CONNECTIONS ON PRINCIPAL G-BUNDLES

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1 CONNECTIONS ON PRINCIPAL G-BUNDLES RAHUL SHAH Abstract. We will describe connections on principal G-bundles via two perspectives: that of distributions and that of connection 1-forms. We will show that the two concepts are the same. 1. Fiber bundles Let M be a manifold. In many areas of mathematics we find that we can associate to each point in the manifold another space. We often have that this space varies smoothly. For example, the tangent space is such a construction. However, we may want to consider attaching a space to each point that is not a vector space. For example, if we delete the - section of the top exterior power of a real vector bundle, we get a space whose connectivity answers whether or not the vector bundle is orientable or not. There are many other uses for such bundles. While there may be more structure to the space attached to each point, other than it being a manifold for example, a vector structure, or a group action, we define the most general concept below and then specialize later. Definition 1.1. A fiber bundle over M with fiber F is a manifold E, and a submersion π : E X s.t. there exists an open cover {U α} of X and isomorphisms φ α : π U α U α F s.t. π 1 φ α = π, where π 1 is the projection onto the first coordinate. In the above definition, isomorphism means an isomorphism in the appropriate category of F. For example, if F is a vector space, then φ α should be a diffeomorphism or biholomorphism which, when restricted to each fiber, is a linear isomorphism. If F is a G-space and G acts fiberwise on E for some group G, then φ α should be equivariant we will define this notion as well later with respect to the G action. We call an open cover with the property above a trivializing cover. Given a fiber bundle π : E M, choose a trivializing cover {U α}. On U α U β, φ α φ β = 1 ψ for ψ u AutF, for each u U α U β. We thus think of ψ as a map ψ : U α U β AutF. The class of this map is the same as the class of the fiber bundle i.e. smooth, holomorphic, etc. It is easy to see that 1.1 ψ αα = 1 ψ ψ βγ ψ γα = 1, where defined. We call ψ the transition maps. A map of fiber bundles between π : E M and π : E M, fiber bundles over M, is a map f : E E s.t. π f = π. An ismorphism of fiber bundles is an invertible map of fiber bundles the inverse must also be a map of fiber bundles. Suppose E and E are isomorphic. Notice that we can always choose an open cover {U α} that is a trivializing cover for both E and E by choosing a mutual refinement of trivializing covers for E and E. Let φ α and φ α be the trivializations into U α F of course the fibers in E and E must be isomorphic, so we can identify the fibers as F and surpress the isomorphism. Let f Uα = f α and set π U α = E α. Then φ α f α φ α = 1 h α, where h α u AutF for each u U α. So let h α : U α AutF be the map given above. It is of the required class. Note that 1.2 ψ = h α ψ h β. 1

2 2 RAHUL SHAH Now suppose that we were given an open cover {U α} of M, and collection of maps ψ : U α U β AutF that satisfy Equations 1.1. Then we can define a fiber bundle π : E M that has the given transition maps. Let [ ] E = U α F /, α where U α F u α, f α u β, f β U β F if and only if u α = u β U α U β and f α = ψ u β [f β ]. One can check that E is a smooth manifold and the obvious map π : E M makes E a fiber bundle with fiber F. We can also check that given two such collections of transition functions that satisfy Equations 1.1, they define isomorphic fiber bundles if and only if there exist h α : U α AutF that make Equation 1.2 hold. 2. Principal G-bundles We often want to consider fiber bundles where the fibers are acted on by some group. The fibers themselves need not be naturally isomorphic to the group if the action is simply transitive, a.k.a. regular or free and transitive, but they might be G-torsors: a space which can be identified with G if we pick an identity. For a familiar analog, think of the difference between an affine space and a vector space. An affine space is a space on which a vector space acts simply transitively and linearly. It can be identified with the vector space if we pick a. Definition 2.1. A manifold X is a right G-space if there is a smooth, real-analytic or holomorphic map m: X G X s.t. mmx, g, h = mx, gh and mx, e = x. We say the group G acts on the space X on the right. In addition, this action is simple if for any x X, if mx, g = x g = e. The action is transitive if given any x, y X, there exists g G s.t. mx, g = y. A space on which G acts simply and transitively is called a G-torsor. One can define a left G-space by requiring that m: G X X and mg, mh, x = mgh, x. A map of G-spaces is a map f : X Y where X and Y are G-spaces s.t. f m = m f 1. Such a map is often called equivariant with respect to the G action. Now suppose that π : P M is a fiber bundle with fibers F. Further suppose that G acts on P on the right. Definition 2.2. We call P a principal G-bundle if π : P M is a fiber bundle and π mp, g = πp for all p P, and that G whose action restricts to an action on π m for m M by the previous statement acts on π m simply transitively for all m M. Notice that for P a principal G-bundle with fibers F, F is a G-torsor. While the automorphism group of an arbitrary smooth manifold can be quite large, the automorphism group of G-torsors is rather simple. Let f : X Y be a map of G-torsors. Fix x X and let y x = fx. Given any other x X, x = mx, g. Since fx = fmx, g = mfx, g = my x, g, we can see that the map f is determined by where we send a single element. In fact, by choosing any element x X, we can identify X with G as G-torsors G is obviously a G-torsor with the obvious right action by multiplication by sending x e and x = mx, g g. Of course, all maps of G-torsors are invertible. Now suppse that we had a map of G-torsors from G to itself. Then this map is determined by where it sends e. Let g = fe. Then fg = fe g = fe g = g g. Hence every map of G-torsors from G to itself is given by left multiplication by some element g. Note that this is not a group homomorphism, but rather a map of G-spaces. We will use the analysis of G-torsors above to help us understand pricipal G-bundles. Recall that since π : P M is a fiber bundle with fibers F for F a G-torsor, we have local trivializations φ α : P α U α F, which are equivariant with respect to the G action on P and on F. We also showed how any G-torsor can be identified with G by

3 CONNECTIONS ON PRINCIPAL G-BUNDLES 3 choosing an identity element. While this choice is not canonical, it does not matter how we choose it; hence we may as well assume that φ α : P α U α G, equivariant. Now notice that ψ : U α U β AutG where AutG means automorphisms as G-torsors. Hence ψ u is given by left multiplication by some element of G, g u. Hence we can define g : U α U β G as g u h = ψ u[h] since every map of right G-spaces from G G is given by left multiplication. Thus we find that the following is an equivalent definition of a principal G bundle: Definition 2.3. A manifold P with a submersion π : P M is called a principal G bundle, if G acts on P on the right, preserving the fibers π m, and the action restricted to any fiber is simply transitive, and if there exists an open cover {U α} of M and maps φ α : P α U α G s.t. φ αmp, g = φ αp g, where π 1 φ α = π for π 1 the projection onto U α. Notice that giving transition maps ψ is equivalent to giving maps g : U α U β G satisfying Equation 1.1. Definition 2.4. We say that two principal G-bundles are isomorphic if there is a G- equivariant, fiber preserving diffeomorphism between them. Proposition 2.5. If f : P P is a smooth, G-equivairant, fiber preserving map for P, P principal G-bundles, then f is an isomorphism. 3. Horizontal spaces and sections A section is a map s: M P s.t. π s = 1. A local section is a section s: U P U for U M, an open subset. While some fiber bundles always have a section eg. vector bundles always have a zero section, principal G-bundles only have sections over open sets U on which the bundle is isomorphic to the trivial bundle U G. Proposition 3.1. Let π : P U be a principal G bundle. Then there exists s: U P, a section, P U G as G-bundles. Proof. Given a section s, we want to define a map of G-bundles f : P U G. By Proposition 2.5, this map is an isomorphism. Define f : U G P by setting f u, g = msu, g. Notice that f u, gh = msu, gh = mmsu, g, h = mf u, g, h. Thus f is a map of G-bundles and thus an isomorphism. We can let f = f. If P U G via isomorphism f, set su = f u, e. Thus sections give us a way to locally find horizontal copies of U in P. While we cannot get a global copy of M sitting inside P unless P is trivial, we can do better on the level of tangent spaces. Essentially what we are looking for is a way to take directional derivatives. If we think about functions on R n, we can take the derivative in the direction of a vector since a tanget vector at one point is naturally identified with a particular tangent vector at another point. We say that the tangent bundle to R n has a natural connection. We want an equivalent notion on fiber bundles. On a vector bundle, this leads to the theory of affine connections. On principal G-bundles, we get principal connections. These notions are intimately connected via the notions of associated bundles and frame bundles. We will develop the theory of connections in this framework. First, some preliminaries. Let X g, the Lie algebra to G. Recall that there exists a smooth group homomorphism from R with coordinate t exptx to G s.t. 3.1 t exptx = X.

4 4 RAHUL SHAH Since G acts on P to the right, for any point p P, p exptx is a smooth curve in P. Define 3.2 σxp = t p exptx = X. We call σx the fundamental vector field associated to X. To see that this is a smooth vector field, note that near p, it is a map from U R n T R T P, and we can fix the R n factor to be the vector, so σx is a smooth section of the tangent bundle to P and is hence smooth. Also recall that there is a map ad: G Endg, defined as follows: For each fixed g G, define a map Ad g : G G by h ghg. Then Ad g e : g g. Hence Ad g e Endg. Thus define ad: G Endg as g Ad g e. That this function is smooth is obvious; recall that Endg g g and hence choosing a basis for g, it suffices to show that for any smooth curve g t in G and for all fixed X g, adg tx is a smooth curve in g, which itself follows as it is the derivative of a smooth function. Proposition 3.2. R g [σxp] = σadg Xp g. Proof. Notice that R g [σxp] = t p exptx g = t p g g exptx g = t p g expad tg X = σadg Xp g. Recall that a smooth distribution of dimension k is a subspace of the tangent space at each point, s.t. there exists a neighbourhood around any point restricted to which the subspace is spanned by k linearly independent, smooth sections. We are now ready to give the most intuitive and geometric definition of a connection on a principal G-bundle π : P M. For p P, let V p T pp be the kernel of π : T pp T πp M. We call V p the vertical vectors. Definition 3.3. A horizontal distribution, H, on P is a principal G-connection if: H is smooth, H p V p = T pp, for any g G, H p g = R g H p. We can understand the vertical vectors intuitively as follows: Proposition 3.4. A tangent vector Y p V p Y p = σxp for some X g. Proof. Let X g and p P be fixed. Let γt = p exptx be a curve. Note that Y p = γ by the definition of the fundamental vector field. Since πγt = πp, constant, we have that π γ = and hence Y p V p. Notice that the rank of π is the dimension of M, the base. By the rank-nullity theorem, the dimension of P equals the dimension of M plus the dimension of the kernel of π. By the first part of this proof we have already shown that the kernel of π is at least the dimension of G. Since the dimension of P equals the dimension of G plus the dimension of M, we have that the kernel of π is exactly σx for some X g.

5 CONNECTIONS ON PRINCIPAL G-BUNDLES 5 While smooth distributions are certainly intuitive, they are not necessarily the easiest kind of data to deal with. The data for a k-distribution are locally k linearly independent, smooth vector fields. Beyond the choices that come from coordinates on P, there are also choices as to which basis for the k-distribution to choose. Hence it can be difficult to work with this data. Instead, we will try to find a more compact way of describing a connection. This notion comes from the original motivation, namely taking directional derivatives. Any object that eats a vector and spits out some other vector is essentially a 1-form valued in some space. Hence we will describe a connection as a 1-form valued in g, the Lie algebra and show that this is equivalent to the distribution given above. Definition 3.5. A connection 1-form, ω, is a 1-form valued in g s.t. for p P, g G and X g, ω is smooth, ω pσxp = X, and ω p ar a Y p = ada ω py p. We construct a smooth distribution H p out of ω by setting H p = ker ω p. We will show that H p, as defined here, is a G-connection on P. Theorem 3.6. Every connection 1-form uniquely determines a G-connection on a principal G-bundle and vice-versa. Proof. We will prove that H p satisfies all of the criteria in the definition: Let p P and choose a trivial coordinate neighbourhood U of p. Let Y 1,..., Y dim M,... Y dim P be smooth vector fields spanning T qp for all q U. Define Y i = Y i ω j Y iσx j, where the X j is a basis of g and ω j are the coordinate functions of ω valued in g. Of course, ωy i =. We note that the Y i span H write any vector in H as a linear combination of the Y is and note that the same linear combination of the Y i equals the chosen vector and hence H p is smooth. For all v p ker π, v p = σxp for some X g. Hence rank ω = dim g. Thus the nullity of ω equals dim H p. So it suffices to show that H p ker π = { }. However, this follows from the fact that ωσx = X. Lastly, R a is one-to-one since R a is invertible. Let v p H p. Then ω p ar a v p = ada ω pv p =. Hence R a H p H p a. But the same holds true for R a and hence we have equality. Given a connection, H p, we need to define a 1-form, ω, and show that it is a connection 1-form. We define ω as follows: if v p ker π, and hence v p = σxp for some X a unique X by dimension count, define ω pv p = X. If v p H p, define ω pv p = and extend this definition linearly. Notice that this gives us a possibly discontinuous 1-form valued in g. Let ω = ω j X j, where X j are a basis for g. Recall that near p, σx 1,..., σx k and Y 1,..., Y l are a smooth basis for T P, and Y 1,..., Y l are a basis for H. We have that ω j σx k = δ j k, C, and ω j Y k =, C and hence ω is smooth. ωσx = X by definition. Let Y p = σxp. Then ω p ar a Y p = ω p aσada Xp a. However, ada X = ada ω py p, which is what we want. Instead, if Y p H p,

6 6 RAHUL SHAH then R a Y p H p a and hence ω py p = and ω p ar a Y p =, and the result follows. This connection 1-form ω is a global 1-form on P. Since for most purposes our data lives on M, we would prefer that the data of a connection also lives on M. What we sacrifice, however, is the globality of the 1-form. Since our bundle is locally trivial, and a trivialization is equivalent to a section, locally we can find a section s α : U α P. We can pull back ω under s to get a 1-form, ω α, valued in g, defined on U α. However, on the overlap two pullbacks do not agree, and this local 1-form does not transform under change of coordinates as a global 1-form. However, this local 1-form transforms in a way that is not too difficult to understand. Let {U α} be a trivializing cover with trivializations defined by sections {s α}. Define the transition maps as the left multiplication by g on the overlaps. Lemma 3.7. We have that 3.3 s β = s α g. Proof. Notice that s β u = φ u, e. Hence, β φ αs β u = φ αφ β u, e = u, g u = u, e g u Since φ α is an isomorphism, we have the result. = φ αs αu g u = φ αs αu g u. On G, there is a canonical left-invariant g-valued 1-form, θ. It is defined as θ h = L h. Notice that L g θ h = θ h L g = L h Lg = L h g = θ g h. Hence θ is actually left-invariant. It is easy to see that every left-invariant differential form on a Lie group is smooth, since the left-invariant vector fields are smooth. On a principal G-bundle, we have maps g : U α U β G, transition maps. We can thus define θ = g θ. Theorem 3.8. The local 1-forms transform as: 3.4 ω β = adg ω α + θ. Proof. Let p P and let u = πp where u U α U β. Let Y u be a vector through u. Let γt be a curve s.t. γ = u and γ t = Y u. Since s β = m s α g, we have that s β Y u = m s α γ t, g γ t = m s α γ t, g u + m p, g γ t.

7 CONNECTIONS ON PRINCIPAL G-BUNDLES 7 Notice that the first term above is which can be rewritten as On the other hand, So Hence m p, g γ t t γt g u, R g u s α Y u. g γ t = L g u L g s βωy u = ωs β Y u u g Y u. = m p, L g u L g g u Y u = t p g u g ug γt = t p g u g ug γt = σ L g g u Y u p g = σ θ g ug Y u p g = σ gθ g p g uyu = σθ Y up g u. = ωr g u s α Y u + ωσθ Y up g u = adg uω αy u + θ Y u. 4. Associated bundles and affine connections The best way to describe the relation between associated and principal bundles is through an example. Let π : E M be a vector bundle with fibers linearly isomorphic to a vector space V. The simplest example of a principal bundle associated to E is its frame bundle, F E. The fiber over each point, u, of M is the collection of bases for the vector space π u. GLV acts naturally to the right on F E, and this action preserves the fibers of F E. This makes F E a principal GLV -bundle. In fact, we can look at the oriented frame bundle for an oriented vector bundle an SLV -bundle, SLE, and if given a metric on E, the orthonormal frame bundle an OV -bundle, OE in the real case or UV -bundle, UE in the complex case. We can use any standard matrix groups to construct such a principal G-bundle. Suppose we are given a principal G-bundle, π : P M. Let ρ: G GLV be a representation of G. Then we can construct a vector bundle associated to the representation ρ whose fibers are V. The most compact way to describe it as follows: Definition 4.1. The associated vector bundle associated to the representation ρ: G GLV and the principal G-bundle, π : P M is the vector bundle P ρ V or sometimes denoted P GV if there is no confusion over which representation of G is used, constructed as the vector bundle with transition maps given by 4.1 {ρ g : U α U β GLV }. We give an alternate description below.

8 8 RAHUL SHAH Proposition 4.2. The vector bundle associated to the G-bundle, π : P M and the representation ρ: G GLV, is isomorphic to the quotient P V /G, with the G-action p, v g = p g, ρg v, where the projection onto M is given by π : [p, v] πp. Proof. It is easy to show that π : P V /G M is a vector bundle with fibers linearly isomorphic to V. Now recall that P ρ V can be constructed as U α V/, α where U α V u α, v α v β, v β U β V if and only if u α = u β and v α = ρg u β v β. Let U α be a trivial neighbourhood for P with trivialization map φ α : P Uα U α G. Recall that we can define a section s α of P by setting s αu α = φ α u α, e. Define a map f α : U α V P V /G by f αu α, v α = [s αu α, v α]. Any other representative of the class of u α, v α in P ρ V will be given by u β, v β, where u α = u β and v α = ρg u β v β. Hence f αu α, v α = f αu α, ρg u αv β = [s αu α, ρg u αv β ]. We now note that g, p g, ρg v p, v g, p g, ρgv p, v g, p g, v p, ρgv. We can thus take the last statement as the equivalence condition defining the quotient P V /G. Thus [s αu α, ρg u αv β ] = [s αu α ρg u α, v β ] = [s β u β, v β ] = f β u β, v β. Hence the local maps {f α} glue to give a global map f : P ρ V P V /G. That this map is an isomorphism of vector bundles is obvious. Notice that by Definition 4.1, it is clear that the frame bundle is the principal G- bundle with the same local transition functions as the original vector bundle. By choosing a metric, for example, and orthonormal transition functions, one can get an OV bundle, etc. We will now take a short detour to describe affine connections from the perspective of covariant derivatives and the perspective of connection 1-forms. Let π : E M be a vector bundle with fibers isomorphic to V. Definition 4.3. An affine connection is a map : ΓE ΓE T M s.t. σ ΓE and f C M, 4.2 σf = σf + σ df. for all This generalizes to an operator, d : Ω r E Ω r+1 E by the Leibniz rule for σ ΓE and ω Ω E: 4.3 d σ ω = d σ ω + σ dω. We call d the exterior covariant derivative. Unlike for the regular exterior derivative, which squares to, the covariant exterior derivative does not square to, but d 2 is related to the curvature of the connection, which we will describe later. Choose a local trivialization of E over U α, open in M. Suppose e 1,..., e n are a frame for E over U α. We

9 CONNECTIONS ON PRINCIPAL G-BUNDLES 9 have that e i = e j A j i, where Aj i ΓT M. For a section of E, σ = σ i e i where σ i are smooth functions, σ = σ i e i = e iσ i + e i dσ i = A j i σi e j + e i dσ i = d + Aσ. We notice an interesting property about these local 1-forms, A, associated to each trivialization of the vector bundle E, namely that: Proposition 4.4. If A α is the 1-form associated to the trivialized neighbourhood U α, then in U α U β for another trivialized neighbourhood U β, the local 1-form A β is given by the formula: 4.4 A β = g A αg + g dg, where A α, A β are matrices of 1-forms defined on U α and U β respectively, and g : U α U β GLE are the change of frame maps. Proof. We will surpress the indices indicating the trivialization in this proof. Let {e i} denote the frame in the U α trivialiation and {e i} denote the frame in the U β trivialization. Similarly, let A and A be the 1-forms in the two trivializations. We will denote g simply by g. Then, A j i g k j e k = A j i e j = e i = e jg j i = dg j i ej + gj i Ak j e k = dg k i e k + g j i Ak j e k. Thus for all k, A j i gj k = dgi k + g j i Ak j and hence ga k i = dgi k + Ag k i. We have thus shown that A = g dg + g Ag, which is the result. It is easy to see that any collection of local 1-forms that satisfies the above transformation law gives us a connection. We notice that this transformation law is the same as Equation 3.4 if G were a matrix Lie group. This motivates the essential point that connections on principal bundles and associated bundles are essentially the same thing: they are both defined by local 1-forms that transform in a certain manner. Theorem 4.5. If ρ: G GLV is a representation and π : P M is a principal G- bundle with a fixed connection ω, {ρ ω α} defines a collection of local 1-forms on M that define a connection on P ρ V. Proof. We have that ω αx u, for any vector X u T um, is the derivative of some curve γ :, 1 G s.t. γ = e. Also recall that adg γ = [ t g uγtg u ], and hence ρ adg ωα = ρg ρ ωαρg. Further, given a curve γ :, 1 M s.t. γ = u and γ = X u, gθx u = t L g ug γt. Thus ρ gθ = ρg dρg. Putting these two results together, we find that ρ ω α obeys the transformation laws for the 1-forms associated to an affine connection, thus completing the proof.

10 1 RAHUL SHAH 5. Curvature We will begin by defining curvature as a global 2-form valued in a certain vector bundle. We will then build a geometric intuition for this 2-form. Definition 5.1. Given a connection ω, interpreted as a global 1-form on the total space P of a principal G-bundle, we define the curvature, Ω, to be dω+ 1 [ω, ω], where [ω, ω]x, Y = 2 [ωx, ωy ]. We can pull back Ω by a local section s α to get a 2-form on U α. Unlike the local 1-forms that do not transform as a globally defined form, the pullback of the curvature does define a global 2-form. We will, for the lack of better notation, also call the pullback Ω. We note that ad P is the vector bundle associated to P by the representation ad : G Endg. Lemma 5.2. For θ, the canonical 1-form on G, dθx, Y = 1 θ[x, Y ]. 2 Lemma 5.3. Proof. We have that [θ, θ ]X, Y = [g θ, g θ]x, Y = [θg X, θg Y ] = Proposition 5.4. The pullback of the curvature transforms as a globally defined 2-form valued in ad P. Proof. Define Ω α = s αω. Recall that pullbacks commute with both wedge products and exterior differentiation. Also recall that the connection form is invariant under the action of G. Thus, Ω β = dω β [ω β, ω β ] = dad g ω α + θ + [ad g ω α + θ, ad g ω α + θ ] = ad g dω α + gdθ + ad g [ω α, ω α] + [θ, θ ] = ad g Ω α.