A CRASH COURSE ON CONNECTIONS DOMINGO TOLEDO 1. INTRODUCTION These notes are meant to give a quick view of the theory of connections and curvature for vector bundles. Basically we follow the outline of x2 of [1], see also Appendix C of [3]. The main goal is to explain that the characteristic classes of a Riemannian manifold have local expressions in terms of the metric. These notes are very preliminary. Watch for updates by looking at the date. Comments, corrections, requests for explanations, are all welcome. 2. DEFINITION OF CONNECTIONS AND CURVATURE Everything will be C1. Let E ! M be a smooth bundle over a smooth manifold. We write A0(E) for the space of smooth sections of E and Ak(E) for the space of smooth sections of the vector bundle Λk(T ∗M) ⊗ E. In other words, Ak(E) Is the space of smooth k-forms with coefficients in E. Alterna- k k 0 tively, A (E) = A (M) ⊗A0(M) A (E). 0 1 Definition 1. A connection on E is an R-linear map r : A (E) ! A (E) with the property that (1) r(fs) = df ⊗ s + frs 0 0 holds for all f 2 A (M) and for all s 2 A (E). If p 2 M and X 2 TpM, and ∗ iX : Tp M ! R denotes evaluation at X, then we write (2) rX s = (iX ⊗ id)rs; called the covariant derivative of s at p in the direction of X. The defining equation (1) becomes (3) r(fs) = (Xf)s + frX s In other words, rs puts together the covariant derivatives rX s in the direction ∗ of tangent vectors X 2 TpM into a single element of T M ⊗ E. This element is evaluated on each X 2 TpM by means of (2). The equation (1) means, in particular, that r is a first-order differential operator A0(E) ! A1(E). In other words, (rs)(p) = 0 whenever s is a section that vanishes to second order at p 2 M. A section vanishes to second order at p if and P only if it can be written as s = fisi where fi(p) = 0 and si(p) = 0. This is a Date: March 20, 2016. 1 2 DOMINGO TOLEDO coordinate-free way of saying that in any local coordinate system, s and all its first order partial derivatives vanish at p. Then using (1) we see that X (rs)(p) = (dfi(p) ⊗ si(p) + fi(p)rs(p)) = 0: In particular, this implies that (rs)(p) depends just on the value of s on a neigh- borhood of p, in other words, it is a local operator. But it says much more: Define 0 an equivalence relation on A (E) by saying s1 ∼ s2 if s1 − s2 vanishes to second order at p. Then (rs1)(p) = (rs2)(p). 2.1. Digression into jets. It can be checked that the collection of these equiva- lence classes, as we vary p, forms a vector bundle over M, called the bundle of one-jets of sections of E, denoted J 1(E). There is a well defined evaluation map 1 0 evp : J (E) ! E induced by evp : A (E) ! Ep. The latter has kernel the space of sections that vanish at p, thus the former has kernel the space of one-jets of sec- ∗ tions that vanish at p. This last space is isomorphic to Tp M ⊗ Ep. Thus the bundle J 1(E) fits into an exact sequence ev (4) T ∗M ⊗ E ! J 1(E) −! E 1 0 1 Let jp : A (E) ! J E denote the map that sends s to its equivalence class. A first order differential operator D : A0(E) ! A0(F ) (E and F vector bundles over M) is equivalent to a bundle map or zeroth-order differential operator D~ : J 1E ! F by the rule Ds = Dj~ 1s. Under this equivalence a connection is a bundle map 1 ∗ (5) r~ : J (E) ! T M ⊗ E such that rj~ T ∗M⊗E = id; in other words, a splitting of (4). 2.2. Space of connections. The point of connections is that in a general vector bundle E ! M there is no canonical way to differentiate sections. A connec- tion gives a way to differentiate sections in all directions. First we should prove existence: n (1) In the trivial bundle M × R , or better, in the trivialized bundle, since we are choosing a particular trivialization, A0(E) is the same as (A0(M))n, n vector functions M ! R , and we can define rs = ds, component- wise exterior derivative. More precisely, a section s = (f1; : : : ; fn) where fi : M ! R and we define rs by (6) rs = ds = (df1; : : : dfn) ∗ (2) The difference of two connections is a section of T M ⊗ End(E): If r1 and r2 are connections, then (r1 − r2)(fs) = f(r1 − r2)(s), thus 0 r1 − r2 is linear over A (M), in other words, is a bundle map E ! T ∗M ⊗E, in other words, a section of T ∗M ⊗End(E) = T ∗M ⊗E ⊗E∗, where E∗ is the dual bundle of E. The same computation shows that if r is a connection and A 2 A1(End(E)), then r + A is a connection. So, if a connection exists, the space of connections is an affine space for the infinite-dimensional vector space A1(M; End(E)). A CRASH COURSE ON CONNECTIONS 3 (3) Given E ! M and an open cover fUαg of M with trivializations(bundle n 0 isomorphisms) φα : EjUα ! Uα × R , define a connection rα on EjUα by 0 −1 (7) rαs = (id ⊗ φα )d(φαs) where d(φαs) is as in (6). (4) Let fλαg be a partition of unity subordinate to fUαg. Then define rs = P 0 α rαλαs In summary, connections exist, they form an affine space for the vector space A1(End(E)). 2.3. Connection one-forms. Let E ! M be a vector bundle and choose an open n cover fUαg with local trivializations φα : EjUα ! Uα × R as above. Let feig n denote the standard basis for R and let ei also denote the constant section ei of n −1 Uα × R . Let si = φα (ei). Then fs1; : : : ; sng is a collection of sections of EjUα that forms a basis of Ep for each p 2 Uα. This collection is called a frame of EjUα , or, more briefly, a local frame for E. 0 Recall the connection rα defined in (7) which corresponds to d under φα. In 0 0 terms of the frame fsig, rα is characterized by rαsi = 0 for i = 1; : : : ; n. Definition 2. Let r be a connection on E, and let rα denote its restriction to 0 0 Uα (more precisely, its restriction to A (Uα;EjUα )). Then rα = rα + θα for a 1 unique θα 2 A (Uα; End(E)), called the connection one-form of r with respect to fsig. Let us drop the subscript α and assume that we are working on EjU for which a frame fsig exists. Then we can write j j 1 (8) θ = fθi g; i; j = 1; : : : ; n; where θi 2 A (U) j 1 In other words, fθi g is a matrix of scalar one-forms representing θ 2 A (End(E)). 0 0 j Since r = r + θ and r si = 0, the θi are characterized by n X j (9) rsi = θi sj for i = 1; : : : ; n j=1 Once we have these forms we can find the covariant derivative of any section on U. Namely, any section s on U can be uniquely written as n X i 0 (10) s = ξ si for some ξi 2 A (U): i=1 Then (1) and (9) give us n n n X i i X i i X j rs = (dξ ⊗ si + ξ rsi) = (dξ ⊗ si + ξ θi sj) i=1 i=1 j=1 4 DOMINGO TOLEDO P i j Relabeling the indices in the second term i;j ξ θi sj we get the final formula n n X i X j i (11) rs = (dξ ⊗ si + ξ θjsi) i=1 j=1 for the covariant derivative of any section over U. We can go one step further in (8) and assume that U ⊂ M is the domain of a m 1 m coordinate chart ~x : U ! R where ~x = (x ; : : : ; x ). Then we can write m j X j µ j 0 (12) θi = Γi,µdx ; where Γi,µ 2 A (U); µ=1 and (11) becomes n n m X i X X i j µ (13) rs = (dξ ⊗ si + Γj,µξ dx si) i=1 j=1 µ=1 j The functions Γi,µ are usually called the Christoffel symbols. 2.4. Connections on induced bundles. Let r be a connection on E ! M, let N be a smooth manifold and let f : N ! M be a smooth map. Then we have the induced bundle (or pull-back) f ∗E ! N defined by the requirement that it be a fiber product: f ∗E −! E ## f N −! M ∗ A section s of s E is a map s : N ! E satisfying s(x) 2 Ef(x) for all x 2 N. If r is a connection on E we want to define an induced connection f ∗r on f ∗E. Since f can be quite arbitrary, it is difficult to give a short definition that works in all generality. It seems best to use local frames. Namely, let fUαg be an open cover α −1 of M so that there is a frame fsi g for each EjUα .