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CONNECTIONS AND CURVATURE NOTES EUGENE LERMAN Contents 1. Connections on vector bundles 1 1.1. Connections 1 1.2. Parallel Transport 6 2. Riemannian geometry 8 2.1. Levi-Civita connection 8 Fiber metrics 8 2.2. Connections induced on submanifolds 11 2.3. The second fundamental form of an embedding 13 3. Geodesics as critical points of the energy functional 16 1. Connections on vector bundles 1.1. Connections. If X is a vector field on an open subset U of Rm, then X is determined by m-tuple (a1; : : : am) of functions: X @ X = a i @x i i Therefore we know how to take directional derivatives of X at a point q 2 U in the direction of a vector m v 2 TqU = R | we simply differentiate the coefficients: X @ (D X) = (D a ) j v q v i q @x q i i where Dvai is the directional derivative of the function ai in the direction v. Consequently we know when a vector field does not change along a curve γ: Dγ_ X = 0: Covariant derivatives generalize the directional derivatives allowing us to differentiate vector fields on arbi- trary manifolds and, more generally, sections of arbitrary vector bundles. Definition 1.1 (Covariant derivative of sections of a vector bundle). Let π : E ! M be a vector bundle. A covariant derivative (also knows as a connection) is an R-bilinear map r : Γ(TM) × Γ(E) ! Γ(E); (X; s) 7! rX s such that (1) rfX s = frX s (2) rX (fs) = X(f) · s + frX s: for all f 2 C1(M), all X 2 Γ(TM), and all s 2 Γ(E). Example 1.2. Let U ⊂ Rm be an open set and E = TU ! U the tangent bundle. Define a connection D on TU ! U by X @ X @ DX ( ai ) = X(ai) : @xi @xi I leave it to the reader to check that this is indeed a connection. 1 Remark 1.3. Lie derivative (X; Y ) 7! LX Y is not a connection on the tangent bundle (why not?). Example 1.4. Let π : E ! M be a trivial bundle of rank k. Then there exist global sections fs1; : : : ; skg of E such that fsj(x)g is a basis for Ex for all points x 2 M (fsig is a frame of EjU ). So for any s 2 Γ(E), P 1 we have s = j fjsj, for some C functions fj. We define a bilinear map r : Γ(TM) × Γ(E) ! Γ(E) by X X rX s = rX ( fjsj) := X(fj)sj: j j It is easy to check that r is indeed a connection on E: X X X rfX s = rfX ( fjsj) = fX(fj)sj = f fjsj = frX s; j j j and X X X X rX (fs) = rX (f fjsj) = X(ffj)sj = X(f) fjsj + f X(fj)sj = X(f)s + frX s: j j j j Lemma 1.5. Any convex linear combination of two connections on a vector bundle E ! M is a connection. 1 2 1 More precisely, let r , r be two connections on E and ρ1; ρ2 2 C (M) be two functions with ρ1 + ρ2 = 1. Then 1 2 Γ(TM) × Γ(E) 3 (X; s) 7! rX s := ρ1rX s + ρ2rX s 2 Γ(E) is a connection. Proof. Exercise. Check that the two properties of the connection hold. As a corollary we get: Proposition 1.6. Any vector bundle π : E ! M has connection. α Proof. Choose a cover fUαg on M such that EjUα is trivial. Let r be a connection on EjUα , as in Example 1.4. Let fρβg be a partition of unity subordinate to fUαg. Then supp ρβ ⊂ Uα for some α = α(β). Define a map r : Γ(TM) × Γ(E) ! Γ(E) by X r s = ρ (rα sj ): X β XU Uα β This is indeed a connection, since a convex linear combination of any finite number of connections is a connection | see Lemma 1.5 above. Proposition 1.7. Let r be a connection on a vector bundle π : E ! M. Then r is local: for any open set 0 0 U and any vector fields X and Y , and for any sections s and s of E such that XjU = Y jU and sjU = s jU , we have 0 (rX s)jU = (rY s )jU : Proof. Since r is bilinear, it is enough to show two things: (a) if XjU = 0, then (rX s)jU = 0 for any s 2 Γ(E); and (b) if sjU = 0, then (rX s)jU = 0 for any X 2 Γ(TM). Fix a point x0 2 U. Then there is a smooth function ρ : U ! [0; 1] with supp ρ ⊂ U and ρjV = 1 for some open neighborhood V of x0. If XjU = 0 then ρX = 0, and hence for any section s of E, 0 = (rρX s)(x0) = ρ(x0)(rX s)(x0) = (rX s)(x0): Since x0 2 U is arbitrary, (a) follows. If sjU = 0 then ρs = 0 on M. This in turn implies that 0 = (rX ρs)(x0) = (X(ρ)s + ρrX s)(x0) = 0 + ρ(x0)(rX s)(x0) = (rX s)(x0): 2 Remark 1.8. It follows that if r is a connection on a vector bundle E ! M then r induces a connection U r : Γ(TU) × Γ(EU ) ! Γ(EjU ) on the restriction EjU for any open set U ⊂ M. Namely, for any x0 2 U let ρ : U ! [0; 1] be a bump function as in the proof above. Then for any X 2 Γ(TU) and any s 2 Γ(EjU ) we have ρX 2 Γ(TM) and ρs 2 Γ(E) (with ρX and ρs extended to all of M by 0). We define: U (r X s)(x0) = (rρX ρs)(x0): By Proposition 1.7, the right hand side does not depend on the choice of the function ρ. We leave it to the reader to check that rU is a connection. Definition 1.9 (Christoffel symbols). Let E ! M be a vector bundle with a connection r. Let (x1; : : : ; xn): m U ! R be a coordinate chart on M small enough so that EjU is trivial. Let fsαg be a frame of EjU : for each x 2 U we require that fsα(x)g is a basis of the fiber Ex. Then any local section s 2 Γ(EjU ) can be written as a linear combination of sα's. In particular, for each index i and β U X α r @ sβ = Γiβsα @x i α β 1 for some functions Γiα 2 C (U). These functions are the Christoffel symbols of the connection r relative to the coordinates (x1; : : : ; xn) and the frame fsαg. It follows easily that the Christoffel symbols determine the connection rU on the coordinate chart U. It is customary not to distinguish between r and its restriction rU . Proposition 1.10. Let r be a connection on on a vector bundle π : E ! M. For any X 2 Γ(TM), any s 2 Γ(E) and any point q the value of the connection (rX s)(q) at a point q 2 M depends only on the vector Xq (and not on the value of X near q). Proof. It's enough to show that if Xq = 0 then (rX s)(q) = 0. Since connections are local we can argue in m coordinates. Choose a coordinate chart (x1; : : : ; xn): U ! R on M with q 2 U such that EjU is trivial. P i @ P k Pick a local frame fsjg of EjU . Then, if X = X , s = fjsj, and Γ denote the associated Christoffel @xi ij symbols, X X i X rX s = rP Xi @ ( fjsj) = X r @ ( fjsj) @xi @xi X i @fj X = X sj + Xifjr @ sj @xi @xi X i X @fj X k = X ( sj + fjΓijsk): @xi i If Xq = 0 then X (q) = 0 for all i. Hence (rX s)(q) = 0 and we are done. As a corollary of the proof computation above we get an expression for the connection in terms of the Christoffel symbols. m Corollary 1.10.1. Let r be a connection on on a vector bundle π : E ! M and (x1; : : : ; xn): U ! R a coordinate chart on M with EjU being trivial. Let fsjg be a frame of EjU . Then X X i @fk X k (1.1) rP Xi @ ( fjsj) = X ( + fjΓij)sk: i @xi @xi j i;k j We note one more corollary that will be useful when we try to define connections induced on submanifolds. Corollary 1.10.2. Let r be a connection on on a vector bundle π : E ! M. For any X 2 Γ(TM), any s 2 Γ(E) and any point q the value of the connection (rX s)(q) at a point q 2 M depends only on the values of s along the integral curve of X through q 3 P i @ P Proof. By the previous corollary, for X = X and s = fjsj i @xi j X i k (rX s)(q) = (Xfk)(q) sk(q) + X (q)fj(q)Γij(q)sk(q): i;k;j And (Xfk)(q) depends only on the values of fk along the integral curve of X. The proof that connections are local has an important generalization to maps of sections of vector bundles. Definition 1.11. Let E ! M and F ! M be two vector bundles. We say that a map T : Γ(E) ! Γ(F ) is tensorial if T is R-linear and for any f 2 C1(M) T (fs) = fT (s) for all sections s 2 Γ(E).
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