THE GEOMETRY OF VECTOR BUNDLES AND AN INTRODUCTION TO GAUGE THEORY LECTURE 17 Professor Steven Bradlow Class Notes From Math 433 University of Illinois at Urbana-Champaign March 2, 1998

Existence of Connections

Say that a bundle is described by

E = (∐ U_{α} × ℝ^{n})/{g_{αβ}}

Here are two ways to construct a connection on E. Both methods use connections, D_{α} defined on E|U_{α}, the local trivializations ψ_{α} : E|U_{α} → U_{α} × ℝ^{n} and a partition of unity, {ρ_{α}} subordinate to the cover {U_{α}}. The two connections are defined by

(a) D_{1} = ∑_{α} D_{α} ∘ ρ_{α}

(b) D_{2} = ∑_{α} ρ_{α}D_{α} Note. D_{α} was defined on E|U_{α} by:

(a) Picking a connection, D̃_{α} on U_{α} × ℝ^{n}.

(b) Setting D_{α} = ψ_{α}^{−1} ∘ D̃_{α} ∘ ψ_{α}, i.e, D_{α}(s) = ψ_{α}^{−1}(D̃_{α}(ψ_{α}(s))) Exercise 1. Show that this is a connection.

In fact, given a bundle isomorphism, h : E → F and a connection D on F, we define a connection, h^{∗}D, on E by h^{∗}D = h^{−1} ∘ D ∘ h.

In the definition of D_{2}, the terms in the sum are of the form

(ρ_{α}D_{α}(s))(b) = ρ_{α}(b)D_{α}(s)(b) = \{ \begin{matrix}ρ_{α}(b)D_{|}α(s)(b) & b ∈ U_{α} \\ 0 & otherwise \end{matrix}

Exercise 2. Check that D_{2} does define a connection.

We now examine the connection, D_{1}. We define

\begin{aligned}(D_{α} ∘ ρ_{α})(s) & = D_{α}(ρ · α(s)) \\ & = dρ_{α} ⊗ s + ρ_{α}D_{α}(s) \end{aligned}

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Using this, we see that

D_{1}(s) = D_{2}(s) + ∑_{α} dρ_{α} ⊗ s

[Note that off of U_{α}, dρ_{α} = 0 since the support of ρ_{α} is in U_{α}.] Exercise 3. Show that D_{1} defines a connection.

The diffrerence between D_{1} and D_{2} is instructive. Note. D_{1} − D_{2} is a End(E) valued 1-form. In fact

(D_{1} − D_{2}) = ∑_{α} dρ_{α} ⊗ Id
i.e, for any s ∈ Ω^{0}(E) (∑_{α} dρ_{α} ⊗ Id)(s) = ∑_{α} dρ_{α} ⊗ s

This illustrates a general fact. Proposition. If D_{1} and D_{2} are connections on a vector bundle, π : E → B, then

D_{1} − D_{2} ∈ Ω^{0}(B, T^{∗}B ⊗ End(E))
= Ω^{1}(B, End(E))

Proof: We check:

(a) This is linear with respect to constants.

(b) This is also linear with repsect to functions:

\begin{aligned}(D_{1} − D_{2})(fs) & = (df ⊗ s + fD_{1}(s)) − (df ⊗ s + fD_{2}(s)) \\ & = f(D_{1} − D_{2})(s) \end{aligned}

Note. With respect to local frames of E, say {e_{i}^{α}}_{i}^{rank E}, locally a section of T^{∗}M ⊗ End(E) is a matrix of 1-forms, A^{α}. Over U_{α}, A^{α} and A^{β} must be related by the transition functions,

A^{α} = g_{αβ}A^{β}g_{βα}

If (D_{1} − D_{2})(fs) = f(D_{1} − D_{2})(s), then the local descriptions (with respect to local frames) will also have this property. Exercise 4. Repeat the computation in the proof of (b) for connection 1-forms.

Summary: D_{1} − D_{2} ∈ Ω^{1}(B, End(E)). Conversely, if A ∈ Ω^{1}(B, End(E)), then we can define (D_{1} + A)(s) = D_{1}(s) + A(s). Exercise 5. Show that D_{1} + A is a connection.

Conclusion: If A(E) is the space of all connections on E, then A(E) = D_{0} + Ω^{1}(B, End(E)) where D_{0} is any fixed connection. That is, A(E) is an infinite dimensional affinve space based on Ω^{1}(E, End(E)). Parallel Transport Definition 1. Given a connection, D, and a vector field, X, then D_{X}s is called the covariant derivative of s along X. We can write it locally: D_{X}s = d_{X}s + (A(X))s, where D_{x}s ∈ Ω^{0}(B, E) and (D_{X}s)(b) = (d_{X_{b}}s) + A(X_{b})s(b), that is, it depends only on X_{b} ∈ T_{b}B.

Definition 2. If s ∈ Ω^{0}(B, E) satisfies Ds = 0, then we say that s is parallel.

Question. Can we find solutions to Ds = 0?

With respect to a local frame, if

s = ( \begin{matrix}s_{1} \\ s_{2} \\ … \\ s_{n} \end{matrix} )

and D = d + A_{1}, then the condition for being parallel is ds_{i} + A_{ij}s_{j} = 0 for all i. With local coordinates, (x_{1}, …, x_{k}), on B, we are thus looking at trying to solve the system of equations,

∑_{α=1}^{k} \frac{∂s_{i}}{∂x_{α}} + (A_{ij}^{α}s_{j})dx_{α} = 0 i = 1, …, n

where A_{ij} = A_{ij}^{α}dx_{α}. This is a system of partial differential equations for which existence of solutions is NOT guarenteed.

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