THE GEOMETRY OF VECTOR BUNDLES AND

AN INTRODUCTION TO GAUGE THEORY

LECTURE 16

Professor Steven Bradlow Class Notes From Math 433

University of Illinois at Urbana-Champaign

February 27, 1998

Connections on Vector Bundles II

We now begin a detailed treatment of connections, starting with:

Definition 1. For a vector bundle E → B, a connection is a map

D : Ω^{0}(B, E) → Ω^{0}(B, T^{∗}B ⊗ E)

such that (i) D is linear and (ii) D(fs) = df ⊗ s + fDs, for f ∈ C^{∞}(B, ℝ) and s ∈ Ω^{0}(B, E).

Lemma. (ii) guarantees that D is a local operator, that is, Ds(b) depends only on s “near b”.

Proof: Take a open neighborhood U of b and define a smooth bump function f on U such that f ≡ 1 on V (⊂ U) near b and f ≡ 0 outside U. Then D(fs) = df ⊗ s + fDs implies D(fs)(b) = Ds(b), but fs is zero outside U! □

Hence local descriptions of D are possible, that is, we can use local frames to get a local description of D.

Say E|_{U} ⟶^{≃^{Ψ}} U × ℝ is a local trivialization, with corresponding local frame {e_{i}}_{i=1}^{n}. Thus e_{i}(b) = Ψ^{−1}(b, ê_{i}), for ê_{i} a standard basis element for ℝ^{n}. Then, over U, with respect to this section, s has a local description s = ∑ s_{i}e_{i} with s_{i} : U → ℝ. Applying (i) and (ii) in Definition 1, we can thus write Ds = ∑ ds_{i}⊗e_{i}+s_{i}(De_{i}).

Claim.

De_{i} = ∑_{j} A_{ji} ⊗ e_{j}

where A_{ji} are 1-forms defined on U.

Proof: De_{i} is a (local) section of T^{∗}U ⊗ E and every section of T^{∗}U ⊗ E has the form ∑_{j} A_{ji}e_{j}. (Note. A_{ji} depend only on D and {e_{i}}, NOT on s!)

Then

Ds = ∑_{i,j}(ds_{j} + A_{ji}s_{j}) ⊗ e_{i}.

If we identify s = (s_{1}, ⋯, s_{n})^{t}, we see that

D ( s_{1} ; ⋮ ; s_{n} ) = (d + A) ( s_{1} ; ⋮ ; s_{n} ) ,

where A = [A_{ji}] is an n × n matrix of 1-forms on U. That is, D = d + A. □

Typeset by AMS-TEX

Note. If E = B × ℝ^{n}, then only one patch is required, that is, we can pick {e_{i}} to be a global frame. In this case, any matrix of 1-forms A will produce a connection D = d + A. In particular, A ≡ 0 yields D = d (The difference between D = d and D = d + A will be clear later).

If E = ∐ U_{α} × ℝ^{n}/{g_{αβ}} is not trivial and {e_{i}^{α}} is a local frame on U_{α}, then we must investigate compatibility of local descriptions:

Suppose D = d + A^{α} over U_{α} and D = d + A^{β} over U_{β}, then

(1)  De_{i}^{α} = A_{ji}^{α}e_{j}^{α}.

Also e_{i}^{α} = g_{ji}^{αβ}e_{j}^{β}, so

\begin{aligned}De_{i}^{α} & = D(g_{ji}^{αβ}e_{j}^{β}) \\ & = dg_{ji}^{αβ} ⊗ e_{j}^{β} + g_{ji}^{αβ}De_{j}^{β} \\ & = dg_{ji}^{αβ}g_{kj}^{βα} ⊗ e_{k}^{α} + g_{ji}^{αβ}A_{kj}^{β}g_{lk}^{βα} ⊗ e_{l}^{α} \\ & = ([g^{βα} · dg^{αβ}]_{li} + [g^{βα}A^{β}g^{αβ}]_{li}) ⊗ e_{l}^{α}. \end{aligned} (2)

Therefore the compatibility implies (1)=(2). Hence

(∗)  A^{α} = g^{βα}A^{β}g^{αβ} + g^{βα} · dg^{αβ}.

\underline{Conclusion}: D is specified by \{A^{α}\}, where A^{α} is a matrix value 1-form on U_{α} and A^{α} and A^{β} related by (∗) on U_{α} ∩ U_{β}.

Note. We cannot take A^{α} ≡ 0, because this does not in general satisfy (∗). \underline{Exception}: If dg^{βα} = 0, that is, g^{βα} : U_{α} ∩ U_{β} → GL(n) is (local) constant. For reason which will become clear, such bundles are called flat bundles:

Definition 2. If E admits description E = ∐ U_{α} × ℝ^{n}/{g_{αβ}} such that dg^{αβ} = 0, then E is called flat.

If E is a flat bundle, we can define a connection D = d with respect to corresponding flat local frames.

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801

E-mail address: bradlow@math.uiuc.edu