unnatural estrangement, by a divorce which no subsequent *connection* can justify, if such precious remains of the earliest attachments are ever entirely outlived.”

—Jane Austen (1775–1817)

“Heaven has a Sea of Glass on which angels go sliding every afternoon. There are many golden streets, but the *principal* thoroughfares are Amen Street and Hallelujah Avenue, which intersect in front of the Throne. These streets play tunes when walked on, and all shoes have songs in them.”

—For the State of Florida, U.S. public relief program (1935-1943)