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Smooth principal bundle from two-chart gluing data

Statement

Let $X$ be a Hausdorff, second-countable smooth manifold covered by open sets $U$ and $V$ . Let $G$ be a finite-dimensional, Hausdorff, second-countable real Lie group, and let $τ : U \cap V \to G$ be smooth. On the disjoint union $Q = (U \times G)_{U} ⊔ (V \times G)_{V}$ , impose the equivalence relation generated, for $x \in U \cap V$ and $h \in G$ , by

$(x, h)_{V} \sim (x, τ (x) h)_{U} .$

Let $P : = Q / \sim$ with the quotient topology, write $[x, h]_{U}$ and $[x, h]_{V}$ for its classes, and define $π : P \to X$ by $π ([x, h]_{U}) = π ([x, h]_{V}) = x$ . Define the right action by $[x, h]_{i} \cdot k = [x, h k]_{i}$ for $k \in G$ and $i \in {U, V}$ . Then $P$ is a smooth right principal $G$ -bundle over $X$ . Its local sections $s_{U} (x) = [x, e]_{U}$ and $s_{V} (x) = [x, e]_{V}$ satisfy $s_{V} (x) = s_{U} (x) τ (x)$ on $U \cap V$ .

Statement references

Tong, Lecture 1 §1.1.2, equations (1.9)–(1.11)

Facts & Assumptions

Given: The Hausdorff, second-countable smooth manifold $X$ , its open cover $U, V$ , the finite-dimensional Hausdorff, second-countable real Lie group $G$ , and the smooth transition map $τ$ .

[A1]

The spaces $U$ and $V$ are open subsets of $X$ , cover $X$ , and inherit its Hausdorff, second-countable smooth-manifold structure; $τ$ is smooth on their overlap.

[F1]

A smooth right principal $G$ -bundle has an underlying topological right principal bundle and a smooth projection and right action, with equivariant diffeomorphisms from the inverse images of base charts to product charts. Smooth principal $G$ -bundle

[F2]

Multiplication and inversion in a finite-dimensional real Lie group are smooth. Lie group

Proof

1.1givenA1F2algebra

The generated equivalence classes contain one representative from each chart copy exactly when the base point is in $U \cap V$ : the transition $(x, h)_{V} \mapsto (x, τ (x) h)_{U}$ is bijective there, with inverse $(x, k)_{U} \mapsto (x, τ (x)^{- 1} k)_{V}$ by [F2]. Thus the canonical maps $j_{U} : U \times G \to P$ and $j_{V} : V \times G \to P$ are injective. For every open $W \subseteq U \times G$ , the preimage of $j_{U} (W)$ in $Q$ is the open subset $W_{U}$ together with the inverse transition image of $W ∣_{(U \cap V) \times G}$ in the $V$ copy; hence $j_{U} (W)$ is open in the quotient. The same argument applies to $j_{V}$ , so these maps are open embeddings and their images give product charts covering $P$ . On their overlap $j_{U}^{- 1} j_{V} (x, h) = (x, τ (x) h)$ , a diffeomorphism with inverse $(x, k) \mapsto (x, τ (x)^{- 1} k)$ by [F2]. Therefore these product charts define a smooth atlas on $P$ .

2.1givenA1step 1.1

The induced projection is continuous because the two product projections agree on the equivalence relation. If two points of $P$ project to distinct points of $X$ , disjoint base neighborhoods separate them since $X$ is Hausdorff. If they project to the same $x$ , choose $U$ or $V$ containing $x$ ; both points lie in that product chart, where distinct group coordinates have disjoint neighborhoods because the chart is Hausdorff. Hence $P$ is Hausdorff. Each product chart is second-countable, and the union of countable bases from the two charts is a countable base for $P$ . The coordinate changes are smooth by step 1.1, so the resulting smooth manifold structure has the required separation and countability properties.

3.1givenA1F1F2step 1.1step 2.1algebra∎

The formula $[x, h]_{i} \cdot k = [x, h k]_{i}$ is well-defined because $(x, τ (x) h)_{U} \cdot k = (x, τ (x) h k)_{U} \sim (x, h k)_{V}$ ; associativity and the identity law follow from the group law. In each chart it is the smooth product action by [F2], and it is free and transitive on each fibre because right multiplication on the $G$ coordinate has those properties. By steps 1.1 and 2.1, the charts are smooth product charts and $P$ is Hausdorff and second-countable. The map $π$ is smooth and locally the product projection, so [F1] applies and $P \to X$ is a smooth right principal $G$ -bundle. The sections $s_{U}, s_{V}$ are smooth in these charts, and $[x, e]_{V} = [x, τ (x)]_{U}$ gives $s_{V} = s_{U} τ$ .

Depends on

Dependency tree · 2 levels

Sources

David Tong, TASI Lectures on Solitons: Instantons, Monopoles, Vortices and Kinks