Definition 2.5. Let M and F be manifolds and let AutF be any Lie subgroup of DiffF . A fiber bundle modeled on (F, AutF ) over M is a smooth map π : E M such that
(i) E is a smooth manifold,
(ii) every point x∈ M has a neighborhood O ⊆ M for which
π−1O ∼= O× F
Any such local diffeomorphism is called a local trivialization.
(iii) Given local trivializations over O, O′ ⊆ M with O ∩O′ nonempty, the transition function
φ : O∩ O′ Diff(F )
takes its values in AutF .
We call AutF the structure group of the fiber bundle E.
For the benefit of the reader, we note that it is not the case that every definition of a fiber bundle utilizes the structure group AutF . Our construction might be reasonably termed a structured fiber bundle; however, we will adhere to the present terminology. We typically relax notation and refer to a fiber bundle E modeled on
2.2. FIBER AND PRINCIPAL BUNDLES
Definition 2.6. Define the automorphism bundle AutE of the F -fiber bundle E to be the group of fiberwise automorphisms of E. That is,
AutxE = Aut(Ex)
This bundle is occasionally called the Adjoint bundle of E and denoted AdE, about which we will have more to say below. The gauge group of E is defined to be the group GE of sections of AutE.
The natural action ofGE on the space ΓE of sections of E will be important later on.
Definition 2.7. Let G be a Lie group. A G-principal bundle P is a G-topological fiber equipped with a free right action of G, the orbits of which coincide with the fibers of P .
Note in the definition above that a G-topological fiber bundle is fiberwise equiv-alent to G in the category of manifolds. This does not endow the fibers of P with a group structure.
Definition 2.8. Let P be a G-principal bundle and let F be a topological space equipped with an action λ : G↷ F . Define the associated bundle P ×λF to P with typical fiber F to be the bundle
P ×λF = (P × F )/G
where G acts diagonally on P × F by
g· (u, f) = (ug−1, gf )
Lemma 2.1. 1. Let X be a right G-principal homogeneous space. Then Aut X ∼= G and the isomorphism is canonical precisely if and only if G is abelian.
2. Suppose that the action λ : G Aut G is effective, and that λ commutes with the right regular representation r : Gop Aut G. Then λ is equivalent to the left regular representation ℓ : G Aut G.
3. The left regular action ℓ : G Aut G is a group isomorphism.
Proof. 1. Fix x∈ X. We will show that φ : Aut G G, given by
α(x) = x· φ(α)
is a group homomorphism. The map φ is well-defined since the right action of G on X is free and transitive. Moreover, φ is a homomorphism since
x· φ(αβ) = αβ(x)
= α
x· φ(β)]
= α(x)· φ(β)
2.2. FIBER AND PRINCIPAL BUNDLES
Injectivity follows as α(x) = x· 1G implies α(x· g) = x · g for every g ∈ G, whence α = 1Aut(X). Finally, since the assignment αg : x· h x · gh determines an automorphism of X, and since φ(αg) = g, we deduce that φ is surjective.
2. We will show that the map φ : G G given by
φ(g) = λg(1)
intertwines λ and ℓ. First note that φ is a homomorphism, since
λgh(1) = λg
λh(1)
= λg(1) λh(1)
using in turn the facts that λ is a homomorphism and that λ commutes with right multiplication. As
λg φ(h)
= λgh(1) = φ ℓg(h)
we deduce that φ is a morphism from λ to ℓ. Since λ is effective, it follows that φ is injective. We conclude that φ is an automorphism of G and, consequently, an intertwiner from λ to ℓ.
3. By part 1. there is an isomorphism φ : G − Aut G. Under this identification,∼ the usual action of Aut(G) on G is also given by φ. By part 2. φ is equivalent
to the left regular action ℓ, yielding an isomorphism ψ : Aut G− Aut G such
Since φ and ψ are isomorphisms, we conclude that that ℓ is an isomorphism as well.
We deduce from Lemma 2.1 that the structure group of a G-principal bundle is G.
Proposition 2.1. Let P be a G-principal bundle. The automorphism bundle AutP is naturally isomorphic to P ×cG, where c : G Aut G is the action of conjugation.
Proof. Let X be a right G-principal space and observe that the assignment x [x, 1]ℓ
determines a canonical isomorphism i : X X×ℓG of right G-principal spaces from X to X×ℓG. Since X×c G acts naturally on X ×ℓG by
2.2. FIBER AND PRINCIPAL BUNDLES
where i∗ = Aut(i−1). Since φ, ψ and ℓ are isomorphisms and since m and i are natural, we deduce that
i∗◦ m : X ×cG− Aut X∼
is a natural isomorphism. By applying this fact to each fiber of P , we obtain a family of isomorphisms
χx : Px×c G− Aut∼ xP
which is easily seen to be smoothly varying and thus to yield a canonical identification
χ : P ×cG− AutP∼
Henceforth, we shall identify the bundles P ×c G and AutP . In the literature, this bundle is sometimes called the Adjoint bundle of P and denoted by AdP [42].
The Adjoint bundle AdP is equipped with a natural group structure; its Lie algebra adP = P ×Adg is called the adjoint bundle. Note that the initial letter of the former bundle is capitalized, while that of the latter is not.
Example 2.1. Let π : E M be a fiber bundle modeled on F . The frame bundle P E of E is the AutF -principal bundle of fiberwise identifications of F with E. That is,
PxE =
u : F − E∼ x
The right action of AutF on P E is given by
(u· g)(f) = u(gf)
for u∈ P E, g ∈ AutF , and f ∈ F .
Example 2.2. If E is a V -vector bundle, then the frame bundle P E is isomorphic as an AutV -principal bundle to the bundle F E of fiberwise bases of E,
FxE ={(σi)i ⊆ Ex
(σi)i is a basis of Ex}
If V =Rk then this isomorphism is natural and we identify P E and F E.
The following example shows that the class of frame bundles and the class principal bundles coincide.
Example 2.3. If P′ is a G-principal bundle on a manifold M , then the frame bundle P P′ is canonically isomorphic to P′. To see this, observe that for any x ∈ M the fiber PxP′ consists of all isomorphisms φx : G P∼ x′ of G-principal homogeneous spaces.
Since this collection is itself a principal homogeneous space under the action of the structure group G ∼= Aut G, we deduce that P′ and P P′ have equivalent fibers. As