THE GEOMETRY OF VECTOR BUNDLES AND AN INTRODUCTION TO GAUGE THEORY LECTURE 8

Professor Steven Bradlow Class Notes From Math 433 University of Illinois at Urbana-Champaign

February 9, 1998

Recall from last time that given

E = (∐ U_{α} × ℝ^{n})/{g_{αβ}},

we can use {(g_{αβ}^{t})^{−1}} to define E^{∗} = (∐ U_{α} × ℝ^{n})/{(g_{αβ}^{t})^{−1}}. Then E_{b}^{∗} = Hom(E_{b}, ℝ). All we need: If a vector space V has a basis {v_{1}, …, v_{n}}, then we can define the dual basis {v_{1}^{∗}, …, v_{n}^{∗}} for V^{∗} = Hom(V, ℝ), where v_{i}^{∗} : V → ℝ defined by

v_{i}^{∗}(v_{j}) = δ_{ij}.

Then any λ ∈ V^{∗} is given by ∑_{i} λ_{i}v_{i}^{∗}.

For complex bundles:

E = (∐ U_{α} × ℂ^{n})/{g_{αβ}}

and

g_{αβ} : U_{α} ∩ U_{β} → GL(n, ℂ).

Exactly the same proof shows that {g_{αβ}^{t}^{−1}} defines E^{∗} and we can identify Hom(ℂ^{n}, ℂ) ≅ ℂ^{n} in the same way and see that E_{b}^{∗} = Hom(E_{b}, ℂ). Remarks. There are some points to note:

(1) We can use a Hermitian metric to identify

Hom(ℂ^{n}, ℂ) ⟶^{≃} ℂ^{n},

but this is NOT a ℂ-linear map, so it is NOT the identification we want!

(2) Real case: Suppose that we can pick a local trivialization E|_{U_{α}} →^{≃} U_{α} × ℝ^{n} such that

g_{αβ} : U_{α} ∩ U_{β} → O(n) ⊂ GL(n, ℝ).

(so structure groups of E can be reduced to O(n)) Then

g_{αβ}^{t}g_{αβ} = I.

So

g_{αβ}^{t}^{−1} = g_{αβ}.

Conclusion:

E^{∗} ≅ E.

(3) Complex case: Even if

g_{αβ} : U_{α} ∩ U_{β} → U(n) ⊂ GL(n, ℂ).

Typeset by AMS-TEX

Then

\overline{g_{αβ}}^{t}g_{αβ} = I.

So

g_{αβ}^{t}^{−1} = \overline{g_{αβ}} ≠ g_{αβ}.

Therefore E^{∗} ≇ E as complex bundles. In fact E^{∗} ≅ \overline{E}.

Metrics on Bundles

Suppose that g_{αβ} : U_{α} ∩ U_{β} → O(n) for E = (∐ U_{α} × ℝ^{n})/{g_{αβ}}.

Claim. We can define a fiberwise inner product on each E_{b}.

Proof: Define ⟨· , ·⟩_{b} on E_{b}, for each b ∈ B as follows: Let (· , ·) be the standard inner product on ℝ^{n}, that is, (e_{i}, e_{j}) = δ_{ij}, where {e_{i}} is the standard basis for ℝ^{n}. Then we can identify E_{b} ⟶^{≃^{Ψα}} ℝ^{n} for any α such that b ∈ U_{α} and define e_{i}^{α}(b) = Ψ_{α}^{−1}(b, e_{i}). Exercise 1. Show that {e_{i}^{α}(b)} is a basis for E_{b}.

Now we define ⟨· , ·⟩_{b} by declairing {e_{i}^{α}(b)} to be an orthonormal frame, that is, by defining ⟨e_{i}^{α}(b), e_{j}^{α}(b)⟩_{b} = δ_{ij}.

Thus if v = Σv_{i}e_{i}^{α}(b) and u = Σu_{i}e_{i}^{α}(b), then

\begin{aligned}⟨u, v⟩_{b} & = (Ψ_{α}(b, v_{1}), Ψ_{α}(b, v_{2})) \\ & = ∑_{i=1}^{n} u_{i}v_{i}. \end{aligned}

□ Exercise 2. Show that this is well-defined if g_{αβ} ∈ O(n). Exercise 3. Show that if σ_{1} and σ_{2} are C^{∞} sections, then b ↦ ⟨σ_{1}(b), σ_{2}(b)⟩_{b} is a smooth map. Exercise 4. Suppose that Ψ : E|_{U} →^{≃} U×ℝ^{n} is a local trivialization and that we define {f_{i}(b) = Ψ^{−1}(b, e_{i})}_{i=1}^{n}. (This is a basis for E_{b} for every b ∈ U and {f_{i}} is called a local frame.)

Define

H_{ij}(b) = ⟨f_{j}(b), f_{i}(b)⟩_{b},

show that the map b ↦ H_{ij}(b) defines a smooth map from U to GL(n). Remark. An assignment of an inner product ⟨· , ·⟩_{b} to each fiber of E is called a smooth bundle metric if either (hence both!) of the maps in Exercises 3 and 4 are smooth. Exercise 5. Suppose E admits a smooth bundle metric, that is, ⟨· , ·⟩_{b} on each E_{b}, smoothly varying with b. Show that we can pick E|_{U} →^{≃} U × ℝ^{n} such that g_{αβ} : U_{α} ∩ U_{β} → O(n). Examples of existence of metrics.

1. On

\begin{matrix}B × ℝ^{n} \\ \\ ℝ^{n} \end{matrix}

Take ⟨· , ·⟩_{b} = (· , ·) (or any other metric on ℝ^{n}.)

2. On E = ∐ U_{α} × ℝ^{n}, take a partition of unity subordinate to {U_{α}}—Need {U_{α}} locally finite here! That is, {ρ_{α} : U_{α} → ℝ is smooth} such that Supp(ρ_{α}) ⊂ U_{α} and Σ_{α}ρ_{α}(b) = 1, ∀ b. At b ∈ U_{α}, we

can define ⟨· , ·⟩_{b,α} using Ψ_{α} : E|_{U_{α}} →^{≃} U_{α} × ℝ^{n}. Then we can set ⟨· , ·⟩_{b} = Σ_{α}ρ_{α}(b)⟨· , ·⟩_{b,α}.

Exercise 6. Show that this definition defines an inner product. Note. Similarly for Hermitian metrics on complex vector bundles. Replacing “orthonormal” by “unitary” and O(n) by U(n), all the expeted analogs of the above results apply. Exercise 7. Fill in the details.

273 Altgeld Hall, 1409 W. Green Street, Urbana, IL 61801 E-mail address: bradlow@math.uiuc.edu