Connections and Curvature 1.1 Connections on principal bundles 1.1.1 The various definitions of a connection Suppose π : P ! M is a G-principal bundle. The differential of the right action ρ : P × G ! P induces maps ρ∗ : TpP × TgG ! TpgP: Precomposing ρ∗ with the zero sections of TG and TP respectively gives a G-action TP × G ! TP and a bundle map j : P × g ,! TP . The image of j is called the vertical subbundle V ⊂ TP and is isomorphic to P × g. Geometrically, the subspace Vp is the tangent space to the fiber at p. Using constant sections, every X 2 g induces a fundamental vector field X∗ = j(−;X) on P . By differentiating the condition for the action ρ one can verify the formula ∗ ∗ Xpg = (Ad(g)X)p · g: (1) Since the differential of jp : G ! P is Lie homomorphisms we obtain another useful formula [X; Y ]∗ = [X∗;Y ∗]: (2) Definition 1. A connection on P is one of the following equivalent items: (i) A horizontal subbundle H ⊂ TP such that for all p 2 P; g 2 G: Hp ⊕ Vp = TpP; Hpg = Hp · g (ii) A connection 1-form ! 2 Ω1(P ; g) satisfying ∗ !(Xp ) = X (8X 2 g); −1 !(v · g) = Ad(g) !(v)(8v 2 TpP; g 2 G): (iii) A parallel lifting G-equivariant bundle map χ : P ×M TM ! TP with π∗χ(p; v) = v (8v 2 Tπ(p)M): (iv) A split in the short exact sequence of G-vector bundles over P 0 ! V ! TP ! P ×M TM ! 0: 1 Recall here that a G-vector bundle is a vector bundle π : E ! B such that E; B are G-spaces, π is G-equivariant, and G acts linearly on each fiber. The bundle P ×M TM is taken to have the G-action (p; v) · g = (p · g; v). Proof. Conditions (i)–(iii) are all equivalent to (iv): (i) is a decomposition −1 of TP , (ii) is basically, up to pr2 ◦ j , a section of V ! TP , and (iii) is a section of TP ! P ×M TM. In a short exact sequence a section canonically determines the other section as well as a subspace decomposition of the middle term (the two factors being isomorphic to the outer terms) and vice versa. In formulas: Hp = ker(!p) = im (χ(p; −)) ∗ !p(v) = X if v = w + Xp for v 2 TpP; w 2 Hp ∗ vvert = !(v)p for v 2 TpP ∗ χ(p; v) = whoriz = w − !(w)p; where π∗w = v; w 2 TpP; v 2 Tπ(p)M Here we use the notation w = whoriz + wvert for the decomposition of (i). The equivariance of the section TP ! V is related to the transformation formula of ! via (1). ∼ From the proof we also obtain the isomorphism χ : P ×M TM = H. 1.1.2 Horizontal vector fields and differential forms To every vector field X 2 X(M) we may associate a unique horizontal vector field X¯ 2 X(P ) with ¯ ¯ Xp 2 Hp; π∗Xp = Xπ(p) (8p 2 P ): ¯ ¯ Namely, Xp = χ(p; Xπ(p)), and X is automatically G-equivariant. Conversely, every G-equivariant horizontal vector field Y 2 X(P ) equals X¯ for a unique X 2 X(M). 1.1.3 The covariant derivative. Curvature k A differential form η 2 Ω (P ) is called horizontal if η(v1; : : : ; vk) = 0 as soon k as some vi is vertical. The set of these forms is denoted Ω (P )horiz. There is a retraction k k r :Ω (P ) ! Ω (P )horiz; (rη)(v1; : : : ; vk) = η ((v1)horiz;::: (vk)horiz) : 2 The covariant derivative associated to a connection is the composition ∗ d ∗+1 r ∗+1 D :Ω (P ) −! Ω (P ) −! Ω (P )horiz: Since r preserves wedge-products, we have D(! ^ η) = D(!) ^ η + (−1)j!j! ^ D(η), but generally D2 6= 0. The curvature of the connection is defined by 2 R = D! 2 Ω (P ; g)horiz: Proposition 2. For all v; w 2 TpP we have 1 R(v; w) = d!(v; w) + [!; !](v; w): (3) 2 Here, [!; η] denotes a wedge-product of g-valued 1-forms. It is formed using the Lie bracket: [!; η](v; w) = [!(v); η(w)] − [!(w); η(v)]. For instance, [!; !](v; w) = 2[!(v);!(w)]. Proof. To exhibit (3) it is sufficient to find local extensions of v; w to vector fields X; Y such that R(X; Y ) = d!(X; Y ) + [!X; !Y ]. By Cartan’s formula for the exterior derivative and since ! is vertical, this is equivalent to −![Xhoriz;Yhoriz] = X(!Y ) − Y (!X) − ![X; Y ] + [!X; !Y ]: If v; w are horizontal pick horizontal local extensions X; Y and both sides become −![X; Y ]. If v; w are vertical, picking X = A∗;Y = B∗ for A; B 2 g reduces the formula to (2). Finally, suppose X is horizontal and Y vertical. Since !Y is constant, the formula reduces to −![X; Y ] = 0: Pick Y = B∗ for B 2 g and a horizontal G-equivariant extension X. It suffices to show that tB [X; Y ] is vertical: 't(p) = p · e is a local flow for Y , so tB X tB − X · e [X; Y ] = lim p·e p = 0; t!0 t by the equivariance of X. From the proposition one may check R(v · g; w · g) = Ad(g−1)R(v; w). ^ 2 One may therefore identify R with a 2-form R 2 Ω (M; gP ) on the base with values in the adjoint bundle gP = P ×Ad g: For v; w 2 Tπ(p)M set ^ R(v; w) = [p; R(¯v; w¯)] for horizontal lifts v;¯ w¯ 2 TpP of v; w. Conversely ^ R(v; w) = pr2 ◦ R(π∗v; π∗w). 3 1.1.4 Local description of connections Suppose (Uα)α2A is an open cover of M over which P is trivial so that we have a presentation by sections and transition functions sα : Uα ! P; tαβ : Uαβ ! G; where tαβ is chosen so that sα · tαβ = sβ. The “local connection 1-forms” ∗ 1 !α = sα! 2 Ω (Uα; g) obey the transformation rule −1 −1 −1 ∗ !β = Ad(tαβ )!α + tαβ · dtαβ = Ad(tαβ )!α + tαβθ: (4) Here θ 2 Ω1(G; g) denotes the left-invariant Maurer-Cartan form of G which −1 is given by θ(v 2 TgG) = (lg)∗ v. The proof is the product rule (sβ)∗;x = −1 (sα)∗;x · tαβ(x) + sα(x)tαβ(x)tαβ (x)(tαβ)∗;x along with the two rules for the −1 connection 1-form. tαβ dtαβ denotes the g-valued 1-form which on v 2 TxUαβ takes the value (l −1 )∗(tαβ)∗;x(v). tαβ (x) Conversely, a family !α of 1-forms that satisfy (4) piece together a con- ∼ nection: Let v 2 TpP and jp : g = Vp. If sα(x) · g = p, then −1 −1 !p(v) = Ad(g )!α(π∗v) + jp [v − (sα)∗π∗v] : 1.1.5 The Bianchi Identity Proposition 3. The curvature satisfies the Bianchi Identity DR = 0. i Proof. Let cjk 2 R be the structure constants with respect to a basis ei 2 g P i P i P i of the Lie algebra: [ej; ek] = i cjkei. Write ! = i ! ei and R = i R ei for ordinary 1-forms !i;Ri. Then (3) asserts 1 X d!i + ci !j ^ !k = Ωi: 2 jk j;k i 1 P i j k j k Applying d yields dΩ = 2 cjk(d! ^ ! − ! ^ d! ), which is vertical since j;k each !i is vertical. 4 1.1.6 The holonomy group. Parallel transport A curve γ¯ : I ! P is said to be a horizontal lift of γ : I ! M in case π ◦ γ¯ = γ; γ¯_ (t) 2 Hγ¯(t) (8t 2 I): Proposition 4. For every curve γ : I ! M and p 2 Pγ(0) there exists a unique horizontal lift γ¯ of γ with γ¯(0) = p. Proof. P is trivial over γ(I), so there is a curve κ : I ! P with πκ = γ. For γ¯, if it exists, we may write γ¯ = κ · f for a map f : I ! G. Then, as in (4), γ¯∗! = Ad(f)−1κ∗! + f −1df: The condition for γ¯ being horizontal is f_(t) · f(t)−1 = −!(_κ(t)). To solve uniquely, it suffices to show that a solution f is an integral curve of some d vector field: Y(g;t) = −!(_κ(t)) · g + dt t on G × I. Note that G-invariant vector fields on G × I have a global flow due to compactness of I and the group structure on G. Remark. There is a similar result for piecewise Ck curves γ: One applies Proposition 4 inductively to the intervals of differentiability, picking the ini- tial points according to continuity. This generality is needed for composition. We define the path category P(M) as having objects M and morphisms P(M)(x; y) = fγ :[a; b] ! M piecewise smooth j a ≤ b; γ(a) = x; γ(b) = yg. Parallel transport allows us to define a functor f : P(M) ! G−Spaces; x 7! Px: ∗ It assigns to a curve γ :[a; b] ! M the map γ from Pγ(a) to Pγ(b) which takes p 2 Pγ(a) to the endpoint of the lift γ¯ using the initial value γ¯(a) = p: f(γ)(p) =γ ¯(b); γ¯(a) = p: Uniqueness of the horizontal lift implies functoriality as well as invariance under reparameterization of the curve. This functor may be viewed as an analog of fiber transport Π1(M) ! Set for a covering map.