MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS WEIMIN CHEN Contents 1. Lie Groups 1 2. Principal Bundles 3 3. Connections and curvature 6 4. Covariant derivatives 12 References 13 1. Lie Groups A Lie group G is a smooth manifold such that the multiplication map G × G ! G, (g; h) 7! gh, and the inverse map G ! G, g 7! g−1, are smooth maps. A Lie subgroup H of G is a subgroup of G which is at the same time an embedded submanifold. A Lie group homomorphism is a group homomorphism which is a smooth map between the Lie groups. The Lie algebra, denoted by Lie(G), of a Lie group G consists of the set of left-invariant vector fields on G, i.e., Lie(G) = fX 2 X (G)j(Lg)∗X = Xg, where Lg : G ! G is the left translation Lg(h) = gh. As a vector space, Lie(G) is naturally identified with the tangent space TeG via X 7! X(e). A Lie group homomorphism naturally induces a Lie algebra homomorphism between the associated Lie algebras. Finally, the universal cover of a connected Lie group is naturally a Lie group, which is in one to one correspondence with the corresponding Lie algebras. Example 1.1. Here are some important Lie groups in geometry and topology. • GL(n; R), GL(n; C), where GL(n; C) can be naturally identified as a Lie sub- group of GL(2n; R). • SL(n; R), O(n), SO(n) = O(n) \ SL(n; R), Lie subgroups of GL(n; R). • SL(n; C), U(n), SU(n) = U(n) \ SL(n; C), Lie subgroups of GL(n; C). • Sp(2n), Lie subgroup of GL(2n; R), defined as the subgroup preserving the 2n standard symplectic form on R . Sp(2n) \ O(2n) = U(n), under the natural identification of GL(n; C) as a subgroup of GL(2n; R). 1 3 • S ⊂ C, S ⊂ H, the spin group Spin(n), n > 2, which is the universal cover of SO(n). Note that Spin(n) ! SO(n) is a double cover as π1(SO(n)) = Z2 c 1 for n > 2; Spin (n) = Spin(n) ×{±1g S . 1 3 3 • Lie groups of low dimensions: S = SO(2) = U(1), S ={±1g = SO(3), S = c 3 3 SU(2) = Spin(3), Spin (3) = U(2), Spin(4) = S × S = SU(2) × SU(2). 1 2 WEIMIN CHEN 1-parameter subgroups: Given any X 2 Lie(G), one can associate a 1-parameter X subgroup of G to X as follows. Let φt : G ! G denote the flow generated by the X X vector field X. Then due to X being left-invariant, we have Lg ◦ φt = φt ◦ Lg for any X g 2 G. This easily implies the following: (1) the flow φt is complete, i.e., it is defined X X X X for all t 2 R, (2) for any s; t 2 R, φs (e)φt (e) = φs+t(e), (3) the flow φt : G ! G is X given by the right translation R X . By (2), φ (e) 2 G is a 1-parameter subgroup φt (e) t of G, which we associate to X 2 Lie(G). Example 1.2. Consider G = GL(n; R). In this case, TeG = M(n; R) is the space of all n × n matrices, and Lie(G) consists of maps A~ : GL(n; R) ! M(n; R), where ~ A A 2 M(n; R), and A : X 2 GL(n; R) 7! XA 2 M(n; R). Hence the flow φt generated d A A A tA by A 2 Lie(G) obeys the ODE dt φt = φt A, which implies easily that φt (e) = e . It is clear that one can replace G by any other matrix groups. X Exponential Map: We define exp : Lie(G) ! G by exp(X) = φ1 (e). It is X clear that φt (e) = exp(tX). One can check easily that exp : Lie(G) ! G is a local diffeomorphism from a neighborhood of 0 2 Lie(G) onto a neighborhood of e 2 G, with d(exp) equaling identity at 0 2 Lie(G). By Example 1.2, for G = GL(n; R), the exponential map exp is given by exp(X) = eX for any X 2 Lie(G). The Adjoint Representation: For any g 2 G, the map Ad(g): G ! G defined by h 7! ghg−1 is an automorphism of G. There is the induced Lie algebra automorphism, which is also denoted by Ad(g): Lie(G) ! Lie(G) for simplicity. The corresponding homomorphism Ad : G ! GL(Lie(G)) sending g to Ad(g) is called the Adjoint Rep- resentation of G. Note that Ad is a Lie group homomorphism. The corresponding Lie algebra representation is denoted by ad : Lie(G) ! M(Lie(G)), where M(Lie(G)) stands for the Lie algebra of GL(Lie(G)). To determine Ad, we let X 2 Lie(G) and consider the corresponding 1-parameter subgroup exp(tX). Then for any g 2 G, Ad(g): Lie(G) ! Lie(G) is given by Xe 2 d −1 d X TeG 7! dt (g · exp(tX) · g )jt=0 = dt (Rg−1 (φt (g)))jt=0 = (Rg−1 )∗(Xg) 2 TeG. To de- d termine ad, for any X; Y 2 Lie(G), ad(X)(Ye) is given by dt (Ad(exp(tX))(Ye))jt=0 = d d X (R )∗(Y )jt=0 = (φ )∗(Y X )jt=0 = (LX Y )e = [X; Y ]e, which im- dt exp(−tX) exp(tX) dt −t φt (e) plies that ad(X)(Y ) = [X; Y ] for any X; Y 2 Lie(G). Note that ad is a Lie algebra homomorphism, i.e., ad([X; Y ]) = ad(X)ad(Y ) − ad(Y )ad(X), which is equivalent to the Jacobi identity [X; [Y; Z]] + [Y; [Z; X]] + [Z; [X; Y ]] = 0. −1 Exercise: For G = GL(n; R), show that Ad(X)(A) = XAX for any X 2 GL(n; R) and A 2 M(n; R), and ad(A)(B) = AB − BA for any A; B 2 M(n; R). The Canonical 1-Form: The canonical 1-form θ on G is the Lie(G)-valued, left invariant 1-form determined by θ(X) = X for any X 2 Lie(G). If we pick a basis i ∗ P i fXig of Lie(G) and let fθ g be the dual basis of Lie(G) , then θ = i θ Xi. Let P i i [Xj;Xk] = i cjkXi (here cjk are called the structure constants which completely i 1 P i j k determine the Lie bracket). Then dθ = − 2 j;k cjkθ ^ θ , called the Maurer-Cartan equation. MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS 3 Exercise: Show that for G = GL(n; R), the canonical 1-form θ is given by A 7! −1 A dA, A 2 GL(n; R). Let ρ : G ! G0 be any Lie group homomorphism, and let θ; θ0 be the canonical 0 ∗ 0 0 1-forms on G; G respectively. We denote by ρ θ , ρ∗θ the Lie(G )-valued 1-forms on G defined as follows: for any tangent vector X on G, ∗ 0 0 (ρ θ )(X) = θ (ρ∗(X)); (ρ∗θ)(X) = ρ∗(θ(X)): Then by taking X to be a left-invariant vector field on G, it follows easily that ∗ 0 ρ θ = ρ∗θ: For reference see Chapter 20 of J. Lee [2] 2. Principal Bundles Fix a smooth manifold M (assume M is connected without loss of generality) and a Lie group G, a principal G-bundle over M is a smooth manifold P with a surjective smooth map π : P ! M (called projection), which satisfies the following conditions: • The space P is equipped with a smooth, free, right action of G: P × G ! P , denoted by (p; g) 7! pg. • For any (p; g) 2 P × G, π(pg) = π(p), and furthermore, π induces a diffeomor- phism between the quotient space P=G and M. • P is locally trivial, i.e., for any x 2 M, there is a neighborhood U and a G- −1 equivariant diffeomorphism 'U : π (U) ! U × G, 'U (p) = (π(p); (p)), such that 'U (pg) = (π(p); (p)g). The space P is called the total space, M is called the base, and G is called the the structure group of the principal G-bundle. Two principal G-bundles P; P 0 over M are called isomorphic if there is a G-equivariant diffeomorphism from P to P 0 which induces the identity map on the base M. (Note: P being locally trivial is equivalent to the G-action being proper.) An alternative definition via transition functions: Suppose fUαg is an open −1 cover of M such that P is trivial over each Uα, with a trivialization 'Uα : π (Uα) ! U × G. For any α; β such that U \ U 6= ;, ' ◦ '−1 : U \ U × G ! U \ U × G α α β Uβ Uα α β α β is given by (x; g) 7! (x; 'βα(x)g) for some smooth map 'βα : Uα \ Uβ ! G. It is easy to check that f'βαg, called the transition functions associated to fUαg, satisfy the cocycle conditions 'γβ(x)'βα(x) = 'γα(x) for any x 2 Uα \ Uβ \ Uγ. On the other hand, if we are given with an open cover fUαg of M and a set of smooth functions f'βα : Uα \Uβ ! Gg satisfying the cocycle conditions 'γβ(x)'βα(x) = 'γα(x) for any x 2 Uα \Uβ \Uγ, we can construct a principal G-bundle over M canonically as follows: let P = tαUα × G= ∼, where for any x 2 Uα \ Uβ,(x; g) 2 Uα × G ∼ (x; 'βα(x)g) 2 Uβ × G.