Abstract
In this chapter we will first give several equivalent constructions of a connection on a principal bundle, and then generalize the notion curvature to a principal bundle, paving the way to a generalization of characteristic classes to principal bundles. Along the way, we also generalize covariant derivatives to principal bundles.
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A principal bundle is a locally trivial family of groups. It turns out that the theory of connections on a vector bundle can be subsumed under the theory of connections on a principal bundle. The latter, moreover, has the advantage that its connection forms are basis-free.
In this chapter we will first give several equivalent constructions of a connection on a principal bundle, and then generalize the notion curvature to a principal bundle, paving the way to a generalization of characteristic classes to principal bundles. Along the way, we also generalize covariant derivatives to principal bundles.
§27 Principal Bundles
We saw in Section 11 that a connection ∇ on a vector bundle E over a manifold M can be represented by a matrix of 1-forms over a framed open set. For any frame e = [e 1 ⋯ e r ] for E over an open set U, the connection matrix ω e relative to e is defined by
for all C ∞ vector fields X over U. If \(\bar{e} = [\bar{e}_{1}\ \cdots \ \bar{e}_{r}] = ea\) is another frame for E over U, where \(a: U \rightarrow \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) is a matrix of C ∞ transition functions, then by Theorem 22.1 the connection matrix ω e transforms according to the rule
Associated to a vector bundle is an object called its frame bundle π: Fr(E) → M; the total space \(\mathop{\mathrm{Fr}}\nolimits (E)\) of the frame bundle is the set of all ordered bases in the fibers of the vector bundle E → M, with a suitable topology and manifold structure. A section of the frame bundle \(\pi: \mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) over an open set U ⊂ M is a map \(s: U \rightarrow \mathop{\mathrm{Fr}}\nolimits (E)\) such that \(\pi \circ s =\mathbb{1}_{U}\), the identity map on U. From this point of view a frame e = [e 1 ⋯ e r ] over U is simply a section of the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) over U.
Suppose ∇ is a connection on the vector bundle E → M. Miraculously, there exists a matrix-valued 1-form ω on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) such that for every frame e over an open set U ⊂ M, the connection matrix ω e of ∇ is the pullback of ω by the section \(e: U \rightarrow \mathop{\mathrm{Fr}}\nolimits (E)\) (Theorem 29.10). This matrix-valued 1-form, called an Ehresmann connection on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\), is determined uniquely by the connection on the vector bundle E and vice versa. It is an intrinsic object of which a connection matrix ω e is but a local manifestation. The frame bundle of a vector bundle is an example of a principal G-bundle for the group \(G =\mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R})\). The Ehresmann connection on the frame bundle generalizes to a connection on an arbitrary principal bundle.
Charles Ehresmann (1905–1979)
This section collects together some general facts about principal bundles.
27.1 Principal Bundles
Let E, M, and F be manifolds. We will denote an open cover \(\mathfrak{U}\) of M either as {U α } or more simply as an unindexed set {U} whose general element is denoted by U. A local trivialization with fiber F for a smooth surjection π: E → M is an open cover \(\mathfrak{U} =\{ U\}\) for M together with a collection \(\{\phi _{U}: \pi ^{-1}(U) \rightarrow U \times F\mid U \in \mathfrak{U}\}\) of fiber-preserving diffeomorphisms ϕ U : π −1(U) → U × F:
where η is projection to the first factor. A fiber bundle with fiber F is a smooth surjection π: E → M having a local trivialization with fiber F. We also say that it is locally trivial with fiber F. The manifold E is the total space and the manifold M the base space of the fiber bundle.
The fiber of a fiber bundle π: E → M over x ∈ M is the set E x : = π −1(x). Because π is a submersion, by the regular level set theorem ([21], Th. 9.13, p. 96) each fiber E x is a regular submanifold of E. For x ∈ U, define \(\phi _{U,x}:=\phi _{U}\vert _{E_{x}}: E_{x} \rightarrow \{ x\} \times F\) to be the restriction of the trivialization ϕ U : π −1(U) → U × F to the fiber E x .
Proposition 27.1.
Let π: E → M be a fiber bundle with fiber F. If ϕ U : π −1 (U) → U × F is a trivialization, then ϕ U,x : E x →{ x} × F is a diffeomorphism.
Proof.
The map ϕ U, x is smooth because it is the restriction of the smooth map ϕ U to a regular submanifold. It is bijective because ϕ U is bijective and fiber-preserving. Its inverse ϕ U, x −1 is the restriction of the smooth map ϕ U −1: U × F → π −1(U) to the fiber {x} × F and is therefore also smooth. □
A smooth right action of a Lie group G on a manifold M is a smooth map
denoted by x ⋅ g: = μ(x, g), such that for all x ∈ M and g, h ∈ G,
- (i):
-
x ⋅ e = x, where e is the identity element of G,
- (ii):
-
(x ⋅ g) ⋅ h = x ⋅ (gh).
We often omit the dot and write more simply xg for x ⋅ g. If there is such a map μ, we also say that G acts smoothly on M on the right. A left action is defined similarly. The stabilizer of a point x ∈ M under an action of G is the subgroup
The orbit of x ∈ M is the set
Denote by \(\mathop{\mathrm{Stab}}\nolimits (x)\setminus G\) the set of right cosets of \(\mathop{\mathrm{Stab}}\nolimits (x)\) in G. By the orbit-stabilizer theorem, for each x ∈ M the map\(: G \rightarrow \mathop{\mathrm{Orbit}}\nolimits (x)\), g ↦ x ⋅ g induces a bijection of sets:
The action of G on M is free if the stabilizer of every point x ∈ M is the trivial subgroup {e}.
A manifold M together with a right action of a Lie group G on M is called a right G-manifold or simply a G-manifold . A map f: N → M between right G-manifolds is right G-equivariant if
for all (x, g) ∈ N × G. Similarly, a map f: N → M between left G-manifolds is left G-equivariant if
for all (g, x) ∈ G × N.
A left action can be turned into a right action and vice versa; for example, if G acts on M on the left, then
is a right action of G on M. Thus, if N is a right G-manifold and M is a left G-manifold, we say a map f: N → M is G-equivariant if
for all (x, g) ∈ N × G.
A smooth fiber bundle π: P → M with fiber G is a smooth principal G-bundle if G acts smoothly and freely on P on the right and the fiber-preserving local trivializations
are G-equivariant, where G acts on U × G on the right by
Example 27.2 (Product G-bundles).
The simplest example of a principal G-bundle over a manifold M is the product G-bundle η: M × G → M. A trivialization is the identity map on M × G.
Example 27.3 (Homogenous manifolds).
If G is a Lie group and H is a closed subgroup, then the quotient G∕H can be given the structure of a manifold such that the projection map π: G → G∕H is a principal H-bundle. This is proven in [22, Th. 3.58, p. 120].
Example 27.4 (Hopf bundle).
The group S 1 of unit complex numbers acts on the complex vector space \(\mathbb{C}^{n+1}\) by left multiplication. This action induces an action of S 1 on the unit sphere S 2n+1 in \(\mathbb{C}^{n+1}\). The complex projective space \(\mathbb{C}P^{n}\) may be defined as the orbit space of S 2n+1 by S 1. The natural projection \(S^{2n+1} \rightarrow \mathbb{C}P^{n}\) with fiber S 1 turn out to be a principal S 1-bundle. When n = 1, \(S^{3} \rightarrow \mathbb{C}P^{1}\) with fiber S 1 is called the Hopf bundle.
Definition 27.5.
Let π Q : Q → N and π P : P → M be principal G-bundles. A morphism of principal G-bundles is a pair of maps \((\bar{f}: Q \rightarrow P,f: N \rightarrow M)\) such that \(\bar{f}: Q \rightarrow P\) is G-equivariant and the diagram
commutes.
Proposition 27.6.
If π: P → M is a principal G-bundle, then the group G acts transitively on each fiber.
Proof.
Since G acts transitively on {x} × G and the fiber diffeomorphism ϕ U, x : P x → { x} × G is G-equivariant, G must also act transitively on the fiber P x . □
Lemma 27.7.
For any group G, a right G-equivariant map f: G → G is necessarily a left translation.
Proof.
Suppose that for all x, g ∈ G,
Setting x = e, the identity element of G, we obtain
where ℓ f(e): G → G is left translation by f(e). □
Suppose {U α } α ∈ A is a local trivialization for a principal G-bundle π: P → M. Whenever the intersection U α β : = U α ∩ U β is nonempty, there are two trivializations on π −1(U α β ):
Then ϕ α ∘ϕ β −1: U α β × G → U α β × G is a fiber-preserving right G-equivariant map. By Lemma 27.7, it is a left translation on each fiber. Thus,
where (x, h) ∈ U α β × G and g α β (x) ∈ G. Because ϕ α ∘ϕ β −1 is a C ∞ function of x and h, setting h = e, we see that g α β (x) is a C ∞ function of x. The C ∞ functions g α β : U α β → G are called transition functions of the principal bundle π: P → M relative to the trivializing open cover {U α } α ∈ A. They satisfy the cocycle condition: for all α, β, γ ∈ A,
From the cocycle condition, one can deduce other properties of the transition functions.
Proposition 27.8.
The transition functions g αβ of a principal bundle π: P → M relative to a trivializing open cover {U α } α∈A satisfy the following properties: for all α,β ∈A,
- (i):
-
g αα = the constant map e,
- (ii):
-
g αβ = g βα −1 if \(U_{\alpha } \cap U_{\beta }\neq \varnothing\).
Proof.
- (i):
-
If α = β = γ, the cocycle condition gives
$$\displaystyle{g_{\alpha \alpha }g_{\alpha \alpha } = g_{\alpha \alpha }.}$$Hence, g α α = the constant map e.
- (ii):
-
if γ = α, the cocycle condition gives
$$\displaystyle{g_{\alpha \beta }g_{\beta \alpha } = g_{\alpha \alpha } = e}$$or
$$\displaystyle{g_{\alpha \beta } = g_{\beta \alpha }^{-1}\quad \text{for }U_{\alpha } \cap U_{\beta }\neq \varnothing.}$$□
In a principal G-bundle P → M, the group G acts on the right on the total space P, but the transition functions g α β in (27.2) are given by left translations by g α β (x) ∈ G. This phenomenon is a consequence of Lemma 27.7.
27.2 The Frame Bundle of a Vector Bundle
For any real vector space V, let \(\mathop{\mathrm{Fr}}\nolimits (V )\) be the set of all ordered bases in V. Suppose V has dimension r. We will represent an ordered basis v 1, …, v r by a row vector v = [v 1 ⋯ v r ], so that the general linear group \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) acts on \(\mathop{\mathrm{Fr}}\nolimits (V )\) on the right by matrix multiplication
Fix a point \(v \in \mathop{\mathrm{Fr}}\nolimits (V )\). Since the action of \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) on \(\mathop{\mathrm{Fr}}\nolimits (V )\) is clearly transitive and free, i.e., \(\mathop{\mathrm{Orbit}}\nolimits (v) =\mathop{ \mathrm{Fr}}\nolimits (V )\) and \(\mathop{\mathrm{Stab}}\nolimits (v) =\{ I\}\), by the orbit-stabilizer theorem there is a bijection
Using the bijection ϕ v , we put a manifold structure on \(\mathop{\mathrm{Fr}}\nolimits (V )\) in such a way that ϕ v becomes a diffeomorphism.
If v′ is another element of \(\mathop{\mathrm{Fr}}\nolimits (V )\), then v′ = va for some \(a \in \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) and
Since left multiplication \(\ell_{a}: \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R}) \rightarrow \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) is a diffeomorphism, the manifold structure on \(\mathop{\mathrm{Fr}}\nolimits (V )\) defined by ϕ v is the same as the one defined by ϕ va . We call \(\mathop{\mathrm{Fr}}\nolimits (V )\) with this manifold structure the frame manifold of the vector space V.
Remark 27.9.
A linear isomorphism ϕ: V → W induces a C ∞ diffeomorphism \(\widetilde{\phi }:\) \(\mathop{\mathrm{Fr}}\nolimits (V ) \rightarrow \mathop{\mathrm{Fr}}\nolimits (W)\) by
Define an action of \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) on \(\mathop{\mathrm{Fr}}\nolimits (\mathbb{R}^{r})\) by
Thus, if \(\phi: \mathbb{R}^{r} \rightarrow \mathbb{R}^{r}\) is given by left multiplication by \(g \in \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\), then so is the induced map \(\widetilde{\phi }\) on the frame manifold \(\mathop{\mathrm{Fr}}\nolimits (\mathbb{R}^{r})\).
Example 27.10 (The frame bundle).
Let η: E → M be a C ∞ vector bundle of rank r. We associate to the vector bundle E a C ∞ principal \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\)-bundle \(\pi: \mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) as follows. As a set the total space \(\mathop{\mathrm{Fr}}\nolimits (E)\) is defined to be the disjoint union
There is a natural projection map \(\pi: \mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) that maps \(\mathop{\mathrm{Fr}}\nolimits (E_{x})\) to {x}.
A local trivialization \(\phi _{\alpha }: E\vert _{U_{\alpha }}\mathop{ \rightarrow }\limits^{\sim } U_{\alpha } \times \mathbb{R}^{r}\) induces a bijection
Via \(\widetilde{\phi _{\alpha }}\) one transfers the topology and manifold structure from \(U_{\alpha } \times \mathop{\mathrm{Fr}}\nolimits (\mathbb{R}^{r})\) to \(\mathop{\mathrm{Fr}}\nolimits (E)\vert _{U_{\alpha }}\). This gives \(\mathop{\mathrm{Fr}}\nolimits (E)\) a topology and a manifold structure such that π: Fr(E) → M is locally trivial with fiber \(\mathop{\mathrm{Fr}}\nolimits (\mathbb{R}^{r})\). As the frame manifold \(\mathop{\mathrm{Fr}}\nolimits (\mathbb{R}^{r})\) is diffeomorphic to the general linear group \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\), it is easy to check that \(\mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) is a C ∞ principal \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\)-bundle. We call it the frame bundle of the vector bundle E.
On a nonempty overlap U α β : = U α ∩ U β , the transition function for the vector bundle E is the C ∞ function \(g_{\alpha \beta }: U_{\alpha \beta } \rightarrow \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) given by
Since the local trivialization for the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) is induced from the trivialization {U α , ϕ α } for E, the transition functions for \(\mathop{\mathrm{Fr}}\nolimits (E)\) are induced from the transition functions {g α β } for E. By Remark 27.9 the transition functions for the open cover \(\{\mathop{\mathrm{Fr}}\nolimits (E)\vert _{U_{\alpha }}\}\) of \(\mathop{\mathrm{Fr}}\nolimits (E)\) are the same as the transition functions \(g_{\alpha \beta }: U_{\alpha \beta } \rightarrow \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) for the vector bundle E, but now of course \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) acts on \(\mathop{\mathrm{Fr}}\nolimits (\mathbb{R}^{r})\) instead of on \(\mathbb{R}^{r}\).
27.3 Fundamental Vector Fields of a Right Action
Suppose G is a Lie group with Lie algebra \(\mathfrak{g}\) and G acts smoothly on a manifold P on the right. To every element \(A \in \mathfrak{g}\) one can associate a vector field A on P called the fundamental vector field on P associated to A: for p in P, define
To understand this equation, first fix a point p ∈ P. Then c p : t ↦ p ⋅ e tA is a curve in P with initial point p. By definition, the vector A p is the initial vector of this curve. Thus,
As a tangent vector at p is a derivation on germs of C ∞ functions at p, in terms of a C ∞ function f at p,
Proposition 27.11.
For each \(A \in \mathfrak{g}\) , the fundamental vector field A is C∞ on P.
Proof.
It suffices to show that for every C ∞ function f on P, the function A f is also C ∞ on P. Let μ: P × G → P be the C ∞ map defining the right action of G on P. For any p in P,
Since e tA is a C ∞ function of t, and f and μ are both C ∞, the derivative
is C ∞ in p and in t. Therefore, A p f is a C ∞ function of p. □
Recall that \(\mathfrak{X}(P)\) denotes the Lie algebra of C ∞ vector fields on the manifold P. The fundamental vector field construction gives rise to a map
For p in P, define j p : G → P by j p (g) = p ⋅ g. Computing the differential j p∗ using the curve c(t) = e tA, we obtain the expression
This alternate description of fundamental vector fields, A p = j p∗(A), shows that the map \(\sigma: \mathfrak{g} \rightarrow \mathfrak{X}(P)\) is linear over \(\mathbb{R}\). In fact, σ is a Lie algebra homomorphism (Problem 27.1).
Example 27.12.
Consider the action of a Lie group G on itself by right multiplication. For p ∈ G, the map j p : G → G, j p (g) = p ⋅ g = ℓ p (g) is simply left multiplication by p. By (27.3), for \(A \in \mathfrak{g}\), A p = ℓ p∗(A). Thus, for the action of G on G by right multiplication, the fundamental vector field A on G is precisely the left-invariant vector field generated by A. In this sense the fundamental vector field of a right action is a generalization of a left-invariant vector field on a Lie group.
For g in a Lie group G, let c g : G → G be conjugation by g: c g (x) = gxg −1. The adjoint representation is defined to be the differential of the conjugation map: \(\mathop{\mathrm{Ad}}\nolimits (g) = (c_{g})_{{\ast}}: \mathfrak{g} \rightarrow \mathfrak{g}\).
Proposition 27.13.
Suppose a Lie group G acts smoothly on a manifold P on the right. Let r g : P → P be the right translation r g (p) = p ⋅ g. For \(A \in \mathfrak{g}\) the fundamental vector field A on P satisfies the following equivariance property:
Proof.
We need to show that for every p in P, \(r_{g{\ast}}(\underline{A}_{p}) =\underline{ (\mathop{\mathrm{Ad}}\nolimits g^{-1})A}_{pg}\). For x in G,
By the chain rule,
□
27.4 Integral Curves of a Fundamental Vector Field
In this section suppose a Lie group G with Lie algebra \(\mathfrak{g}:=\mathrm{ Lie}(G)\) acts smoothly on the right on a manifold P.
Proposition 27.14.
For p ∈ P and \(A \in \mathfrak{g}\) , the curve c p (t) = p ⋅ e tA, \(t \in \mathbb{R}\) , is the integral curve of the fundamental vector field A through p.
Proof.
We need to show that \(c_{p}^{\prime}(t) =\underline{ A}_{c_{p}(t)}\) for all \(t \in \mathbb{R}\) and all p ∈ P. It is essentially a sequence of definitions:
□
Proposition 27.15.
The fundamental vector field A on a manifold P vanishes at a point p in P if and only if A is in the Lie algebra of \(\mathop{\mathrm{Stab}}\nolimits (p)\).
Proof.
( ⇐ ) If \(A \in \mathrm{Lie}\big(\mathop{\mathrm{Stab}}\nolimits (p)\big)\), then \(e^{tA} \in \mathop{\mathrm{Stab}}\nolimits (p)\), so
( ⇒ ) Suppose A p = 0. Then the constant map γ(t) = p is an integral curve of A through p, since
On the other hand, by Proposition 27.14, c p (t) = p ⋅ e tA is also an integral curve of A through p. By the uniqueness of the integral curve through a point, c p (t) = γ(t) or p ⋅ e tA = p for all \(t \in \mathbb{R}\). This implies that \(e^{tA} \in \mathop{\mathrm{Stab}}\nolimits (p)\) and therefore \(A \in \mathrm{Lie}\big(\mathop{\mathrm{Stab}}\nolimits (p)\big)\). □
Corollary 27.16.
For a right action of a Lie group G on a manifold P, let p ∈ P and j p : G → P be the map j p (g) = p ⋅ g. Then the kernel ker j p∗ of the differential of j p at the identity
is \(\mathrm{Lie}\big(\mathop{\mathrm{Stab}}\nolimits (p)\big)\).
Proof.
For \(A \in \mathfrak{g}\), we have A p = j p∗(A) by (27.3). Thus,
□
27.5 Vertical Subbundle of the Tangent Bundle TP
Throughout this section, G is a Lie group with Lie algebra \(\mathfrak{g}\) and π: P → M is a principal G-bundle. On the total space P there is a natural notion of vertical tangent vectors. We will show that the vertical tangent vectors on P form a trivial subbundle of the tangent bundle TP.
By the local triviality of a principal bundle, at every point p ∈ P the differential π ∗, p : T p P → T π(p) M of the projection π is surjective. The vertical tangent subspace \(\mathcal{V}_{p} \subset T_{p}P\) is defined to be kerπ ∗, p . Hence, there is a short exact sequence of vector spaces
and
An element of \(\mathcal{V}_{p}\) is called a vertical tangent vector at p.
Proposition 27.17.
For any \(A \in \mathfrak{g}\) , the fundamental vector field A is vertical at every point p ∈ P.
Proof.
With j p : G → P defined as usual by j p (g) = p ⋅ g,
Since A p = j p∗(A) by (27.3), and π ∘ j p is a constant map,
□
Thus, in case P is a principal G-bundle, we can refine Corollary 27.16 to show that j p∗ maps \(\mathfrak{g}\) into the vertical tangent space:
In fact, this is an isomorphism.
Proposition 27.18.
For p ∈ P, the differential at e of the map j p : G → P is an isomorphism of \(\mathfrak{g}\) onto the vertical tangent space: \(j_{p{\ast}} = (j_{p})_{{\ast},e}: \mathfrak{g}\mathop{ \rightarrow }\limits^{\sim }\mathcal{V}_{p}\).
Proof.
By Corollary 27.16, \(\ker j_{p{\ast}} = \mathrm{Lie}\big(\mathop{\mathrm{Stab}}\nolimits (p)\big)\). Since G acts freely on P, the stabilizer of any point p ∈ P is the trivial subgroup {e}. Thus, kerj p∗ = 0 and j p∗ is injective. By Proposition 27.17, the image j p∗ lies in the vertical tangent space \(\mathcal{V}_{p}\). Since \(\mathfrak{g}\) and \(\mathcal{V}_{p}\) have the same dimension, the injective linear map \(j_{p{\ast}}: \mathfrak{g} \rightarrow \mathcal{V}_{p}\) has to be an isomorphism. □
Corollary 27.19.
The vertical tangent vectors at a point of a principal bundle are precisely the fundamental vectors.
Let B 1, …, B ℓ be a basis for the Lie algebra \(\mathfrak{g}\). By the proposition, the fundamental vector fields B 1, …, B ℓ on P form a basis of \(\mathcal{V}_{p}\) at every point p ∈ P. Hence, they span a trivial subbundle \(\mathcal{V}:=\coprod _{p\in P}\mathcal{V}_{p}\) of the tangent bundle TP. We call \(\mathcal{V}\) the vertical subbundle of TP.
As we learned in Section 20.5, the differential π ∗: TP → TM of a C ∞ map π: P → M induces a bundle map \(\tilde{\pi }_{{\ast}}: TP \rightarrow \pi ^{{\ast}}TM\) over P, given by
The map \(\tilde{\pi }_{{\ast}}\) is surjective because it sends the fiber T p P onto the fiber (π ∗ TM) p ≃ T π(p) M. Its kernel is precisely the vertical subbundle \(\mathcal{V}\) by (27.4). Hence, \(\mathcal{V}\) fits into a short exact sequence of vector bundles over P:
27.6 Horizontal Distributions on a Principal Bundle
On the total space P of a smooth principal bundle π: P → M, there is a well-defined vertical subbundle \(\mathcal{V}\) of the tangent bundle TP. We call a subbundle \(\mathcal{H}\) of TP a horizontal distribution on P if \(TP = \mathcal{V}\oplus \mathcal{H}\) as vector bundles; in other words, \(T_{p}P = \mathcal{V}_{p} + \mathcal{H}_{p}\) and \(\mathcal{V}_{p} \cap \mathcal{H}_{p} = 0\) for every p ∈ P. In general, there is no canonically defined horizontal distribution on a principal bundle.
A splitting of a short exact sequence of vector bundles \(0 \rightarrow A\mathop{ \rightarrow }\limits^{ i}B\mathop{ \rightarrow }\limits^{ j}C \rightarrow 0\) over a manifold P is a bundle map k: C → B such that \(j \circ k =\mathbb{1}_{C}\), the identity bundle map on C.
Proposition 27.20.
Let
be a short exact sequence of vector bundles over a manifold P. Then there is a one-to-one correspondence
Proof.
If H is a subbundle of B such that B = i(A) ⊕ H, then there are bundle isomorphisms H ≃ B∕i(A) ≃ C. Hence, C maps isomorphically onto H in B. This gives a splitting k: C → B.
If k: C → B is a splitting, let H: = k(C), which is a subbundle of B. Moreover, if i(a) = k(c) for some a ∈ A and c ∈ C, then
Hence, i(A) ∩ k(C) = 0.
Finally, to show that B = i(A) + k(C), let b ∈ B. Then
By the exactness of (27.6), b − kj(b) = i(a) for some a ∈ A. Thus,
This proves that B = i(A) + k(C) and therefore B = i(A) ⊕ k(C). □
As we just saw in the preceding section, for every principal bundle π: P → M the vertical subbundle \(\mathcal{V}\) fits into a short exact sequence (27.5) of vector bundles over P. By Proposition 27.20, there is a one-to-one correspondence between horizontal distributions on P and splittings of the sequence (27.5).
Problems
27.1. Lie bracket of fundamental vector fields
Let G be a Lie group with Lie algebra \(\mathfrak{g}\) and let P be a manifold on which G acts on the right. Prove that for \(A,B \in \mathfrak{g}\),
Hence, the map \(\sigma: \mathfrak{g} \rightarrow \mathfrak{X}(P)\), A ↦ A is a Lie algebra homomorphism.
27.2. ∗ Short exact sequence of vector spaces
Prove that if \(0 \rightarrow A\mathop{ \rightarrow }\limits^{ i}B\mathop{ \rightarrow }\limits^{ j}C \rightarrow 0\) is a short exact sequence of finite-dimensional vector spaces, then dimB = dimA + dimC.
27.3. Splitting of a short exact sequence
Suppose \(0 \rightarrow A\mathop{ \rightarrow }\limits^{ i}B \rightarrow C \rightarrow 0\) is a short exact sequence of vector bundles over a manifold P. A retraction of i: A → B is a map \(r: B\mathop{ \rightarrow }\limits^{ j}A\) such that \(r \circ i =\mathbb{1}_{A}\). Show that i has a retraction if and only if the sequence has a splitting.
27.4. ∗ The differential of an action
Let μ: P × G → P be an action of a Lie group G on a manifold P. For g ∈ G, the tangent space T g G may be identified with \(\ell_{g{\ast}}\mathfrak{g}\), where ℓ g : G → G is left multiplication by g ∈ G and \(\mathfrak{g} = T_{e}G\) is the Lie algebra of G. Hence, an element of the tangent space T (p, g)(P × G) is of the form (X p , ℓ g∗ A) for X p ∈ T p P and \(A \in \mathfrak{g}\). Prove that the differential
is given by
27.5. Fundamental vector field under a trivialization
Let ϕ α : π −1 U α → U α × G
be a trivialization of π −1 U ga in a principal bundle P. Let \(A \in \mathfrak{g}\), the Lie algebra of G and A the fundamental vector field on P that it induces. Prove that
27.6. Trivial principal bundle
Prove that a principal bundle π: P → M is trivial if and only if it has a section.
27.7. Pullback of a principal bundle to itself
Prove that if π: P → M is a principal bundle, then the pullback bundle π ∗ P → P is trivial.
27.8. Quotient space of a principal bundle
Let G be a Lie group and H a closed subgroup. Prove that if π P → M is a principal G-bundle, then P → P∕H is a principal H-subbundle.
27.9. Fundamental vector fields
Let N and M be G-manifolds with G acting on the right. If \(A \in \mathfrak{g}\) and f: N → M is G-equivariant, then
§28 Connections on a Principal Bundle
Let G be a Lie group with Lie algebra \(\mathfrak{g}\). As we saw in the preceding section, on a principal G-bundle P → M, the notion of a vertical tangent vector is well defined, but not that of a horizontal tangent vector. A connection on a principal bundle is essentially the choice of a horizontal complement to the vertical tangent bundle on P. Alternatively, it can be given by a \(\mathfrak{g}\)-valued 1-form on P. In this section we will study these two equivalent manifestations of a connection:
- (i):
-
a smooth right-invariant horizontal distribution on P,
- (ii):
-
a smooth G-equivariant \(\mathfrak{g}\)-valued 1-form ω on P such that on the fundamental vector fields,
$$\displaystyle{ \omega (\underline{A}) = A\quad \text{for all }A \in \mathfrak{g}. }$$(28.1)
Under the identification of \(\mathfrak{g}\) with a vertical tangent space, condition (28.1) says that ω restricts to the identity map on vertical vectors.
The correspondence between (i) and (ii) is easy to describe. Given a right-invariant horizontal distribution \(\mathcal{H}\) on P, we define a \(\mathfrak{g}\)-valued 1-form ω on P to be, at each point p, the projection with kernel \(\mathcal{H}_{p}\) from the tangent space to the vertical space. Conversely, given a right-equivariant \(\mathfrak{g}\)-valued 1-form ω that is the identity on the vertical space at each point p ∈ P, we define a horizontal distribution \(\mathcal{H}\) on P to be kerω p at each p ∈ P.
28.1 Connections on a Principal Bundle
Let G be a Lie group with Lie algebra \(\mathfrak{g}\), and let π: P → M be a principal G-bundle. A distribution on a manifold is a subbundle of the tangent bundle. Recall that a distribution \(\mathcal{H}\) on P is horizontal if it is complementary to the vertical subbundle \(\mathcal{V}\) of the tangent bundle TP: for all p in P,
Suppose \(\mathcal{H}\) is a horizontal distribution on the total space P of a principal G-bundle π: P → M. For p ∈ P, if j p : G → P is the map j p (g) = p ⋅ g, then the vertical tangent space \(\mathcal{V}_{p}\) can be canonically identified with the Lie algebra \(\mathfrak{g}\) via the isomorphism \(j_{p{\ast}}: \mathfrak{g} \rightarrow \mathcal{V}_{p}\) (Proposition 27.18). Let \(v: T_{p}P = \mathcal{V}_{p} \oplus \mathcal{H}_{p} \rightarrow \mathcal{V}_{p}\) be the projection to the vertical tangent space with kernel \(\mathcal{H}_{p}\). For Y p ∈ T p P, v(Y p ) is called the vertical component of Y p . (Although the vertical subspace \(\mathcal{V}_{p}\) is intrinsically defined, the notion of the vertical component of a tangent vector depends on the choice of a horizontal complement \(\mathcal{H}_{p}\).) If ω p is the composite
then ω is a \(\mathfrak{g}\)-valued 1-form on P. In terms of ω, the vertical component of Y p ∈ T p P is
Theorem 28.1.
If \(\mathcal{H}\) is a smooth right-invariant horizontal distribution on the total space P of a principal G-bundle π: P → M, then the \(\mathfrak{g}\) -valued 1-form ω on P defined above satisfies the following three properties:
- (i):
-
for any \(A \in \mathfrak{g}\) and p ∈ P, we have ω p(A p) = A;
- (ii):
-
(G-equivariance) for any g ∈ G, \(r_{g}^{{\ast}}\omega = (\mathop{\mathrm{Ad}}\nolimits g^{-1})\omega\);
- (iii):
-
ω is C ∞.
Proof.
(i) Since A p is already vertical (Proposition 27.17), the projection v leaves it invariant, so
(ii) For p ∈ P and Y p ∈ T p P, we need to show
Since both sides are \(\mathbb{R}\)-linear in Y p and Y p is the sum of a vertical and a horizontal vector, we may treat these two cases separately.
If Y p is vertical, then by Proposition 27.18, Y p = A p for some \(A \in \mathfrak{g}\). In this case
If Y p is horizontal, then by the right-invariance of the horizontal distribution \(\mathcal{H}\), so is r g∗ Y p . Hence,
(iii) Fix a point p ∈ P. We will show that ω is C ∞ in a neighborhood of p. Let B 1, …, B ℓ be a basis for the Lie algebra \(\mathfrak{g}\) and B 1, …, B ℓ the associated fundamental vector fields on P. By Proposition 27.11, these vector fields are all C ∞ on P. Since \(\mathcal{H}\) is a C ∞ distribution on P, one can find a neighborhood W of p and C ∞ horizontal vector fields X 1, …, X n on W that span \(\mathcal{H}\) at every point of W. Then B 1, …, B ℓ , X 1, …, X n is a C ∞ frame for the tangent bundle TP over W. Thus, any C ∞ vector field X on W can be written as a linear combination
with C ∞ coefficients a i, b j on W. By the definition of ω,
This proves that ω is a C ∞ 1-form on W. □
Note that in this theorem the proof of the smoothness of ω requires only that the horizontal distribution \(\mathcal{H}\) be smooth; it does not use the right-invariance of \(\mathcal{H}\).
Definition 28.2.
An Ehresmann connection or simply a connection on a principal G-bundle P → M is a \(\mathfrak{g}\)-valued 1-form ω on P satisfying the three properties of Theorem 28.1.
A \(\mathfrak{g}\)-valued 1-form α on P can be viewed as a map \(\alpha: TP \rightarrow \mathfrak{g}\) from the tangent bundle TP to the Lie algebra \(\mathfrak{g}\). Now both TP and \(\mathfrak{g}\) are G-manifolds: the Lie group G acts on TP on the right by the differentials of right translations and it acts on \(\mathfrak{g}\) on the left by the adjoint representation. By (27.1), \(\alpha: TP \rightarrow \mathfrak{g}\) is G-equivariant if and only if for all p ∈ P, X p ∈ T p P, and g ∈ G,
or
Thus, \(\alpha: TP \rightarrow \mathfrak{g}\) is G-equivariant if and only if \(r_{g}^{{\ast}}\alpha = (\mathop{\mathrm{Ad}}\nolimits g^{-1})\alpha\) for all g ∈ G. Condition (ii) of a connection ω on a principal bundle says precisely that ω is G-equivariant as a map from TP to \(\mathfrak{g}\).
28.2 Vertical and Horizontal Components of a Tangent Vector
As we noted in Section 27.5, on any principal G-bundle π: P → M, the vertical subspace \(\mathcal{V}_{p}\) of the tangent space T p P is intrinsically defined:
By Proposition 27.18, the map j p∗ naturally identifies the Lie algebra \(\mathfrak{g}\) of G with the vertical subspace \(\mathcal{V}_{p}\).
In the presence of a horizontal distribution on the total space P of a principal bundle, every tangent vector Y p ∈ T p P decomposes uniquely into the sum of a vertical vector and a horizontal vector:
These are called, respectively, the vertical component and horizontal component of the vector Y p . As p varies over P, this decomposition extends to a decomposition of a vector field Y on P:
We often omit the parentheses in v(Y ) and h(Y ), and write vY and hY instead.
Proposition 28.3.
If \(\mathcal{H}\) is a C ∞ horizontal distribution on the total space P of a principal bundle, then the vertical and horizontal components v(Y ) and h(Y ) of a C ∞ vector field Y on P are also C ∞.
Proof.
Let ω be the \(\mathfrak{g}\)-valued 1-form associated to the horizontal distribution \(\mathcal{H}\) by (28.2). It is C ∞ by Theorem 28.1(iii). In terms of a basis B 1, …, B ℓ for \(\mathfrak{g}\), we can write ω = ∑ ω i B i , where ω i are C ∞ 1-forms on P. If Y p ∈ T p P, then by (28.3) its vertical component v(Y p ) is
As p varies over P,
Since ω i, Y, and B i are all C ∞, so is v(Y ). Because h(Y ) = Y − v(Y ), the horizontal component h(Y ) of a C ∞ vector field Y on P is also C ∞. □
On a principal bundle π: P → M, if r g : P → P is right translation by g ∈ G, then π ∘ r g = π. It follows that π ∗∘ r g∗ = π ∗. Thus, the right translation r g∗: T p P → T pg P sends a vertical vector to a vertical vector. By hypothesis, \(r_{g{\ast}}\mathcal{H}_{p} = \mathcal{H}_{pg}\) and hence the right translation r g∗ also sends a horizontal vector to a horizontal vector.
Proposition 28.4.
Suppose \(\mathcal{H}\) is a smooth right-invariant horizontal distribution on the total space of a principal G-bundle π: P → M. For each g ∈ G, the right translation r g∗ commutes with the projections v and h.
Proof.
Any X p ∈ T p P decomposes into vertical and horizontal components:
Applying r g∗ to both sides, we get
Since r g∗ preserves vertical and horizontal subspaces, r g∗ v(X p ) is vertical and r g∗ h(X p ) is horizontal. Thus, (28.4) is the decomposition of r g∗ X p into vertical and horizontal components. This means for every X p ∈ T p P,
□
28.3 The Horizontal Distribution of an Ehresmann Connection
In Section 28.1 we showed that a smooth, right-invariant horizontal distribution on the total space of a principal bundle determines an Ehresmann connection. We now prove the converse.
Theorem 28.5.
If ω is a connection on the principal G-bundle π: P → M, then \(\mathcal{H}_{p}:=\ker \omega _{p}\) , p ∈ P, is a smooth right-invariant horizontal distribution on P.
Proof.
We need to verify three properties:
- (i):
-
At each point p in P, the tangent space T p P decomposes into a direct sum \(T_{p}P = \mathcal{V}_{p} \oplus \mathcal{H}_{p}\).
- (ii):
-
For p ∈ P and g ∈ G, \(r_{g{\ast}}(\mathcal{H}_{p}) \subset \mathcal{H}_{pg}\).
- (iii):
-
\(\mathcal{H}\) is a C ∞ subbundle of the tangent bundle TP.
(i) Since \(\mathcal{H}_{p} =\ker \omega _{p}\), there is an exact sequence
The map \(j_{p{\ast}}: \mathfrak{g} \rightarrow \mathcal{V}_{p} \subset T_{p}P\) provides a splitting of the sequence. By Proposition 27.20, there is a sequence of isomorphisms
(ii) Suppose \(Y _{p} \in \mathcal{H}_{p} =\ker \omega _{p}\). By the right-equivariance property of an Ehresmann connection,
Hence, \(r_{g{\ast}}Y _{p} \in \mathcal{H}_{pg}\).
(iii) Let B 1, …, B ℓ be a basis for the Lie algebra \(\mathfrak{g}\) of G. Then ω = ∑ ω i B i , where ω 1, …, ω ℓ are smooth \(\mathbb{R}\)-valued 1-forms on P and for p ∈ P,
Since \(\omega _{p}: T_{p}P \rightarrow \mathfrak{g}\) is surjective, ω 1, …, ω ℓ are linearly independent at p.
Fix a point p ∈ P and let x 1, …, x m be local coordinates near p on P. Then
for some C ∞ functions f j i in a neighborhood of p.
Let b 1, …, b m be the fiber coordinates of TP near p, i.e., if v q ∈ T q P for q near p, then
In terms of local coordinates,
Let F i(q, b) = ∑ j = 1 m f j i(q)b j, i = 1, …, ℓ. Since ω 1, …, ω ℓ are linearly independent at p, the Jacobian matrix [∂ F i∕∂ b j] = [f j i], an ℓ × m matrix, has rank ℓ at p. Without loss of generality, we may assume that the first ℓ × ℓ block of [f j i(p)] has rank ℓ. Since having maximal rank is an open condition, there is a neighborhood U p of p on which the first ℓ × ℓ block of [f j i] has rank ℓ. By the implicit function theorem, on U p , b 1, …, b ℓ are C ∞ functions of b ℓ+1, …, b m, say
Let
These are C ∞ vector fields on U p that span \(\mathcal{H}_{q}\) at each point q ∈ U p . By the subbundle criterion (Theorem 20.4), \(\mathcal{H}\) is a C ∞ subbundle of TP. □
28.4 Horizontal Lift of a Vector Field to a Principal Bundle
Suppose \(\mathcal{H}\) is a horizontal distribution on a principal bundle π: P → M. Let X be a vector field on M. For every p ∈ P, because the vertical subspace \(\mathcal{V}_{p}\) is kerπ ∗, the differential π ∗: T p P → T π(p) M induces an isomorphism
of the horizontal subspace \(\mathcal{H}_{p}\) with the tangent space T π(p) M. Consequently, there is a unique horizontal vector \(\tilde{X}_{p} \in \mathcal{H}_{p}\) such that \(\pi _{{\ast}}(\tilde{X}_{p}) = X_{\pi (p)} \in T_{\pi (p)}M\). The vector field \(\tilde{X}\) is called the horizontal lift of X to P.
Proposition 28.6.
If \(\mathcal{H}\) is a C ∞ right-invariant horizontal distribution on the total space P of a principal bundle π: P → M, then the horizontal lift \(\tilde{X}\) of a C ∞ vector field X on M is a C ∞ right-invariant vector field on P.
Proof.
Let x ∈ M and p ∈ π −1(x). By definition, \(\pi _{{\ast}}(\tilde{X}_{p}) = X_{x}\). If q is any other point of π −1(x), then q = pg for some g ∈ G. Since π ∘ r g = π,
By the uniqueness of the horizontal lift, \(r_{g{\ast}}\tilde{X}_{p} =\tilde{ X}_{pg}\). This proves the right-invariance of \(\tilde{X}\). We prove the smoothness of \(\tilde{X}\) by proving it locally. Let {U} be a trivializing open cover for P with trivializations \(\phi _{U}: \pi ^{-1}(U)\mathop{ \rightarrow }\limits^{\sim } U \times G\). Define
Let η: U × G → U be the projection to the first factor. Then Z is a C ∞ vector field on U × G such that η ∗ Z (x, g) = X x , and Y: = (ϕ U∗)−1 Z is a C ∞ vector field on π −1(U) such that π ∗ Y p = X π(p). By Proposition 28.3, hY is a C ∞ vector field on π −1(U). Clearly it is horizontal. Because Y p = v(Y p ) + h(Y p ) and π ∗ v(Y p ) = 0, we have π ∗ Y p = π ∗ h(Y p ) = X π(p). Thus, hY lifts X over U. By the uniqueness of the horizontal lift, \(hY =\widetilde{ X}\) over U. This proves that \(\tilde{X}\) is a smooth vector field on P. □
28.5 Lie Bracket of a Fundamental Vector Field
If a principal bundle P comes with a connection, then it makes sense to speak of horizontal vector fields on P; these are vector fields all of whose vectors are horizontal.
Lemma 28.7.
Suppose P is a principal bundle with a connection. Let A be the fundamental vector field on P associated to \(A \in \mathfrak{g}\).
- (i):
-
If Y is a horizontal vector field on P, then [A ,Y ] is horizontal.
- (ii):
-
If Y is a right-invariant vector field on P, then [A ,Y ] = 0.
Proof.
(i) A local flow for A is \(\phi _{t}(p) = pe^{tA} = r_{e^{tA}}(p)\) (Proposition 27.14). By the identification of the Lie bracket with the Lie derivative of vector fields [21, Th. 20.4, p. 225] and the definition of the Lie derivative,
Since right translation preserves horizontality (Theorem 28.5), both \(\left (r_{e^{-tA}}\right )_{{\ast}}Y _{pe^{tA}}\) and Y p are horizontal vectors. Denote the difference quotient in (28.5) by c(t). For every t near 0 in \(\mathbb{R}\), c(t) is in the vector space \(\mathcal{H}_{p}\) of horizontal vectors at p. Therefore, \([\underline{A},Y ]_{p} =\lim _{t\rightarrow 0}c(t) \in \mathcal{H}_{p}\).
(ii) If Y is right-invariant, then
In that case, it follows from (28.5) that [A, Y ] p = 0. □
Problems
28.1. Maurer–Cartan connection
If θ is the Maurer–Cartan form on a Lie group and π 2: M × G → G is the projection to the second factor, prove that ω: = π 2 ∗ θ is a connection on the trivial bundle π 1: M × G → M. It is called the Maurer–Cartan connection.
28.2. Convex linear combinations of connections
Prove that a convex linear combination ω of connections ω 1, …, ω n on a principal bundle π: P → M is again a connection on P. (ω = ∑ λ i ω i , ∑ λ i = 1, λ i ≥ 0.)
28.3. Pullback of a connection
Let π Q : Q → N and π P : P → M be principal G-bundles, and let \((\bar{f}: Q \rightarrow P,f: N \rightarrow M)\) be a morphism of principal bundles. Prove that if θ is a connection on P, then \(\bar{f}^{{\ast}}\theta\) is a connection on Q.
§29 Horizontal Distributions on a Frame Bundle
In this section we will explain the process by which a connection ∇ on a vector bundle E over a manifold M gives rise to a smooth right-invariant horizontal distribution on the associated frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\). This involves a sequence of steps. A connection on the vector bundle E induces a covariant derivative on sections of the vector bundle along a curve. Parallel sections along the curve are those whose derivative vanishes. Just as for tangent vectors in Section 14, starting with a frame e x for the fiber of the vector bundle at the initial point x of a curve, there is a unique way to parallel translate the frame along the curve. In terms of the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\), what this means is that every curve in M has a unique lift to \(\mathop{\mathrm{Fr}}\nolimits (E)\) starting at e x representing parallel frames along the curve. Such a lift is called a horizontal lift. The initial vector at e x of a horizontal lift is a horizontal vector at e x . The horizontal vectors at a point of \(\mathop{\mathrm{Fr}}\nolimits (E)\) form a subspace of the tangent space \(T_{e_{x}}\big(\mathop{\mathrm{Fr}}\nolimits (E)\big)\). In this way we obtain a horizontal distribution on the frame bundle. We show that this horizontal distribution on \(\mathop{\mathrm{Fr}}\nolimits (E)\) arising from a connection on the vector bundle E is smooth and right-invariant. It therefore corresponds to a connection ω on the principal bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\). We then show that ω pulls back under a section e of \(\mathop{\mathrm{Fr}}\nolimits (E)\) to the connection matrix ω e of the connection ∇ relative to the frame e on an open set U.
29.1 Parallel Translation in a Vector Bundle
In Section 14 we defined parallel translation of a tangent vector along a curve in a manifold with an affine connection. In fact, the same development carries over to an arbitrary vector bundle η: E → M with a connection ∇.
Let c: [a, b] → M be a smooth curve in M. Instead of vector fields along the curve c, we consider smooth sections of the pullback bundle c ∗ E over [a, b]. These are called smooth sections of the vector bundle E along the curve c. We denote by Γ(c ∗ E) the vector space of smooth sections of E along the curve c. If E = TM is the tangent bundle of a manifold M, then an element of Γ(c ∗ TM) is simply a vector field along the curve c in M. Just as in Theorem 13.1, there is a unique \(\mathbb{R}\)-linear map
called the covariant derivative corresponding to ∇, such that
- (i):
-
(Leibniz rule) for any C ∞ function f on the interval [a, b],
$$\displaystyle{\frac{D(fs)} {dt} = \frac{df} {dt}s + f \frac{Ds} {dt};}$$ - (ii):
-
if s is induced from a global section \(\tilde{s} \in \varGamma (M,E)\) in the sense that \(s(t) =\tilde{ s}\big(c(t)\big)\), then
$$\displaystyle{\frac{Ds} {dt} (t) = \nabla _{c^{\prime}(t)}\tilde{s}.}$$
Definition 29.1.
A section s ∈ Γ(c ∗ E) is parallel along a curve c: [a, b] → M if Ds∕dt ≡ 0 on [a, b].
As in Section 14.5, the equation Ds∕dt ≡ 0 for a section s to be parallel is equivalent to a system of linear first-order ordinary differential equations. Suppose c: [a, b] → M maps into a framed open set (U, e 1, …, e r ) for E. Then s ∈ Γ(c ∗ E) can be written as
By properties (i) and (ii) of the covariant derivative,
Hence, Ds∕dt ≡ 0 if and only if
This is a system of linear first-order differential equations. By the existence and uniqueness theorems of differential equations, it has a solution on a small interval about a give point t 0 and the solution is uniquely determined by its value at t 0. Thus, a parallel section is uniquely determined by its value at a point. If s ∈ Γ(c ∗ E) is a parallel section of the pullback bundle c ∗ E, we say that s(b) is the parallel transport of s(a) along c: [a, b] → M. The resulting map: E c(a) → E c(b) is called parallel translation from E c(a) to E c(b).
Theorem 29.2.
Let η: E → M be a C ∞ vector bundle with a connection ∇ and let c: [a,b] → M be a smooth curve in M. There is a unique parallel translation φ a,b from E c(a) to E c(b) along c. This parallel translation φ a,b : E c(a) → E c(b) is a linear isomorphism.
The proof is similar to that of Theorem 14.14.
A parallel frame along the curve c: [a, b] → M is a collection of parallel sections \(\big(e_{1}(t),\ldots,e_{r}(t)\big)\), t ∈ [a, b], such that for each t, the elements e 1(t), …, e r (t) form a basis for the vector space E c(t).
Let \(\pi: \mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) be the frame bundle of the vector bundle η: E → M. A curve \(\tilde{c}(t)\) in \(\mathop{\mathrm{Fr}}\nolimits (E)\) is called a lift of the curve c(t) in M if \(c(t) =\pi (\tilde{c}(t))\). It is a horizontal lift if in addition \(\tilde{c}(t)\) is a parallel frame along c.
Restricting the domain of the curve c to the interval [a, t], we obtain from Theorem 29.2 that parallel translation is a linear isomorphism of E c(a) with E c(t). Thus, if a collection of parallel sections \(\big(s_{1}(t),\ldots,s_{r}(t)\big) \in \varGamma (c^{{\ast}}E)\) forms a basis at one time t, then it forms a basis at every time t ∈ [a, b]. By Theorem 29.2, for every smooth curve c: [a, b] → M and ordered basis (s 1, 0, …, s r, 0) for E c(a), there is a unique parallel frame along c whose value at a is (s 1, 0, …, s r, 0). In terms of the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\), this shows the existence and uniqueness of a horizontal lift with a specified initial point in \(\mathop{\mathrm{Fr}}\nolimits (E)\) of a curve c(t) in M.
29.2 Horizontal Vectors on a Frame Bundle
On a general principal bundle vertical vectors are intrinsically defined, but horizontal vectors are not. We will see shortly that a connection on a vector bundle E over a manifold M determines a well-defined horizontal distribution on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\). The elements of the horizontal distribution are the horizontal vectors. Thus, the notion of a horizontal vector on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) depends on a connection on E.
Definition 29.3.
Let E → M be a vector bundle with a connection ∇, x ∈ M, and \(e_{x} \in \mathop{\mathrm{Fr}}\nolimits (E_{x})\). A tangent vector \(v \in T_{e_{x}}(\mathop{\mathrm{Fr}}\nolimits (E))\) is said to be horizontal if there is a curve c(t) through x in M such that v is the initial vector \(\tilde{c}^{\prime}(0)\) of the unique horizontal lift of \(\tilde{c}(t)\) of c(t) to \(\mathop{\mathrm{Fr}}\nolimits (E)\) starting at e x .
Our goal now is to show that the horizontal vectors at a point e x of the frame bundle form a vector subspace of the tangent space \(T_{e_{x}}\big(\mathop{\mathrm{Fr}}\nolimits (E)\big)\). To this end we will derive an explicit formula for \(\tilde{c}^{\prime}(0)\) in terms of a local frame for E. Suppose c: [0, b] → M is a smooth curve with initial point c(0) = x, and \(\tilde{c}(t)\) is its unique horizontal lift to \(\mathop{\mathrm{Fr}}\nolimits (E)\) with initial point e x = (e 1, 0, …, e r, 0). Let s be a frame for E over a neighborhood U of x with s(x) = e x . Then s(c(t)) is a lift of c(t) to \(\mathop{\mathrm{Fr}}\nolimits (E)\) with initial point e x , but of course it is not necessarily a horizontal lift (see Figure 29.1). For any t ∈ [0, b], we have two ordered bases s(c(t)) and \(\tilde{c}(t)\) for E c(t), so there is a smooth matrix \(a(t) \in \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) such that \(s(c(t)) =\tilde{ c}(t)a(t)\). At t = 0, \(s(c(0)) = e_{x} =\tilde{ c}(0)\), so that a(0) = I, the identity matrix in \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\).
Two liftings of a curve
Lemma 29.4.
In the notation above, let \(s_{{\ast}}: T_{x}(M) \rightarrow T_{e_{x}}(\mathop{\mathrm{Fr}}\nolimits (E))\) be the differential of s and a′(0) the fundamental vector field on \(\mathop{\mathrm{Fr}}\nolimits (E)\) associated to \(a^{\prime}(0) \in \mathfrak{g}\mathfrak{l}(r, \mathbb{R})\). Then
Proof.
Let \(P =\mathop{ \mathrm{Fr}}\nolimits (E)\) and \(G =\mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R})\), and let μ: P × G → P be the right action of G on P. Then
with c(0) = x, \(\tilde{c}(0) = e_{x}\), and a(0) = the identity matrix I. Differentiating (29.1) with respect to t and evaluating at 0 gives
By the formula for the differential of an action (Problem 27.4),
□
Lemma 29.5.
Let E → M be a vector bundle with a connection ∇. Suppose s = (s 1 ,…,s r ) is a frame for E over an open set U, \(\tilde{c}(t)\) a parallel frame over a curve c(t) in U with \(\tilde{c}(0) = s(c(0))\) , and a(t) the curve in \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) such that \(s(c(t)) =\tilde{ c}(t)a(t)\) . If ω s = [ω j i ] is the connection matrix of ∇ with respect to the frame (s 1 ,…,s r ) over U, then a′(0) = ω s (c′(0)).
Proof.
Label c(0) = x and \(\tilde{c}_{i}(0) = s_{i}\big(c(0)\big) = e_{i,x}\). By the definition of the connection matrix,
On the other hand, by the defining properties of the covariant derivative (Section 29.1),
Setting t = 0 gives
Equating (29.2) and (29.3), we obtain (a j i)′(0) = ω j i(c′(0)). □
Thus, Lemma 29.4 for the horizontal lift of c′(0) can be rewritten in the form
Proposition 29.6.
Let π: E → M be a smooth vector bundle with a connection over a manifold M of dimension n. For x ∈ M and e x an ordered basis for the fiber E x , the subset \(\mathcal{H}_{e_{x}}\) of horizontal vectors in the tangent space \(T_{e_{x}}(\mathop{\mathrm{Fr}}\nolimits (E))\) is a vector space of dimension n, and \(\pi _{{\ast}}: \mathcal{H}_{e_{x}} \rightarrow T_{x}M\) is a linear isomorphism.
Proof.
In formula (29.4), ω s (c′(0)) is \(\mathbb{R}\)-linear in its argument c′(0) because ω s is a 1-form at c(0). The operation \(A\mapsto \underline{A}_{e_{x}}\) of associating to a matrix \(A \in \mathfrak{g}\mathfrak{l}(r, \mathbb{R})\) a tangent vector \(\underline{A}_{e_{x}} \in T_{e_{x}}\big(\mathop{\mathrm{Fr}}\nolimits (E)\big)\) is \(\mathbb{R}\)-linear by (27.3). Hence, formula (29.4) shows that the map
is \(\mathbb{R}\)-linear. As the image of a vector space T x M under a linear map, the set \(\mathcal{H}_{e_{x}}\) of horizontal vectors \(\tilde{c}^{\prime}(0)\) at e x is a vector subspace of \(T_{e_{x}}\big(\mathop{\mathrm{Fr}}\nolimits (E)\big)\).
Since \(\pi \big(\tilde{c}(t)\big) = c(t)\), taking the derivative at t = 0 gives \(\pi _{{\ast}}\big(\tilde{c}^{\prime}(0)\big) = c^{\prime}(0)\), so π ∗ is a left inverse to the map ϕ. This proves that \(\phi: T_{x}M \rightarrow T_{e_{x}}\big(\mathop{\mathrm{Fr}}\nolimits (E)\big)\) is injective. Its image is by definition \(\mathcal{H}_{e_{x}}\). It follows that \(\phi: T_{x}M \rightarrow \mathcal{H}_{e_{x}}\) is an isomorphism with inverse \(\pi _{{\ast}}: \mathcal{H}_{e_{x}} \rightarrow T_{e_{x}}M\). □
29.3 Horizontal Lift of a Vector Field to a Frame Bundle
We have learned so far that a connection on a vector bundle E → M defines a horizontal subspace \(\mathcal{H}_{p}\) of the tangent space T p P at each point p of the total space of the frame bundle \(\pi: P =\mathop{ \mathrm{Fr}}\nolimits (E) \rightarrow M\). The horizontal subspace \(\mathcal{H}_{p}\) has the same dimension as M. The vertical subspace \(\mathcal{V}_{p}\) of T p P is the kernel of the surjection π ∗: T p P → T π(p) M; as such, \(\dim \mathcal{V}_{p} =\dim T_{p}P -\dim M\). Hence, \(\mathcal{V}_{p}\) and \(\mathcal{H}_{p}\) have complementary dimensions in T p P. Since \(\pi _{{\ast}}(\mathcal{V}_{p}) = 0\) and \(\pi _{{\ast}}: \mathcal{H}_{p} \rightarrow T_{\pi (p)}M\) is an isomorphism, \(\mathcal{V}_{p} \cap \mathcal{H}_{p} = 0\). It follows that there is a direct sum decomposition
Our goal now is to show that as p varies in P, the subset \(\mathcal{H}:=\bigcup _{p\in P}\mathcal{H}_{p}\) of the tangent bundle TP defines a C ∞ horizontal distribution on P in the sense of Section 27.6.
Since \(\pi _{{\ast},p}: \mathcal{H}_{p} \rightarrow T_{\pi (p)}M\) is an isomorphism for each p ∈ P, if X is a vector field on M, then there is a unique vector field \(\tilde{X}\) on P such that \(\tilde{X}_{p} \in \mathcal{H}_{p}\) and \(\pi _{{\ast},p}(\tilde{X}_{p}) = X_{\pi (p)}\). The vector field \(\tilde{X}\) is called the horizontal lift of X to the frame bundle P.
Since every tangent vector X x ∈ T x M is the initial vector c′(0) of a curve c, formula (29.4) for the horizontal lift of a tangent vector can be rewritten in the following form.
Lemma 29.7 (Horizontal lift formula).
Suppose ∇ is a connection on a vector bundle E → M and ω s is its connection matrix on a framed open set (U,s). For x ∈ U, \(p = s(x) \in \mathop{\mathrm{Fr}}\nolimits (E)\) , and X x ∈ T x M, let \(\tilde{X}_{p}\) be the horizontal lift of X x to p in \(\mathop{\mathrm{Fr}}\nolimits (E)\) . Then
Proposition 29.8.
Let E → M be a C ∞ rank r vector bundle with a connection and \(\pi: \mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) its frame bundle. If X is a C ∞ vector field on M, then its horizontal lift \(\tilde{X}\) to \(\mathop{\mathrm{Fr}}\nolimits (E)\) is a C ∞ vector field.
Proof.
Let \(P =\mathop{ \mathrm{Fr}}\nolimits (E)\) and \(G =\mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R})\). Since the question is local, we may assume that the bundle P is trivial, say P = M × G. By the right invariance of the horizontal distribution,
Let s: M → P = M × G be the section s(x) = (x, 1). By the horizontal lift formula (Lemma 29.7),
Let p = (x, a) ∈ P and let f be a C ∞ function on P. We will prove that \(\tilde{X}_{p}f\) is C ∞ as a function of p. By (29.6) and (29.7),
so it suffices to prove separately that \(\big(r_{a{\ast}}(s_{{\ast},x}X_{x})\big)f\) and \(\big(r_{a{\ast}}\underline{\omega _{s}(X_{x})}_{(x,1)}\big)f\) are C ∞ functions on P.
The first term is
where μ: P × G → P is the action of G on P and π 2: P = M × G → G is the projection π 2(p) = π 2(x, a) = a. The formula (29.9) expresses \(\big(r_{a{\ast}}s_{{\ast},x}(X_{x})\big)f\) as a C ∞ function on P.
By the right equivariance of the connection form ω s , in (29.8) the second term can be rewritten as
where \(\big(\mathop{\mathrm{Ad}}\nolimits \pi _{2}(p)^{-1}\big)\omega _{s}(X_{\pi (p)})\) is a C ∞ function: \(P \rightarrow \mathfrak{g}\mathfrak{l}(r, \mathbb{R})\) that we will denote by A(p). The problem now is to show that p ↦ A(p) p f is a C ∞ function of p.
Let μ: P × G → P be the right action of \(G =\mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R})\) on \(P =\mathop{ \mathrm{Fr}}\nolimits (E)\). Then
Since f, μ, A, and the exponential map are all C ∞ functions, A(p) p f is a C ∞ function of p. Thus, \(\tilde{X}_{p}f\) in (29.8) is a C ∞ function of p. This proves that \(\tilde{X}\) is a C ∞ vector field on P. □
Theorem 29.9.
A connection ∇ on a smooth vector bundle E → M defines a C ∞ distribution \(\mathcal{H}\) on the frame bundle \(\pi: P =\mathop{ \mathrm{Fr}}\nolimits (E) \rightarrow M\) such that at any p ∈ P,
- (i):
-
\(T_{p}P = \mathcal{V}_{p} \oplus \mathcal{H}_{p}\);
- (ii):
-
\(r_{g{\ast}}(\mathcal{H}_{p}) = \mathcal{H}_{pg}\) for any \(g \in G =\mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R})\),
where r g : P → P is the right action of G on P.
Proof.
To prove that \(\mathcal{H}\) is a C ∞ subbundle of TP, let U be a coordinate open set in M and s 1, …, s n a C ∞ frame on U. By Proposition 29.8 the horizontal lifts \(\widetilde{s_{1}},\ldots,\widetilde{s_{n}}\) are C ∞ vector fields on \(\tilde{U}:=\pi ^{-1}(U)\). Moreover, for each \(p \in \tilde{ U}\), since \(\pi _{{\ast},p}: \mathcal{H}_{p} \rightarrow T_{\pi (p)}M\) is an isomorphism, \((\widetilde{s_{1}})_{p},\ldots,(\widetilde{s_{n}})_{p}\) form a basis for \(\mathcal{H}_{p}\). Thus, over \(\tilde{U}\) the C ∞ sections \(\widetilde{s_{1}},\ldots,\widetilde{s_{n}}\) of TP span \(\mathcal{H}\). By Theorem 20.4, this proves that \(\mathcal{H}\) is a C ∞ subbundle of TP.
Equation (29.5) establishes (i).
As for (ii), let \(\tilde{c}^{\prime}(0) \in \mathcal{H}_{p}\), where c(t) is a curve in M and \(\tilde{c}(t) = [v_{1}(t)\ \cdots \ v_{r}(t)]\) is its horizontal lift to P with initial point p. Here we are writing a frame as a row vector so that the group action is simply matrix multiplication on the right. For any \(g = [g_{j}^{i}] \in \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\),
Since Dv i ∕dt ≡ 0 by the horizontality of v i and g j i are constants, D(∑ g j i v i )∕dt ≡ 0. Thus, \(\tilde{c}(t)g\) is the horizontal lift of c(t) with initial point \(\tilde{c}(0)g\). It has initial tangent vector
This proves that \(r_{g{\ast}}\mathcal{H}_{p} \subset \mathcal{H}_{pg}\). Because \(r_{g{\ast}}: \mathcal{H}_{p} \rightarrow \mathcal{H}_{pg}\) has a two-sided inverse r g −1 ∗, it is bijective. In particular, \(r_{g{\ast}}\mathcal{H}_{p} = \mathcal{H}_{pg}\). □
29.4 Pullback of a Connection on a Frame Bundle Under a Section
Recall that a connection ∇ on a vector bundle E can be represented on a framed open set (U, e 1, …, e r ) for E by a connection matrix ω e depending on the frame. Such a frame e = (e 1, …, e r ) is in fact a section \(e: U \rightarrow \mathop{\mathrm{Fr}}\nolimits (E)\) of the frame bundle. We now use the horizontal lift formula (Lemma 29.7) to prove that the Ehresmann connection ω on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) determined by ∇ pulls back under the section e to the connection matrix ω e .
Theorem 29.10.
Let ∇ be a connection on a vector bundle E → M and let ω be the Ehresmann connection on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) determined by ∇. If e = (e 1 ,…,e r ) is a frame for E over an open set U, viewed as a section \(e: U \rightarrow \mathop{\mathrm{Fr}}\nolimits (E)\vert _{U}\) , and ω e is the connection matrix of ∇ relative to the frame e, then ω e = e ∗ ω.
Proof.
Let x ∈ U and \(p = e(x) \in \mathop{\mathrm{Fr}}\nolimits (E)\). Suppose X x is a tangent vector to M at x. If we write ω e, x for the value of the connection matrix ω e at the point x ∈ U, then ω e, x is an r × r matrix of 1-forms at x and ω e, x (X x ) is an r × r matrix of real numbers, i.e., an element of the Lie algebra \(\mathfrak{g}\mathfrak{l}(r, \mathbb{R})\). The corresponding fundamental vector field on \(\mathop{\mathrm{Fr}}\nolimits (E)\) is ω e, x (X x ). By Lemma 29.7, the horizontal lift of X x to \(p \in \mathop{\mathrm{Fr}}\nolimits (E)\) is
Applying the Ehresmann connection ω p to both sides of this equation, we get
Since this is true for all X x ∈ T x M,
□
§30 Curvature on a Principal Bundle
Let G be a Lie group with Lie algebra \(\mathfrak{g}\). Associated to a connection ω on a principal G-bundle is a \(\mathfrak{g}\)-valued 2-form Ω called its curvature. The definition of the curvature is suggested by the second structural equation for a connection ∇ on a vector bundle E. Just as the connection form ω on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) pulls back by a section e of \(\mathop{\mathrm{Fr}}\nolimits (E)\) to the connection matrix ω e of ∇ with respect to the frame e, so the curvature form Ω on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) pulls back by e to the curvature matrix Ω e of ∇ with respect to e. Thus, the curvature form Ω on the frame bundle is an intrinsic object of which the curvature matrices Ω e are but local manifestations.
30.1 Curvature Form on a Principal Bundle
By Theorem 11.1 if ∇ is a connection on a vector bundle E → M, then its connection and curvature matrices ω e and Ω e on a framed open set (U, e) = (U, e 1, …, e r ) are related by the second structural equation (Theorem 11.1)
In terms of the Lie bracket of matrix-valued forms (see (21.12)), this can be rewritten as
An Ehresmann connection on a principal bundle is Lie algebra-valued. In a general Lie algebra, the wedge product is not defined, but the Lie bracket is always defined. This strongly suggests the following definition for the curvature of an Ehresmann connection on a principal bundle.
Definition 30.1.
Let G be a Lie group with Lie algebra \(\mathfrak{g}\). Suppose ω is an Ehresmann connection on a principal G-bundle π: P → M. Then the curvature of the connection ω is the \(\mathfrak{g}\)-valued 2-form
Recall that frames for a vector bundle E over an open set U are sections of the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\). Let ω be the connection form on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) determined by a connection ∇ on E. In the same way that ω pulls back by sections of \(\mathop{\mathrm{Fr}}\nolimits (E)\) to connection matrices, the curvature form Ω of the connection ω on \(\mathop{\mathrm{Fr}}\nolimits (E)\) pulls back by sections to curvature matrices.
Proposition 30.2.
If ∇ is a connection on a vector bundle E → M and ω is the associated Ehresmann connection on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) , then the curvature matrix Ω e relative to a frame e = (e 1 ,…,e r ) for E over an open set U is the pullback e ∗ Ω of the curvature Ω on the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\).
Proof.
□
30.2 Properties of the Curvature Form
Now that we have defined the curvature of a connection on a principal G-bundle π: P → M, it is natural to study some of its properties. Like a connection form, the curvature form Ω is equivariant with respect to right translation on P and the adjoint representation on \(\mathfrak{g}\). However, unlike a connection form, a curvature form is horizontal in the sense that it vanishes as long as one argument is vertical. In this respect it acts almost like the opposite of a connection form, which vanishes on horizontal vectors.
Lemma 30.3.
Let G be a Lie group with Lie algebra \(\mathfrak{g}\) and π: P → M a principal G-bundle with a connection ω. Fix a point p ∈ P.
- (i):
-
Every vertical vector X p ∈ T p P can be extended to a fundamental vector field A on P for some \(A \in \mathfrak{g}\).
- (ii):
-
Every horizontal vector Y p ∈ T p P can be extended to the horizontal lift \(\tilde{B}\) of a C ∞ vector field B on M.
Proof.
(i) By the surjectivity of \(j_{p{\ast}}: \mathfrak{g} \rightarrow \mathcal{V}_{p}\) (Proposition 27.18) and Equation (27.3),
for some \(A \in \mathfrak{g}\). Then the fundamental vector field A on P extends X p .
(ii) Let x = π(p) in M and let B x be the projection π ∗(Y p ) ∈ T x M of the vector Y p . We can extend B x to a smooth vector field B on M. The horizontal lift \(\tilde{B}\) of B extends Y p on P. □
By Proposition 28.6, such a horizontal lift \(\tilde{B}\) is necessarily right-invariant.
Theorem 30.4.
Let G be a Lie group with Lie algebra \(\mathfrak{g}\) . Suppose π: P → M is a principal G-bundle, ω a connection on P, and Ω the curvature form of ω.
- (i):
-
(Horizontality) For p ∈ P and X p ,Y p ∈ T p P,
$$\displaystyle{ \varOmega _{p}(X_{p},Y _{p}) = (d\omega )_{p}(hX_{p},hY _{p}). }$$(30.1) - (ii):
-
(G-equivariance) For g ∈ G, we have \(r_{g}^{{\ast}}\varOmega = (\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega\).
- (iii):
-
(Second Bianchi identity) dΩ = [Ω,ω].
Proof.
(i) Since both sides of (30.1) are linear in X p and in Y p , we may decompose X p and Y p into vertical and horizontal components, and so it suffices to check the equation for vertical and horizontal vectors only. There are three cases.
Case 1. Both X p and Y p are horizontal. Then
Case 2. One of X p and Y p is horizontal; the other is vertical. Without loss of generality, we may assume X p vertical and Y p horizontal. Then [ω p , ω p ](X p , Y p ) = 0 as in Case 1.
By Lemma 30.3 the vertical vector X p extends to a fundamental vector field A on P and the horizontal vector Y p extends to a right-invariant horizontal vector field \(\tilde{B}\) on P. By the global formula for the exterior derivative (Problem 21.105)
On the right-hand side, \(\omega (\tilde{B}) = 0\) because \(\tilde{B}\) is horizontal, and \(\tilde{B}\omega (\underline{A}) =\tilde{ B}A = 0\) because A is a constant function on P. Being the bracket of a fundamental and a horizontal vector field, \([\underline{A},\tilde{B}]\) is horizontal by Lemma 28.7, and therefore \(\omega([\underline{A}, \tilde{B}])=0\), the left-hand side of (30.1) becomes
The right-hand side of (30.1) is also zero because hX p = 0.
Case 3. Both X p and Y p are vertical. As in Case 2, we can write X p = A p and Y p = B p for some \(A,B \in \mathfrak{g}\). We have thus extended the vertical vectors X p and Y p to fundamental vector fields X = A and Y = B on P. By the definition of curvature,
In this sum the first term is
Hence, (30.2) becomes
On the other hand,
(ii) Since the connection form ω is right-equivariant with respect to \(\mathop{\mathrm{Ad}}\nolimits\),
In this computation we used the fact that because \(\mathop{\mathrm{Ad}}\nolimits g^{-1} = (c_{g^{-1}})_{{\ast}}\) is the differential of a Lie group homomorphism, it is a Lie algebra homomorphism.
(iii) Taking the exterior derivative of the definition of the curvature form, we get
□
In case P is the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) of a rank r vector bundle E, with structure group \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\), the second Bianchi identity becomes by Proposition 21.7
where the connection and curvature forms ω and Ω are \(\mathfrak{g}\mathfrak{l}(r, \mathbb{R})\)-valued forms on \(\mathop{\mathrm{Fr}}\nolimits (E)\). It should not be so surprising that it has the same form as the second Bianchi identity for the connection and curvature matrices relative to a frame e for E (Proposition 22.3). Indeed, by pulling back (30.3) by a frame \(e: U \rightarrow \mathop{\mathrm{Fr}}\nolimits (E)\), we get
which is precisely Proposition 22.3.
Problems
30.1. Curvature of the Maurer–Cartan connection
Let G be a Lie group with Lie algebra \(\mathfrak{g}\), and M a manifold. Compute the curvature of the Maurer–Cartan connection ω on the trivial bundle π: M × G → M.
30.2. Generalized second Bianchi identity on a frame bundle
Suppose \(\mathop{\mathrm{Fr}}\nolimits (E)\) is the frame bundle of a rank r vector bundle E over M. Let ω be an Ehresmann connection and Ω its curvature form on \(\mathop{\mathrm{Fr}}\nolimits (E)\). These are differential forms on \(\mathop{\mathrm{Fr}}\nolimits (E)\) with values in the Lie algebra \(\mathfrak{g}\mathfrak{l}(r, \mathbb{R})\). Matrix multiplication and the Lie bracket on \(\mathfrak{g}\mathfrak{l}(r, \mathbb{R})\) lead to two ways to multiply \(\mathfrak{g}\mathfrak{l}(r, \mathbb{R})\)-valued forms (see Section 21.5). We write Ω k to denote the wedge product of Ω with itself k times. Prove that d(Ω k) = [Ω k, ω].
30.3. Lie bracket of horizontal vector fields
Let P → M be a principal bundle with a connection, and X, Y horizontal vector fields on P.
- (a):
-
Prove that Ω(X, Y ) = −ω([X, Y ]).
- (b):
-
Show that [X, Y ] is horizontal if and only if the curvature Ω(X, Y ) equals zero.
§31 Covariant Derivative on a Principal Bundle
Throughout this chapter, G will be a Lie group with Lie algebra \(\mathfrak{g}\) and V will be a finite-dimensional vector space. To a principal G-bundle π: P → M and a representation \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\), one can associate a vector bundle P × ρ V → M with fiber V. When ρ is the adjoint representation \(\mathop{\mathrm{Ad}}\nolimits\) of G on its Lie algebra \(\mathfrak{g}\), the associated bundle \(P \times _{\mathop{\mathrm{Ad}}\nolimits }\mathfrak{g}\) is called the adjoint bundle, denoted by \(\mathop{\mathrm{Ad}}\nolimits P\).
Differential forms on M with values in the associated bundle P × ρ V turn out to correspond in a one-to-one manner to certain V -valued forms on P called tensorial forms of type ρ. The curvature Ω of a connection ω on the principal bundle P is a \(\mathfrak{g}\)-valued tensorial 2-form of type \(\mathop{\mathrm{Ad}}\nolimits\) on P. Under this correspondence it may be viewed as a 2-form on M with values in the adjoint bundle \(\mathop{\mathrm{Ad}}\nolimits P\).
Using a connection ω, one can define a covariant derivative D of vector-valued forms on a principal bundle P. This covariant derivative maps tensorial forms to tensorial forms, and therefore induces a covariant derivative on forms on M with values in an associated bundle. In terms of the covariant derivative D, the curvature form is Ω = D ω, and Bianchi’s second identity becomes D Ω = 0.
31.1 The Associated Bundle
Let π: P → M be a principal G-bundle and \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) a representation of G on a finite-dimensional vector space V. We write ρ(g)v as g ⋅ v or even gv. The associated bundle E: = P × ρ V is the quotient of P × V by the equivalence relation
We denote the equivalence class of (p, v) by [p, v]. The associated bundle comes with a natural projection β: P × ρ V → M, β([p, v]) = π(p). Because
the projection β is well defined.
As a first example, the proposition below shows that an associated bundle of a trivial principal G-bundle is a trivial vector bundle.
Proposition 31.1.
If \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) is a finite-dimensional representation of a Lie group G, and U is any manifold, then there is a fiber-preserving diffeomorphism
Proof.
The map ϕ is well defined because if h is any element of G, then
Define ψ: U × V → (U × G) × ρ V by
It is easy to check that ϕ and ψ are inverse to each other, are C ∞, and commute with the projections. □
Since a principal bundle P → M is locally U × G, Proposition 31.1 shows that the associated bundle P × ρ V → M is locally trivial with fiber V. The vector space structure on V then makes P × ρ V into a vector bundle over M:
It is easy to show that these are well-defined operations not depending on the choice of p ∈ E x and that this makes the associated bundle β: E → M into a vector bundle (Problem 31.2).
Example 31.2.
Let \(\mathop{\mathrm{Ad}}\nolimits: G \rightarrow \mathop{\mathrm{GL}}\nolimits (\mathfrak{g})\) be the adjoint representation of a Lie group G on its Lie algebra \(\mathfrak{g}\). For a principal G-bundle π: P → M, the associated vector bundle \(\mathop{\mathrm{Ad}}\nolimits P:= P \times _{\mathop{\mathrm{Ad}}\nolimits }\mathfrak{g}\) is called the adjoint bundle of P.
31.2 The Fiber of the Associated Bundle
If π: P → M is a principal G-bundle, \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) is a representation, and E: = P × ρ V → M is the associated bundle, we denote by P x the fiber of P above x ∈ M, and by E x the fiber of E above x ∈ M. For each p ∈ P x , there is a canonical way of identifying the fiber E x with the vector space V:
Lemma 31.3.
Let π: P → M be a principal G-bundle, \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) a finite-dimensional representation, and E = P × ρ V the associated vector bundle. For each point p in the fiber P x , the map f p : V → E x is a linear isomorphism.
Proof.
Suppose [p, v] = [p, w]. Then (p, w) = (pg, g −1 v) for some g ∈ G. Since G acts freely on P, the equality p = pg implies that g = 1. Hence, w = g −1 v = v. This proves that f p is injective.
If [q, w] is any point in E x , then q ∈ P x , so q = pg for some g ∈ G. It follows that
This proves that f p is surjective. □
The upshot is that every point p of the total space P of a principal bundle gives a linear isomorphism f p : V → E π(p) from V to the fiber of the associated bundle E above π(p).
Lemma 31.4.
Let E = P × ρ V be the vector bundle associated to the principal G-bundle P → M via the representation \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) , and f p : V → E x the linear isomorphism v ↦ [p,v]. If g ∈ G, then f pg = f p ∘ρ(g).
Proof.
For v ∈ V,
□
Example 31.5.
Let π: P → M be a principal G-bundle. The vector bundle P × ρ V → M associated to the trivial representation \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) is the trivial bundle M × V → M, for there is a vector bundle isomorphism
with inverse map
In this case, for each p ∈ P the linear isomorphism f p : V → E x = V, v ↦ [p, v], is the identity map.
31.3 Tensorial Forms on a Principal Bundle
We keep the same notation as in the previous section. Thus, π: P → M is a principal G-bundle, \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) a finite-dimensional representation of G, and E: = P × ρ V the vector bundle associated to P via ρ.
Definition 31.6.
A V -valued k-form φ on P is said to be right-equivariant of type ρ or right-equivariant with respect to ρ if for every g ∈ G,
What this means is that for p ∈ P and v 1, …, v k ∈ T p P,
In the literature (for example, [12, p. 75]), such a form is said to be pseudo-tensorial of type ρ.
Definition 31.7.
A V -valued k-form φ on P is said to be horizontal if φ vanishes whenever one of its arguments is a vertical vector. Since a 0-form never takes an argument, every 0-form on P is by definition horizontal.
Definition 31.8.
A V -valued k-form φ on P is tensorial of type ρ if it is right-equivariant of type ρ and horizontal. The set of all smooth tensorial V -valued k-forms of type ρ is denoted by Ω ρ k(P, V ).
Example.
Since the curvature Ω of a connection ω on a principal G-bundle P is horizontal and right-equivariant of type \(\mathop{\mathrm{Ad}}\nolimits\), it is tensorial of type \(\mathop{\mathrm{Ad}}\nolimits\).
The set Ω ρ k(P, V ) of tensorial k-forms of type ρ on P becomes a vector space with the usual addition and scalar multiplication of forms. These forms are of special interest because they can be viewed as forms on the base manifold M with values in the associated bundle E: = P × ρ V. To each tensorial V -valued k form φ ∈ Ω ρ k(P, V ) we associate a k-form φ ♭ ∈ Ω k(M, E) as follows. Given x ∈ M and v 1, …, v k ∈ T x M, choose any point p in the fiber P x and choose lifts u 1, …, u k at p of v 1, …, v k , i.e., vectors in T p P such that π ∗(u i ) = v i . Then φ ♭ is defined by
where f p : V → E x is the isomorphism v ↦ [p, v] of the preceding section.
Conversely, if ψ ∈ Ω k(M, E), we define ψ ♯ ∈ Ω ρ k(P, V ) as follows. Given p ∈ P and u 1, …, u k ∈ T p P, let x = π(p) and set
Theorem 31.9.
The map
is a well-defined linear isomorphism with inverse \(\psi ^{\sharp } \leftarrowtail \psi\).
Proof.
To show that φ ♭ is well defined, we need to prove that the definition (31.3) is independent of the choice of p ∈ P x and of u 1, …, u k ∈ T p P. Suppose u′1, …, u′ k ∈ T p P is another set of vectors such that π ∗(u′ i ) = v i . Then π ∗(u′ i − u i ) = 0 so that u′ i − u i is vertical. Since φ is horizontal and k-linear,
This proves that for a given p ∈ P, the definition (31.3) is independent of the choice of lifts of v 1, …, v k to p.
Next suppose we choose pg instead of p as the point in the fiber P x . Because π ∘ r g = π,
so that r g∗ u 1, …, r g∗ u k are lifts of v 1, …, v k to pg. We have, by right equivariance with respect to ρ,
So by Lemma 31.4,
This proves that the definition (31.3) is independent of the choice of p in the fiber P x .
Let ψ ∈ Ω k(M, E). It is clear from the definition (31.4) that ψ ♯ is horizontal. It is easy to show that ψ ♯ is right-equivariant with respect to ρ (Problem 31.4). Hence, ψ ♯ ∈ Ω ρ k(P, V ).
For v 1, …, v k ∈ T x M, choose p ∈ P x and vectors u 1, …, u k ∈ T p P that lift v 1, …, v k . Then
Hence, ψ ♯ ♭ = ψ.
Similarly, φ ♭ ♯ = φ for φ ∈ Ω ρ k(P, V ), which we leave to the reader to show (Problem 31.5). Therefore, the map ψ ↦ ψ ♯ is inverse to the map φ ↦ φ ♭. □
Example 31.10 (Curvature as a form on the base).
By Theorem 31.9, the curvature form Ω of a connection on a principal G-bundle P can be viewed as an element of \(\varOmega ^{2}(M,\mathop{\mathrm{Ad}}\nolimits P)\), a 2-form on M with values in the adjoint bundle \(\mathop{\mathrm{Ad}}\nolimits P\).
When k = 0 in Theorem 31.9, Ω p 0(P, V ) consists of maps f: P → V that are right-equivariant with respect to ρ:
or
On the right-hand side of Theorem 31.9,
Hence, we have the following corollary.
Corollary 31.11.
Let G be a Lie group, P → M a principal G-bundle, and \(\rho: G \rightarrow \mathop{\mathrm{Aut}}\nolimits (V )\) a representation of G. There is a one-to-one correspondence
By the local triviality condition, for any principal bundle π: P → M the projection map π is a submersion and therefore the pullback map π ∗: Ω ∗(M) → Ω ∗(P) is an injection. A differential form φ on P is said to be basic if it is the pullback π ∗ ψ of a form ψ on M; it is G-invariant if r g ∗ φ = φ for all g ∈ G. More generally, for any vector space V, these concepts apply to V -valued forms as well.
Suppose \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) is the trivial representation \(\rho (g) =\mathbb{1}\) for all g ∈ G. Then an equivariant form φ of type ρ on P satisfies
Thus, an equivariant form of type ρ for the trivial representation ρ is exactly an invariant form on P. Unravelling Theorem 31.9 for a trivial representation will give the following theorem.
Theorem 31.12.
Let π: P → M be a principal G-bundle and V a vector space. A V-valued form on P is basic if and only if it is horizontal and G-invariant.
Proof.
Let \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) be the trivial representation. As noted above, Ω ρ k(P, V ) consists of horizontal, G-invariant V -valued k-forms on P.
By Example 31.5, when ρ is the trivial representation, the vector bundle E = P × ρ V is the product bundle M × V over M and for each p ∈ P, the linear isomorphism f p : V → E x = V, where x = π(p), is the identity map. Then the isomorphism
is given by
Therefore,
This proves that horizontal, G-invariant forms on P are precisely the basic forms. □
31.4 Covariant Derivative
Recall that the existence of a connection ω on a principal G-bundle π: P → M is equivalent to the decomposition of the tangent bundle TP into a direct sum of the vertical subbundle \(\mathcal{V}\) and a smooth right-invariant horizontal subbundle \(\mathcal{H}\). For any vector X p ∈ T p P, we write
as the sum of its vertical and horizontal components. This will allow us to define a covariant derivative of vector-valued forms on P. By the isomorphism of Theorem 31.9, we obtain in turn a covariant derivative of forms on M with values in an associated bundle.
Let \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) be a finite-dimensional representation of G and let E: = P × ρ V be the associated vector bundle.
Proposition 31.13.
If φ ∈Ω k (P,V ) is right-equivariant of type ρ, then so is dφ.
Proof.
For a fixed g ∈ G,
since ρ(g −1) is a constant linear map for a fixed g. □
In general, the exterior derivative does not preserve horizontality. For any V -valued k-form φ on P, we define its horizontal component φ h ∈ Ω k(P, V ) as follows: for p ∈ P and v 1, …, v k ∈ T p P,
Proposition 31.14.
If φ ∈Ω k (P,V ) is right-equivariant of type ρ, then so is φ h.
Proof.
For g ∈ G, p ∈ P, and v 1, …, v k ∈ T p P,
□
Propositions 31.13 and 31.14 together imply that if φ ∈ Ω k(P, V ) is right-equivariant of type ρ, then (d φ)h ∈ Ω k+1(P, V ) is horizontal and right-equivariant of type ρ, i.e., tensorial of type ρ.
Definition 31.15.
Let π: P → M be a principal G-bundle with a connection ω and let V be a real vector space. The covariant derivative of a V -valued k-form φ ∈ Ω k(P, V ) is D φ = (d φ)h.
Let \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) be a finite-dimensional representation of the Lie group G. The covariant derivative is defined for any V -valued k-form on P, and it maps a right-equivariant form of type ρ to a tensorial form of type ρ. In particular, it restricts to a map
on the space of tensorial forms.
Proposition 31.16.
Let π: P → M be a principal G-bundle with a connection and \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) a representation of G. The covariant derivative
on tensorial forms of type ρ is an antiderivation of degree + 1.
Proof.
Let ω, τ ∈ Ω ρ ∗(P, V ) be tensorial forms of type ρ. Then
Since τ and ω are horizontal, τ h = τ and ω h = ω. Therefore,
⊓⊔
If E: = P × ρ V is the associated vector bundle via the representation ρ, then the isomorphism of Theorem 31.9 transforms the linear map (31.5) into a linear map
Unlike the exterior derivative, the covariant derivative depends on the choice of a connection on P. Moreover, D 2 ≠ 0 in general.
Example 31.17 (Curvature of a principal bundle).
By Theorem 30.4 the curvature form \(\varOmega \in \varOmega _{\mathop{\mathrm{Ad}}\nolimits }^{2}(P,\mathfrak{g})\) on a principal bundle is the covariant derivative D ω of the connection form \(\omega \in \varOmega ^{1}(P,\mathfrak{g})\). Because ω is not horizontal, it is not in \(\varOmega _{\mathop{\mathrm{Ad}}\nolimits }^{1}(P,\mathfrak{g})\).
31.5 A Formula for the Covariant Derivative of a Tensorial Form
Let π: P → M be a smooth principal G-bundle with a connection ω, and let \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) be a finite-dimensional representation of G. In the preceding section we defined the covariant derivative of a V -valued k-form φ on P: D φ = (d φ)h, the horizontal component of d φ. In this section we derive a useful alternative formula for the covariant derivative, but only for a tensorial form.
The Lie group representation \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) induces a Lie algebra representation \(\rho _{{\ast}}: \mathfrak{g} \rightarrow \mathfrak{g}\mathfrak{l}(V )\), which allows us to define a product of a \(\mathfrak{g}\)-valued k-form τ and a V -valued ℓ-form φ on P: for p ∈ P and v 1, …, v k+ℓ ∈ T p P,
For the same reason as the wedge product, τ ⋅ φ is multilinear and alternating in its arguments; it is therefore a (k + ℓ)-covector with values in V.
Example 31.18.
If \(V = \mathfrak{g}\) and \(\rho =\mathop{ \mathrm{Ad}}\nolimits: G \rightarrow \mathop{\mathrm{GL}}\nolimits (\mathfrak{g})\) is the adjoint representation, then
In this case we also write [τ, φ] instead of τ ⋅ φ.
Theorem 31.19.
Let π: P → M be a principal G-bundle with connection form ω, and \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) a finite-dimensional representation of G. If φ ∈Ω ρ k (P,V ) is a V -valued tensorial form of type ρ, then its covariant derivative is given by
Proof.
Fix p ∈ P and v 1, …, v k+1 ∈ T p P. We need to show that
Because both sides of (31.6) are linear in each argument v i , which may be decomposed into the sum of a vertical and a horizontal component, we may assume that each v i is either vertical or horizontal. By Lemma 30.3, throughout the proof we may further assume that the vectors v 1, …, v k+1 have been extended to vector fields X 1, …, X k+1 on P each of which is either vertical or horizontal. If X i is vertical, then it is a fundamental vector field A i for some \(A_{i} \in \mathfrak{g}\). If X i is horizontal, then it is the horizontal lift \(\tilde{B_{i}}\) of a vector field B i on M. By construction, \(\tilde{B_{i}}\) is right-invariant (Proposition 28.6).
Instead of proving (31.6) at a point p, we will prove the equality of functions
where
and
Case 1. The vector fields X 1 ,…,X k+1 are all horizontal.Then II = 0 because ω(X σ(1)) = 0 for all σ ∈ S k+1. In this case, (31.7) is trivially true.
Case 2. At least two of X 1 ,…,X k+1 are vertical.By the skew-symmetry of the arguments, we may assume that X 1 = A 1 and X 2 = A 2 are vertical. By Problem 27.1, [X 1, X 2] = [A 1, A 2] is also vertical.
The left-hand side of (31.7) is zero because hX 1 = 0. By the global formula for the exterior derivative [21, Th. 20.14, p. 233],
In this expression every term in the first sum is zero because φ is horizontal and at least one of its arguments is vertical. In the second sum at least one of the arguments of φ is X 1, X 2, or [X 1, X 2], all of which are vertical. Therefore, every term in the second sum in I is also zero.
As for II in (31.7), in every term at least one of the arguments of φ is vertical, so II = 0.
Case 3. The first vector field X 1 = A is vertical; the rest X2,…,Xk+1 are horizontal and right-invariant.The left-hand side of (31.7) is clearly zero because hX 1 = 0.
On the right-hand side,
Because φ is horizontal and X 1 is vertical, the only nonzero term in the first sum is
and the only nonzero terms in the second sum are
Since the X j , j = 2, …, k + 1, are right-invariant horizontal vector fields, by Lemma 28.7,
Therefore,
If σ(i) = 1 for any i ≥ 2, then
It follows that the nonzero terms in II all satisfy σ(1) = 1 and
Denote by f the function φ(X 2, …, X k+1) on P. For p ∈ P, to calculate A p f, choose a curve c(t) in G with initial point c(0) = e and initial vector c′(0) = A, for example, c(t) = exp(tA). Then with j p : G → P being the map j p (g) = p ⋅ g,
By the right-invariance of the horizontal vector fields X 2, …, X k+1,
Differentiating this expression with respect to t and using the fact that the differential of the inverse is the negative [21, Problem 8.8(b)], we have
So the right-hand side of (31.7) is
□
If V is the Lie algebra \(\mathfrak{g}\) of a Lie group G and ρ is the adjoint representation of G, then ω ⋅ φ = [ω, φ]. In this case, for any tensorial k-form \(\varphi \in \varOmega _{\mathop{\mathrm{Ad}}\nolimits }^{k}(P,\mathfrak{g})\),
Although the covariant derivative is defined for any V -valued form on P, Theorem 31.19 is true only for tensorial forms. Since the connection form ω is not tensorial, Theorem 31.19 cannot be applied to ω. In fact, by the definition of the curvature form,
By Theorem 30.4, Ω = (d ω)h = D ω. Combining these two expressions for the curvature, one obtains
The factor of 1∕2 shows that Theorem 31.19 is not true when applied to ω.
Since the curvature form Ω on a principal bundle P is tensorial of type \(\mathop{\mathrm{Ad}}\nolimits\), Theorem 31.19 applies and the second Bianchi identity (Theorem 30.4) may be restated as
Problems
Unless otherwise specified, in the following problems G is a Lie group with Lie algebra \(\mathfrak{g}\), π: P → M a principal G-bundle, \(\rho: G \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\) a finite-dimensional representation of G, and E = P × ρ V the associated bundle.
31.1. Transition functions of an associated bundle
Show that if {(U α , ϕ α )} is a trivialization for P with transition functions g α β : U α ∩ U β → G, then there is a trivialization {(U α , ψ α )} for E with transition functions \(\rho \circ g_{\alpha \beta }: U_{\alpha } \cap U_{\beta } \rightarrow \mathop{\mathrm{GL}}\nolimits (V )\).
31.2. Vector bundle structure on an associated bundle
Show that the operations (31.2) on E = P × ρ V are well defined and make the associated bundle β: E → M into a vector bundle.
31.3. Associated bundle of a frame bundle
Let E → M be a vector bundle of rank r and \(\mathop{\mathrm{Fr}}\nolimits (E) \rightarrow M\) its frame bundle. Show that the vector bundle associated to \(\mathop{\mathrm{Fr}}\nolimits (E)\) via the identity representation \(\rho: \mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R}) \rightarrow \mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\) is isomorphic to E.
31.4. Tensorial forms
Prove that if ψ ∈ Ω k(M, P × ρ V ), then ψ ♯ ∈ Ω k(P, V ) is right-equivariant with respect to ρ.
31.5. Tensorial forms
For φ ∈ Ω ρ k(P, V ), prove that φ ♭ ♯ = φ.
§32 Characteristic Classes of Principal Bundles
To a real vector bundle E → M of rank r, one can associate its frame bundle Fr(E) → M, a principal \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\)-bundle. Similarly, to a complex vector bundle of rank r, one can associate its frame bundle, a principal \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{C})\)-bundle and to an oriented real vector bundle of rank r, one can associate its oriented frame bundle, a principal \(\mathop{\mathrm{GL}}\nolimits ^{+}(r, \mathbb{R})\)-bundle, where \(\mathop{\mathrm{GL}}\nolimits ^{+}(r, \mathbb{R})\) is the group of all r × r matrices of positive determinant. The Pontrjagin classes of a real vector bundle, the Chern classes of a complex vector bundle, and the Euler class of an oriented real vector bundle may be viewed as characteristic classes of the associated principal G-bundle for \(G =\mathop{ \mathrm{GL}}\nolimits (r, \mathbb{R}),\mathop{\mathrm{GL}}\nolimits (r, \mathbb{C})\), and \(\mathop{\mathrm{GL}}\nolimits ^{+}(r, \mathbb{R})\), respectively.
In this section we will generalize the construction of characteristic classes to principal G-bundles for any Lie group G. These are some of the most important diffeomorphism invariants of a principal bundle.
32.1 Invariant Polynomials on a Lie Algebra
Let V be a vector space of dimension n and V ∨ its dual space. An element of \(\mathop{\mathrm{Sym}}\nolimits ^{k}(V ^{\vee })\) is called a polynomial of degree k on V. Relative to a basis e 1, …, e n for V and corresponding dual basis α 1, …, α n for V ∨, a function \(f: V \rightarrow \mathbb{R}\) is a polynomial of degree k if and only if it is expressible as a sum of monomials of degree k in α 1, …, α n:
For example, if \(V = \mathbb{R}^{n\times n}\) is the vector space of all n × n matrices, then \(\mathop{\mathrm{tr}}\nolimits X\) is a polynomial of degree 1 on V and detX is a polynomial of degree n on V.
Suppose now that \(\mathfrak{g}\) is the Lie algebra of a Lie group G. A polynomial \(f: \mathfrak{g} \rightarrow \mathbb{R}\) is said to be \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant if for all g ∈ G and \(X \in \mathfrak{g}\),
For example, if G is the general linear group \(\mathop{\mathrm{GL}}\nolimits (n, \mathbb{R})\), then \((\mathop{\mathrm{Ad}}\nolimits g)X = gXg^{-1}\) and \(\mathop{\mathrm{tr}}\nolimits X\) and detX are \(\mathop{\mathrm{Ad}}\nolimits G\)-invariant polynomials on the Lie algebra \(\mathfrak{g}\mathfrak{l}(n, \mathbb{R})\).
32.2 The Chern–Weil Homomorphism
Let G be a Lie group with Lie algebra \(\mathfrak{g}\), P → M a principal G-bundle, ω an Ehresmann connection on P, and Ω the curvature form of ω. Fix a basis e 1, …, e n for \(\mathfrak{g}\) and dual basis α 1, …, α n for \(\mathfrak{g}^{\vee }\). Then the curvature form Ω is a linear combination
where the coefficients Ω i are real-valued 2-forms on P. If \(f: \mathfrak{g} \rightarrow \mathbb{R}\) is the polynomial \(\sum a_{I}\alpha ^{i_{1}}\cdots \alpha ^{i_{k}}\), we define f(Ω) to be the 2k-form
on P. Although defined in terms of a basis for \(\mathfrak{g}\), the 2k-form f(Ω) is independent of the choice of a basis (Problem 32.2).
Recall that the covariant derivative D φ of a k-form φ on a principal bundle P is given by
where v i ∈ T p P and hv i is the horizontal component of v i .
Lemma 32.1.
Let π: P → M be a principal bundle. If φ is a basic form on P, then dφ = Dφ.
Proof.
A tangent vector X p ∈ T p P decomposes into the sum of its vertical and horizontal components:
Here h: T p P → T p P is the map that takes a tangent vector to its horizontal component. Since π ∗ X p = π ∗ hX p for all X p ∈ T p P, we have
Suppose φ = π ∗ τ for τ ∈ Ω k(M). Then
⊓⊔
The Chern–Weil homomorphism is based on the following theorem. As before, G is a Lie group with Lie algebra \(\mathfrak{g}\).
Theorem 32.2.
Let Ω be the curvature of a connection ω on a principal G-bundle π: P → M, and f an \(\mathop{\mathrm{Ad}}\nolimits (G)\) -invariant polynomial of degree k on \(\mathfrak{g}\) . Then
- (i):
-
f(Ω) is a basic form on P, i.e., there exists a 2k-form Λ on M such that f(Ω) = π ∗ Λ.
- (ii):
-
Λ is a closed form.
- (iii):
-
The cohomology class [Λ] is independent of the connection.
Proof.
- (i):
-
Since the curvature Ω is horizontal, so are its components Ω i and therefore so is \(f(\varOmega ) =\sum a_{I}\varOmega ^{i_{1}} \wedge \cdots \wedge \varOmega ^{i_{k}}.\)
To check the G-invariance of f(Ω), let g ∈ G. Then
$$\displaystyle\begin{array}{rcl} r_{g}^{{\ast}}\big(f(\varOmega )\big)& =& r_{ g}^{{\ast}}\big(\sum a_{ I}\varOmega ^{i_{1} } \wedge \cdots \wedge \varOmega ^{i_{k} }\big) {}\\ & =& \sum a_{I}r_{g}^{{\ast}}(\varOmega ^{i_{1} }) \wedge \cdots \wedge r_{g}^{{\ast}}(\varOmega ^{i_{k} }). {}\\ \end{array}$$Since the curvature form Ω is right-equivariant,
$$\displaystyle{r_{g}^{{\ast}}\varOmega = (\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega }$$or
$$\displaystyle{r_{g}^{{\ast}}(\sum \varOmega ^{i}e_{ i}) =\sum \big ((\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega \big)^{i}e_{ i},}$$so that
$$\displaystyle{r_{g}^{{\ast}}(\varOmega ^{i}) =\big ((\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega \big)^{i}.}$$Thus,
$$\displaystyle\begin{array}{rcl} r_{g}^{{\ast}}\big(f(\varOmega )\big)& =& \sum a_{ I}\big((\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega \big)^{i_{1} } \wedge \cdots \wedge \big ((\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega \big)^{i_{k} } {}\\ & =& f\big((\mathop{\mathrm{Ad}}\nolimits g^{-1})\varOmega \big) {}\\ & =& f(\varOmega )\qquad \mbox{ (by the $\mathop{\mathrm{Ad}}\nolimits G$-invariance of $f$)}. {}\\ \end{array}$$Since f(Ω) is horizontal and G-invariant, by Theorem 31.12, it is basic.
- (ii):
-
Since π ∗: T p P → T π(p) M is surjective, π ∗: Ω ∗(M) → Ω ∗(P) is injective. Therefore, to show that d Λ = 0, it suffices to show that
$$\displaystyle{\pi ^{{\ast}}d\varLambda = d\pi ^{{\ast}}\varLambda = df(\varOmega ) = 0.}$$If \(f =\sum a_{I}\alpha ^{i_{1}}\cdots \alpha ^{i_{k}}\), then
$$\displaystyle{f(\varOmega ) =\sum a_{I}\varOmega ^{i_{1} } \wedge \cdots \wedge \varOmega ^{i_{k} }.}$$In this expression, each a I is a constant and therefore by Lemma 32.1
$$\displaystyle{Da_{I} = da_{I} = 0.}$$By the second Bianchi identity (31.8), D Ω = 0. Therefore, D Ω i = 0 for each i. Since the Ω i are right-equivariant of type \(\mathop{\mathrm{Ad}}\nolimits\) and horizontal, they are tensorial forms. By Lemma 32.1 and because D is an antiderivation on tensorial forms (Proposition 31.16)
$$\displaystyle\begin{array}{rcl} d\big(f(\varOmega )\big)& =& D\big(f(\varOmega )\big) = D\big(\sum a_{I}\varOmega ^{i_{1} } \wedge \cdots \wedge \varOmega ^{i_{k} }\big) {}\\ & =& \sum _{I}\sum _{j}a_{I}\varOmega ^{i_{1} } \wedge \cdots \wedge D\varOmega ^{i_{j} } \wedge \cdots \wedge \varOmega ^{i_{2k} } {}\\ & =& 0. {}\\ \end{array}$$ - (iii):
-
Let I be an open interval containing the closed interval [0, 1]. Then P × I is a principal G-bundle over M × I. Denote by ρ the projection P × I → P to the first factor. If ω 0 and ω 1 are two connections on P, then
$$\displaystyle{ \tilde{\omega }= (1 - t)\rho ^{{\ast}}\omega _{ 0} + t\rho ^{{\ast}}\omega _{ 1} }$$(32.2)is a connection on P × I (Check the details). Moreover, if i t : P → P × I is the inclusion p ↦ (p, t), then \(i_{0}^{{\ast}}\tilde{\omega } =\omega _{0}\) and \(i_{1}^{{\ast}}\tilde{\omega } =\omega _{1}\).
Let
$$\displaystyle{\tilde{\varOmega }= d\tilde{\omega } + \frac{1} {2}[\tilde{\omega },\tilde{\omega }]}$$be the curvature of the connection \(\tilde{\omega }\). It pulls back under i 0 to
$$\displaystyle\begin{array}{rcl} i_{0}^{{\ast}}\tilde{\varOmega }& =& d\imath _{ 0}^{{\ast}}\tilde{\omega } + \frac{1} {2}i_{0}^{{\ast}}[\tilde{\omega },\tilde{\omega }] {}\\ & =& d\omega _{0} + \frac{1} {2}[i_{0}^{{\ast}}\tilde{\omega },i_{ 0}^{{\ast}}\tilde{\omega }] {}\\ & =& d\omega _{0} + \frac{1} {2}[\omega _{0},\omega _{0}] {}\\ & =& \varOmega _{0}, {}\\ \end{array}$$the curvature of the connection ω 0. Similarly, \(i_{1}^{{\ast}}\tilde{\varOmega } =\varOmega _{1}\), the curvature of the connection ω 1.
For any \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant polynomial
$$\displaystyle{f =\sum a_{I}\alpha ^{i_{1} }\cdots \alpha ^{i_{k} }}$$of degree k on \(\mathfrak{g}\),
$$\displaystyle\begin{array}{rcl} i_{0}^{{\ast}}f(\tilde{\varOmega })& =& i_{ 0}^{{\ast}}\sum a_{ I}\tilde{\varOmega }^{i_{1} } \wedge \cdots \wedge \tilde{\varOmega }^{i_{k} } {}\\ & =& \sum a_{I}\varOmega _{0}^{i_{1} } \wedge \cdots \wedge \varOmega _{0}^{i_{k} } {}\\ & =& f(\varOmega _{0}) {}\\ \end{array}$$and
$$\displaystyle{i_{1}^{{\ast}}f(\tilde{\varOmega }) = f(\varOmega _{ 1}).}$$Note that i 0 and i 1: P → P × I are homotopic through the homotopy i t . By the homotopy axiom of de Rham cohomology, the cohomology classes \([i_{0}^{{\ast}}f(\tilde{\varOmega })]\) and \([i_{1}^{{\ast}}f(\tilde{\varOmega })]\) are equal. Thus, [f(Ω 0)] = [f(Ω 1)], or
$$\displaystyle{\pi ^{{\ast}}[\varLambda _{ 0}] =\pi ^{{\ast}}[\varLambda _{ 1}].}$$By the injectivity of π ∗, [Λ 0] = [Λ 1], so the cohomology class of Λ is independent of the connection. □
Let π: P → M be a principal G-bundle with curvature form Ω. To every \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant polynomial on \(\mathfrak{g}\), one can associate the cohomology class [Λ] ∈ H ∗(M) such that f(Ω) = π ∗ Λ. The cohomology class [Λ] is called the characteristic class of P associated to f. Denote by \(\mathop{\mathrm{Inv}}\nolimits (\mathfrak{g})\) the algebra of all \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant polynomials on \(\mathfrak{g}\). The map
that maps each \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant polynomial to its characteristic class is called the Chern–Weil homomorphism .
Example 32.3.
If the Lie group G is \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{C})\), then by Theorem B.10 the ring of \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant polynomials on \(\mathfrak{g}\mathfrak{l}(r, \mathbb{C})\) is generated by the coefficients f k (X) of the characteristic polynomial
The characteristic classes associated to f 1(X), …, f k (X) are the Chern classes of a principal \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{C})\)-bundle. These Chern classes generalize the Chern classes of the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) of a complex vector bundle E of rank r.
Example 32.4.
If the Lie group G is \(\mathop{\mathrm{GL}}\nolimits (r, \mathbb{R})\), then by Theorem B.13 the ring of \(\mathop{\mathrm{Ad}}\nolimits (G)\)-invariant polynomials on \(\mathfrak{g}\mathfrak{l}(r, \mathbb{R})\) is also generated by the coefficients f k (X) of the characteristic polynomial
The characteristic classes associated to f 1(X), …, f k (X) generalize the Pontrjagin classes of the frame bundle \(\mathop{\mathrm{Fr}}\nolimits (E)\) of a real vector bundle E of rank r. (For a real frame bundle the coefficients f k (Ω) vanish for k odd.)
Problems
32.1. Polynomials on a vector space
Let V be a vector space with bases e 1, …, e n and u 1, …, u n . Prove that if a function \(f: V \rightarrow \mathbb{R}\) is a polynomial of degree k with respect to the basis e 1, …, e n , then it is a polynomial of degree k with respect to the basis u 1, …, u n . Thus, the notion of a polynomial of degree k on a vector space V is independent of the choice of a basis.
32.2. Chern–Weil forms
In this problem we keep the notations of this section. Let e 1, …, e n and u 1, … u n be two bases for the Lie algebra \(\mathfrak{g}\) with dual bases α 1, …, α n and β 1, …, β n, respectively. Suppose
and
Prove that
This shows that the definition of f(Ω) is independent of the choice of basis for \(\mathfrak{g}\).
32.3. Connection on P × I
Show that the 1-form \(\tilde{\omega }\) in (32.2) is a connection on P × I.
32.4. Chern–Weil homomorphism
Show that the map \(w: \mathop{\mathrm{Inv}}\nolimits (\mathfrak{g}) \rightarrow H^{{\ast}}(M)\) in (32.3) is an algebra homomorphism.
References
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol. I and II, John Wiley and Sons, 1963.
L. W. Tu, An Introduction to Manifolds, 2nd ed., Universitext, Springer, New York, 2011.
F. Warner, Foundations of Differentiable Manifolds and Lie Groups, Springer, New York, 1983.
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Tu, L.W. (2017). Chapter 6 Principal Bundles and Characteristic Classes. In: Differential Geometry. Graduate Texts in Mathematics, vol 275. Springer, Cham. https://doi.org/10.1007/978-3-319-55084-8_6
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