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 ane 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 @ @ = ij , r@xi @xj @xk ✓ ◆ Xk=1 k for some unique smooth functions, ij,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 :[a, b] M be a smooth curve in M.Asmooth vector field along! the curve is a smooth map, X :[a, b] TM,suchthat⇡(X(t)) = (t), for all t [a, b](!X(t) T M). 2 2 (t)

Recall that the curve, :[a, b] M,issmoothi↵ is the restriction to [a, b]ofasmoothcurveonsomeopen! interval containing [a, b].

Since a vector X field along a curve does not neces- sarily extend to an open subset of M (for example, if the image of is dense in M), the covariant derivative

( 0(t0) X)(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 :[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 , 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((t0)) for all t0 [a, b], then DX 2 (t )=( Z) . dt 0 r0(t0) (t0) 606 CHAPTER 12. CONNECTIONS ON MANIFOLDS Proof. Since ([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 :[a, b] U for some ! chart, (U,'). As ' :[a, b] Rn,wecanwrite ! ' (t)=(u (t),...,u (t)), 1 n where each ui = pri ' is smooth. Now, it is easy to see that n dui @ 0(t0)= . dt @xi i=1 (t0) X ✓ ◆ If (s1,...,sn)isaframeoverU,wecanwrite n

X(t)= Xi(t)si((t)), i=1 X for some smooth functions, Xi. 12.2. PARALLEL TRANSPORT 607 Then, conditions (1) and (2) imply that DX n dX = j s ((t)) + X (t) (s ((t))) dt dt j j r0(t) j j=1 X ✓ ◆ and since n dui @ 0(t)= , dt @x i=1 ✓ i◆(t) X k there exist some smooth functions, ij,sothat n dui @ 0(t)(sj((t))) = (sj((t))) r dt r@xi i=1 X du = i k s ((t)). dt ij k Xi,k It follows that

DX n dX du = k + k i X s ((t)). dt 0 dt ij dt j1 k ij Xk=1 X @ A Conversely, the above expression defines a linear operator, D/dt,anditiseasytocheckthatitsatisfies(1)and (2). 608 CHAPTER 12. CONNECTIONS ON MANIFOLDS The operator, D/dt is often called covariant derivative along and it is also denoted by or simply . r0(t) r0

Definition 12.3. Let M be a smooth manifold and let be a connection on M.Foreverycurve, :[a, b] M, inr M,avectorfield,X,along is parallel (along!) i↵ DX (s)=0 foralls [a, b]. dt 2

If M was embedded in Rd,forsomed,thentosaythat X is parallel along would mean that the directional derivative, (D0X)((t)), is normal to T(t)M.

The following proposition can be shown using the exis- tence and uniqueness of solutions of ODE’s (in our case, linear ODE’s) and its proof is omitted: 12.2. PARALLEL TRANSPORT 609 Proposition 12.5. Let M be a smooth manifold and let be a connection on M. For every C1 curve, :[ra, b] M, in M, for every t [a, b] and every ! 2 v T(t)M, there is a unique parallel vector field, X, along2 such that X(t)=v.

For the proof of Proposition 12.5 it is sucient to consider the portions of the curve contained in some chart. In such a chart, (U,'), as in the proof of Proposition 12.4, using a local frame, (s1,...,sn), over U,wehave

DX n dX du = k + k i X s ((t)), dt 0 dt ij dt j1 k ij Xk=1 X @ A with ui = pri ' .Consequently,X is parallel along our portion of i↵the system of linear ODE’s in the unknowns, Xk,

dX du k + k i X =0,k=1,...,n, dt ij dt j ij X is satisfied. 610 CHAPTER 12. CONNECTIONS ON MANIFOLDS Remark: Proposition 12.5 can be extended to piecewise C1 curves.

Definition 12.4. Let M be a smooth manifold and let be a connection on M.Foreverycurve, r:[a, b] M,inM,foreveryt [a, b], the paral- lel transport! from (a) to (t) along2 is the linear map from T(a)M to T(t)M,whichassociatestoany v T M the vector, X (t) T M,whereX is 2 (a) v 2 (t) v the unique parallel vector field along with Xv(a)=v.

The following proposition is an immediate consequence of properties of linear ODE’s:

Proposition 12.6. Let M be a smooth manifold and let be a connection on M. For every C1 curve, :[ra, b] M, in M, the parallel transport along defines for! every t [a, b] a linear isomorphism, P : T M T M2 , between the tangent spaces, (a) ! (t) T(a)M and T(t)M. 12.2. PARALLEL TRANSPORT 611 In particular, if is a closed curve, that is, if (a)=(b)=p,weobtainalinearisomorphism,P,of the tangent space, TpM,calledtheholonomy of .The holonomy group of based at p,denotedHol( ), is r p r the subgroup of GL(n, R)(wheren is the dimension of the manifold M)givenby

Holp( )= P GL(n, R) r { 2 | is a closed curve based at p . }

If M is connected, then Holp( )dependsonthebase- point p M up to conjugationr and so Hol ( )and 2 p r Holq( )areisomorphicforallp, q M.Inthiscase,it makesr sense to talk about the holonomy2 group of .By r abuse of language, we call Holp( )theholonomy group of M. r 612 CHAPTER 12. CONNECTIONS ON MANIFOLDS 12.3 Connections Compatible with a Metric; Levi-Civita Connections

If a Riemannian manifold, M,hasametric,thenitisnat- ural to define when a connection, ,onM is compatible with the metric. r

Given any two vector fields, Y,Z X(M), the smooth function Y,Z is defined by 2 h i Y,Z (p)= Y ,Z , h i h p pip for all p M. 2

Definition 12.5. Given any metric, , ,onasmooth manifold, M,aconnection, ,onMhis compatiblei with the metric,forshort,ametricr connection i↵ X( Y,Z )= Y,Z + Y, Z , h i hrX i h rX i for all vector fields, X, Y, Z X(M). 2 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 613 Proposition 12.7. Let M be a Riemannian manifold with a metric, , . Then, M, possesses metric connections. h i

Proof. For every chart, (U↵,'↵), we use the Gram-Schmidt procedure to obtain an orthonormal frame over U↵ and ↵ we let be the trivial connection over U↵.Bycon- struction,r ↵ is compatible with the metric. We finish the argumemtr by using a partition of unity, leaving the details to the reader.

We know from Proposition 12.7 that metric connections on TM exist. However, there are many metric connec- tions on TM and none of them seems more relevant than the others.

It is remarkable that if we require a certain kind of sym- metry on a metric connection, then it is uniquely deter- mined. 614 CHAPTER 12. CONNECTIONS ON MANIFOLDS Such a connection is known as the Levi-Civita connec- tion.TheLevi-Civitaconnectioncanbecharacterized in several equivalent ways, a rather simple way involving the notion of torsion of a connection.

There are two error terms associated with a connection. The first one is the curvature, R(X, Y )= + . r[X,Y ] rY rX rXrY The second natural error term is the torsion, T (X, Y ), of the connection, ,givenby r T (X, Y )= Y X [X, Y ], rX rY which measures the failure of the connection to behave like the Lie bracket. 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 615 Proposition 12.8. (Levi-Civita, Version 1) Let M be any Riemannian manifold. There is a unique, metric, torsion-free connection, , on M, that is, a connec- tion satisfying the conditionsr

X( Y,Z )= Y,Z + Y, Z h i hrX i h rX i Y X =[X, Y ], rX rY for all vector fields, X, Y, Z X(M). This connec- tion is called the Levi-Civita2 connection (or canoni- cal connection) on M. Furthermore, this connection is determined by the Koszul formula

2 Y,Z = X( Y,Z )+Y ( X, Z ) Z( X, Y ) hrX i h i h i h i Y,[X, Z] X, [Y,Z] Z, [Y,X] . h ih ih i 616 CHAPTER 12. CONNECTIONS ON MANIFOLDS Proof. First, we prove uniqueness. Since our metric is anon-degeneratebilinearform,itsucestoprovethe Koszul formula. As our connection is compatible with the metric, we have X( Y,Z )= Y,Z + Y, Z h i hrX i h rX i Y ( X, Z )= X, Z + X, Z h i hrY i h rY i Z( X, Y )= X, Y X, Y h i hrZ ih rZ i and by adding up the above equations, we get X( Y,Z )+Y ( X, Z ) Z( X, Y )= h i h i h i Y, Z X h rX rZ i + X, Z Y h rY rZ i + Z, Y + X . h rX rY i Then, using the fact that the torsion is zero, we get X( Y,Z )+Y ( X, Z ) Z( X, Y )= h i h i h i Y,[X, Z] + X, [Y,Z] h i h i + Z, [Y,X] +2 Z, Y h i h rX i which yields the Koszul formula.

We will not prove existence here. The reader should con- sult the standard texts for a proof. 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 617 Remark: In a chart, (U,'), if we set

@ @kgij = (gij) @xk then it can be shown that the Christo↵el symbols are given by 1 n k = gkl(@ g + @ g @ g ), ij 2 i jl j il l ij Xl=1 kl where (g )istheinverseofthematrix(gkl).

It can be shown that a connection is torsion-free i↵ k k ij =ji, for all i, j, k.

We conclude this section with various useful facts about torsion-free or metric connections.

First, there is a nice characterization for the Levi-Civita connection induced by a Riemannian manifold over a sub- manifold. 618 CHAPTER 12. CONNECTIONS ON MANIFOLDS Proposition 12.9. Let M be any Riemannian mani- fold and let N be any submanifold of M equipped with the induced metric. If M and N are the Levi-Civita connections on M andrN, respectively,r induced by the metric on M, then for any two vector fields, X and Y in X(M) with X(p),Y(p) TpN, for all p N, we have 2 2 N M Y =( Y )k, rX rX M where ( X Y )k(p) is the orthogonal projection of M Y (pr) onto T N, for every p N. rX p 2

In particular, if is a curve on a surface, M R3,then ✓ avectorfield,X(t), along is parallel i↵ X0(t)isnormal to the tangent plane, T(t)M.

If is a metric connection, then we can say more about ther parallel transport along a curve. Recall from Section 12.2, Definition 12.3, that a vector field, X,alongacurve, ,isparalleli↵ DX =0. dt 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 619 Proposition 12.10. Given any Riemannian mani- fold, M, and any metric connection, , on M, for every curve, :[a, b] M, on M, if rX and Y are two vector fields along!, then d DX DY X(t),Y(t) = ,Y + X, . dt h i dt dt ⌧ ⌧

Using Proposition 12.10 we get

Proposition 12.11. Given any Riemannian mani- fold, M, and any metric connection, , on M, for every curve, :[a, b] M, on M, if rX and Y are two vector fields along! that are parallel, then X, Y = C, h i for some constant, C. In particular, X(t) is con- stant. Furthermore, the linear isomorphism,k k P : T T , is an isometry. (a) ! (b)

In particular, Proposition 12.11 shows that the holonomy group, Hol ( ), based at p,isasubgroupofO(n). p r 620 CHAPTER 12. CONNECTIONS ON MANIFOLDS