Chapter 12 Connections on Manifolds 12.1 Connections on Manifolds Given a manifold, M,ingeneral,foranytwopoints, p, q M,thereisno“natural”isomorphismbetween 2 the tangent spaces TpM and TqM. Given a curve, c:[0, 1] M,onM as c(t)moveson ! M,howdoesthetangentspace,Tc(t)M change as c(t) moves? n n If M = R ,thenthespaces,Tc(t)R ,arecanonically n n n isomorphic to R and any vector, v Tc(0)R ⇠= R ,is simply moved along c by parallel transport2 ,thatis,at n c(t), the tangent vector, v,alsobelongstoTc(t)R . 595 596 CHAPTER 12. CONNECTIONS ON MANIFOLDS However, if M is curved, for example, a sphere, then it is not obvious how to “parallel transport” a tangent vector at c(0) along a curve c. Awaytoachievethisistodefinethenotionofparallel vector field along a curve and this, in turn, can be defined in terms of the notion of covariant derivative of a vector field. Assume for simplicity that M is a surface in R3.Given any two vector fields, X and Y defined on some open sub- set, U R3,foreveryp U,thedirectional derivative, ✓ 2 DXY (p), of Y with respect to X is defined by Y (p + tX(p)) Y (p) DXY (p)=lim − . t 0 ! t If f : U R is a di↵erentiable function on U,forevery p U,the!directional derivative, X[f](p) (or X(f)(p)), of2f with respect to X is defined by f(p + tX(p)) f(p) X[f](p)=lim − . t 0 ! t We know that X[f](p)=df p(X(p)). 12.1. CONNECTIONS ON MANIFOLDS 597 It is easily shown that DXY (p)isR-bilinear in X and Y , is C1(U)-linear in X and satisfies the Leibniz derivation rule with respect to Y ,thatis: Proposition 12.1. The directional derivative of vec- tor fields satisfies the following properties: DX1+X2Y (p)=DX1Y (p)+DX2Y (p) DfXY (p)=fDXY (p) DX(Y1 + Y2)(p)=DXY1(p)+DXY2(p) DX(fY)(p)=X[f](p)Y (p)+f(p)DXY (p), for all X, X ,X ,Y,Y ,Y X(U) and all f C1(U). 1 2 1 2 2 2 Now, if p U where U M is an open subset of M,for 2 ✓ any vector field, Y ,definedonU (Y (p) TpM,forall p U), for every X T M,thedirectionalderivative,2 2 2 p DXY (p), makes sense and it has an orthogonal decom- position, D Y (p)= Y (p)+(D ) Y (p), X rX n X where its horizontal (or tangential) component is Y (p) T M and its normal component is (D ) Y (p). rX 2 p n X 598 CHAPTER 12. CONNECTIONS ON MANIFOLDS The component, XY (p), is the covariant derivative of Y with respect torX T M and it allows us to define 2 p the covariant derivative of a vector field, Y X(U), with 2 respect to a vector field, X X(M), on M. 2 We easily check that XY satisfies the four equations of Proposition 12.1. r In particular, Y ,maybeavectorfieldassociatedwitha curve, c:[0, 1] M. ! A vector field along a curve, c, is a vector field, Y ,such that Y (c(t)) Tc(t)M,forallt [0, 1]. We also write Y (t)forY (c(2t)). 2 Then, we say that Y is parallel along c i↵ c0(t)Y =0 along c. r 12.1. CONNECTIONS ON MANIFOLDS 599 The notion of parallel transport on a surface can be de- fined using parallel vector fields along curves. Let p, q be any two points on the surface M and assume there is a curve, c:[0, 1] M,joiningp = c(0) to q = c(1). ! Then, using the uniqueness and existence theorem for ordinary di↵erential equations, it can be shown that for any initial tangent vector, Y T M,thereisaunique 0 2 p parallel vector field, Y ,alongc,withY (0) = Y0. If we set Y = Y (1), we obtain a linear map, Y Y , 1 0 7! 1 from TpM to TqM which is also an isometry. As a summary, given a surface, M,ifwecandefineano- tion of covariant derivative, : X(M) X(M) X(M), satisfying the properties of Propositionr ⇥ 12.1, then! we can define the notion of parallel vector field along a curve and the notion of parallel transport, which yields a natural way of relating two tangent spaces, TpM and TqM,using curves joining p and q. 600 CHAPTER 12. CONNECTIONS ON MANIFOLDS This can be generalized to manifolds using the notion of connection. We will see that the notion of connection induces the notion of curvature. Moreover, if M has a Riemannian metric, we will see that this metric induces auniqueconnectionwithtwoextraproperties(theLevi- Civita connection). Definition 12.1. Let M be a smooth manifold. A connection on M is a R-bilinear map, : X(M) X(M) X(M), r ⇥ ! where we write XY for (X, Y ), such that the follow- ing two conditionsr hold: r Y = f Y rfX rX (fY)=X[f]Y + f Y, rX rX for all X, Y X(M)andallf C1(M). The vector 2 2 field, XY ,iscalledthecovariant derivative of Y with respectr to X. AconnectiononM is also known as an affine connection on M. 12.1. CONNECTIONS ON MANIFOLDS 601 Abasicpropertyof is that it is a local operator. r Proposition 12.2. Let M be a smooth manifold and let be a connection on M. For every open subset, r U M, for every vector field, Y X(M), if ✓ 2 Y 0 on U, then XY 0 on U for all X X(M), that⌘ is, is a localr operator.⌘ 2 r Proposition 12.2 implies that a connection, ,onM, r restricts to a connection, U,oneveryopensubset, U M. r ✓ It can also be shown that ( Y )(p)onlydependson rX X(p), that is, for any two vector fields, X, Y X(M), if X(p)=Y (p)forsomep M,then 2 2 ( Z)(p)=( Z)(p)foreveryZ X(M). rX rY 2 Consequently, for any p M,thecovariantderivative, ( Y )(p), is well defined2 for any tangent vector, ru u TpM,andanyvectorfield,Y ,definedonsomeopen subset,2 U M,withp U. ✓ 2 602 CHAPTER 12. CONNECTIONS ON MANIFOLDS Observe that on U,then-tuple of vector fields, @ ,..., @ ,isalocalframe. @x1 @xn ⇣ ⌘ We can write n @ k @ @ = Γij , r@xi @xj @xk ✓ ◆ Xk=1 k for some unique smooth functions, Γij,definedonU, called the Christo↵el symbols. We say that a connection, ,isflat on U i↵ r @ =0, for all X X(U), 1 i n. rX @x 2 ✓ i◆ 12.1. CONNECTIONS ON MANIFOLDS 603 Proposition 12.3. Every smooth manifold, M, pos- sesses a connection. Proof. We can find a family of charts, (U↵,'↵), such that U is a locally finite open cover of M.If(f )isa { ↵}↵ ↵ partition of unity subordinate to the cover U↵ ↵ and if ↵ { } is the flat connection on U↵,thenitisimmediately verifiedr that = f ↵ r ↵r ↵ X is a connection on M. Remark: AconnectiononTM can be viewed as a lin- ear map, : X(M) Hom (X(M), X(M)), r ! C1(M) such that, for any fixed Y X(M), the map, 2 Y : X XY ,isC1(M)-linear, which implies that rY is a7! (1, 1) r tensor. r 604 CHAPTER 12. CONNECTIONS ON MANIFOLDS 12.2 Parallel Transport The notion of connection yields the notion of parallel transport. First, we need to define the covariant deriva- tive of a vector field along a curve. Definition 12.2. Let M be a smooth manifold and let γ :[a, b] M be a smooth curve in M.Asmooth vector field along! the curve γ is a smooth map, X :[a, b] TM,suchthat⇡(X(t)) = γ(t), for all t [a, b](!X(t) T M). 2 2 γ(t) Recall that the curve, γ :[a, b] M,issmoothi↵γ is the restriction to [a, b]ofasmoothcurveonsomeopen! interval containing [a, b]. Since a vector X field along a curve γ does not neces- sarily extend to an open subset of M (for example, if the image of γ is dense in M), the covariant derivative ( γ0(t0) X)γ(t0) may not be defined, so we need a propo- sitionr showing that the covariant derivative of a vector field along a curve makes sense. 12.2. PARALLEL TRANSPORT 605 Proposition 12.4. Let M be a smooth manifold, let be a connection on M and γ :[a, b] M be a r ! smooth curve in M. There is a R-linear map, D/dt, defined on the vector space of smooth vector fields, X, along γ, which satisfies the following conditions: (1) For any smooth function, f :[a, b] R, ! D(fX) df DX = X + f dt dt dt (2) If X is induced by a vector field, Z X(M), 2 that is, X(t0)=Z(γ(t0)) for all t0 [a, b], then DX 2 (t )=( Z) . dt 0 rγ0(t0) γ(t0) 606 CHAPTER 12. CONNECTIONS ON MANIFOLDS Proof. Since γ([a, b]) is compact, it can be covered by a finite number of open subsets, U↵,suchthat(U↵,'↵)isa chart. Thus, we may assume that γ :[a, b] U for some ! chart, (U,'). As ' γ :[a, b] Rn,wecanwrite ◦ ! ' γ(t)=(u (t),...,u (t)), ◦ 1 n where each ui = pri ' γ is smooth. Now, it is easy to see that ◦ ◦ n dui @ γ0(t0)= . dt @xi i=1 γ(t0) X ✓ ◆ If (s1,...,sn)isaframeoverU,wecanwrite n X(t)= Xi(t)si(γ(t)), i=1 X for some smooth functions, Xi.