Chapter 12. Connections on Manifolds

Transcription

1 Chapter 12 Connections on Manifolds 12.1 Connections on Manifolds Given a manifold, M, in general, for any two points, p, q 2 M, thereisno natural isomorphismbetween the tangent spaces T p M and T q M. Given a curve, c: [0, 1]! M, onm as c(t) moveson M, howdoesthetangentspace,t c(t) M change as c(t) moves? If M = R n,thenthespaces,t c(t) R n,arecanonically isomorphic to R n and any vector, v 2 T c(0) R n = R n,is simply moved along c by parallel transport, that is,at c(t), the tangent vector, v, alsobelongstot c(t) R n. 595

2 596 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 R 3. Given any two vector fields, X and Y defined on some open subset, U R 3,foreveryp 2 U, thedirectional derivative, D X Y (p), of Y with respect to X is defined by Y (p + tx(p)) Y (p) D X Y (p) =lim. t!0 t If f : U! R is a di erentiable function on U, forevery p 2 U, thedirectional derivative, X[f](p) (or X(f)(p)), of f 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)).

3 12.1. CONNECTIONS ON MANIFOLDS 597 It is easily shown that D X Y (p) isr-bilinear in X and Y, is C 1 (U)-linear in X and satisfies the Leibniz derivation rule with respect to Y,thatis: Proposition The directional derivative of vector fields satisfies the following properties: D X1 +X 2 Y (p) =D X1 Y (p)+d X2 Y (p) D fx Y (p) =fd X Y (p) D X (Y 1 + Y 2 )(p) =D X Y 1 (p)+d X Y 2 (p) D X (fy)(p) =X[f](p)Y (p)+f(p)d X Y (p), for all X, X 1,X 2,Y,Y 1,Y 2 2 X(U) and all f 2 C 1 (U). Now, if p 2 U where U M is an open subset of M, for any vector field, Y,definedonU (Y (p) 2 T p M,forall p 2 U), for every X 2 T p M,thedirectionalderivative, D X Y (p), makes sense and it has an orthogonal decomposition, D X Y (p) =r X Y (p)+(d n ) X Y (p), where its horizontal (or tangential) component is r X Y (p) 2 T p M and its normal component is (D n ) X Y (p).

4 598 CHAPTER 12. CONNECTIONS ON MANIFOLDS The component, r X Y (p), is the covariant derivative of Y with respect to X 2 T p M and it allows us to define the covariant derivative of a vector field, Y 2 X(U), with respect to a vector field, X 2 X(M), on M. We easily check that r X Y satisfies the four equations of Proposition 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)) 2 T c(t) M,forallt 2 [0, 1]. We also write Y (t) fory (c(t)). Then, we say that Y is parallel along c i r c 0 (t)y =0 along c.

5 12.1. CONNECTIONS ON MANIFOLDS 599 The notion of parallel transport on a surface can be defined 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 0 2 T p M,thereisaunique parallel vector field, Y,alongc, withy (0) = Y 0. If we set Y 1 = Y (1), we obtain a linear map, Y 0 7! Y 1, from T p M to T q M which is also an isometry. As a summary, given a surface, M, ifwecandefineanotion of covariant derivative, r: X(M) X(M)! X(M), satisfying the properties of Proposition 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, T p M and T q M,using curves joining p and q.

6 600 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 Let M be a smooth manifold. A connection on M is a R-bilinear map, r: X(M) X(M)! X(M), where we write r X Y for r(x, Y ), such that the following two conditions hold: r fx Y = fr X Y r X (fy)=x[f]y + fr X Y, for all X, Y 2 X(M) andallf 2 C 1 (M). The vector field, r X Y,iscalledthecovariant derivative of Y with respect to X. AconnectiononM is also known as an a on M. ne connection

7 12.1. CONNECTIONS ON MANIFOLDS 601 Abasicpropertyofr is that it is a local operator. Proposition Let M be a smooth manifold and let r be a connection on M. For every open subset, U M, for every vector field, Y 2 X(M), if Y 0 on U, then r X Y 0 on U for all X 2 X(M), that is, r is a local operator. Proposition 12.2 implies that a connection, r, onm, restricts to a connection, r U, oneveryopensubset, U M. It can also be shown that (r X Y )(p) onlydependson X(p), that is, for any two vector fields, X, Y 2 X(M), if X(p) =Y (p) forsomep 2 M, then (r X Z)(p) =(r Y Z)(p) for every Z 2 X(M). Consequently, for any p 2 M, thecovariantderivative, (r u Y )(p), is well defined for any tangent vector, u 2 T p M,andanyvectorfield,Y,definedonsomeopen subset, U M, withp 2 U.

8 602 CHAPTER 12. CONNECTIONS ON MANIFOLDS Observe that on U, then-tuple of @x 1,...,@x n,isalocalframe. We can j = nx k=1 k, for some unique smooth functions, k ij, defined on U, called the Christo el symbols. We say that a connection, r, isflat on U r X =0, for all X 2 X(U), 1 apple i apple i

9 12.1. CONNECTIONS ON MANIFOLDS 603 Proposition Every smooth manifold, M, possesses 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 r is the flat connection on U,thenitisimmediately verified that r = X f r is a connection on M. Remark: AconnectiononTM can be viewed as a linear map, r: X(M)! Hom C 1 (M)(X(M), X(M)), such that, for any fixed Y 2 X(M), the map, ry : X 7! r X Y,isC 1 (M)-linear, which implies that ry is a (1, 1) tensor.

10 604 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 derivative of a vector field along a curve. Definition 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 2 [a, b] (X(t) 2 T (t) M). 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 necessarily extend to an open subset of M (for example, if the image of is dense in M), the covariant derivative (r 0 (t 0 ) X) (t0 ) may not be defined, so we need a proposition showing that the covariant derivative of a vector field along a curve makes sense.

11 12.2. PARALLEL TRANSPORT 605 Proposition Let M be a smooth manifold, let r be a connection on M and :[a, b]! M be a 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 dt dt X + f DX dt (2) If X is induced by a vector field, Z 2 X(M), that is, X(t 0 )=Z( (t 0 )) for all t 0 2 [a, b], then DX dt (t 0)=(r 0 (t 0 ) Z) (t0 ).

12 606 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]! R n,wecanwrite ' (t) =(u 1 (t),...,u n (t)), where each u i = pr i ' is smooth. Now, it is easy to see that nx 0 du (t 0 )=. i i=1 (t 0 ) If (s 1,...,s n )isaframeoveru, wecanwrite nx X(t) = X i (t)s i ( (t)), i=1 for some smooth functions, X i.

13 12.2. PARALLEL TRANSPORT 607 Then, conditions (1) and (2) imply that DX nx dxj = dt dt s j( (t)) + X j (t)r 0 (t)(s j ( (t))) and since j=1 0 (t) = nx i=1 du i k there exist some smooth functions, ij,sothat nx du i r 0 (t)(s j ( (t))) = dt (s j ( i=1 = X i,k du i dt (t), k ijs k ( (t)). It follows that 0 nx = DX dt dx k dt + X ij k ij 1 du i dt X A j s k ( (t)). Conversely, the above expression defines a linear operator, D/dt, anditiseasytocheckthatitsatisfies(1)and (2).

14 608 CHAPTER 12. CONNECTIONS ON MANIFOLDS The operator, D/dt is often called covariant derivative along and it is also denoted by r 0 (t) or simply r 0. Definition Let M be a smooth manifold and let r be a connection on M. Foreverycurve, :[a, b]! M, in M, avectorfield,x, along is parallel (along ) i DX dt (s) =0 foralls 2 [a, b]. If M was embedded in R d,forsomed, thentosaythat X is parallel along would mean that the directional derivative, (D 0X)( (t)), is normal to T (t) M. The following proposition can be shown using the existence and uniqueness of solutions of ODE s (in our case, linear ODE s) and its proof is omitted:

15 12.2. PARALLEL TRANSPORT 609 Proposition Let M be a smooth manifold and let r be a connection on M. For every C 1 curve, :[a, b]! M, in M, for every t 2 [a, b] and every v 2 T (t) M, there is a unique parallel vector field, X, along such that X(t) =v. For the proof of Proposition 12.5 it is su cient 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, (s 1,...,s n ), over U, wehave 0 1 DX nx dx k + X k du i ij dt dt dt X A j s k ( (t)), ij k=1 with u i = pr i '.Consequently,X is parallel along our portion of i the system of linear ODE s in the unknowns, X k, dx k dt + X ij k ij du i dt X j =0, k =1,...,n, is satisfied.

16 610 CHAPTER 12. CONNECTIONS ON MANIFOLDS Remark: Proposition 12.5 can be extended to piecewise C 1 curves. Definition Let M be a smooth manifold and let r be a connection on M. Foreverycurve, :[a, b]! M, inm, foreveryt 2 [a, b], the parallel transport from (a) to (t) along is the linear map from T (a) M to T (t) M, which associates to any v 2 T (a) M the vector, X v (t) 2 T (t) M,whereX v is the unique parallel vector field along with X v (a) =v. The following proposition is an immediate consequence of properties of linear ODE s: Proposition Let M be a smooth manifold and let r be a connection on M. For every C 1 curve, :[a, b]! M, in M, the parallel transport along defines for every t 2 [a, b] a linear isomorphism, P : T (a) M! T (t) M, between the tangent spaces, T (a) M and T (t) M.

17 12.2. PARALLEL TRANSPORT 611 In particular, if is a closed curve, that is, if (a) = (b) =p, weobtainalinearisomorphism,p,of the tangent space, T p M,calledtheholonomy of.the holonomy group of r based at p, denotedhol p (r), is the subgroup of GL(n, R) (wheren is the dimension of the manifold M) givenby Hol p (r) ={P 2 GL(n, R) is a closed curve based at p}. If M is connected, then Hol p (r) dependsonthebasepoint p 2 M up to conjugation and so Hol p (r) and Hol q (r) areisomorphicforallp, q 2 M. Inthiscase,it makes sense to talk about the holonomy group of r. By abuse of language, we call Hol p (r) theholonomy group of M.

18 612 CHAPTER 12. CONNECTIONS ON MANIFOLDS 12.3 Connections Compatible with a Metric; Levi-Civita Connections If a Riemannian manifold, M,hasametric,thenitisnatural to define when a connection, r, onm is compatible with the metric. Given any two vector fields, Y,Z 2 X(M), the smooth function hy,zi is defined by for all p 2 M. hy,zi(p) =hy p,z p i p, Definition Given any metric, h, i,onasmooth manifold, M, aconnection,r, onm is compatible with the metric, forshort,ametric connection i X(hY,Zi) =hr X Y,Zi + hy,r X Zi, for all vector fields, X, Y, Z 2 X(M).

19 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 613 Proposition Let M be a Riemannian manifold with a metric, h, i. Then, M, possesses metric connections. Proof. For every chart, (U,' ), we use the Gram-Schmidt procedure to obtain an orthonormal frame over U and we let r be the trivial connection over U. By construction, r is compatible with the metric. We finish the argumemt 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 connections on TM and none of them seems more relevant than the others. It is remarkable that if we require a certain kind of symmetry on a metric connection, then it is uniquely determined.

20 614 CHAPTER 12. CONNECTIONS ON MANIFOLDS Such a connection is known as the Levi-Civita connection. The Levi-Civita connection can be characterized 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 ] + r Y r X r X r Y. The second natural error term is the torsion, T (X, Y ), of the connection, r, givenby T (X, Y )=r X Y r Y X [X, Y ], which measures the failure of the connection to behave like the Lie bracket.

21 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 615 Proposition (Levi-Civita, Version 1) Let M be any Riemannian manifold. There is a unique, metric, torsion-free connection, r, on M, that is, a connection satisfying the conditions X(hY,Zi) =hr X Y,Zi + hy,r X Zi r X Y r Y X =[X, Y ], for all vector fields, X, Y, Z 2 X(M). This connection is called the Levi-Civita connection (or canonical connection) on M. Furthermore, this connection is determined by the Koszul formula 2hr X Y,Zi = X(hY,Zi)+Y (hx, Zi) Z(hX, Y i) hy,[x, Z]i hx, [Y,Z]i hz, [Y,X]i.

22 616 CHAPTER 12. CONNECTIONS ON MANIFOLDS Proof. First, we prove uniqueness. Since our metric is anon-degeneratebilinearform,itsu cestoprovethe Koszul formula. As our connection is compatible with the metric, we have X(hY,Zi) =hr X Y,Zi + hy,r X Zi Y (hx, Zi) =hr Y X, Zi + hx, r Y Zi Z(hX, Y i) = hr Z X, Y i hx, r Z Y i and by adding up the above equations, we get X(hY,Zi)+Y (hx, Zi) Z(hX, Y i) = hy,r X Z r Z Xi + hx, r Y Z r Z Y i + hz, r X Y + r Y Xi. Then, using the fact that the torsion is zero, we get X(hY,Zi) +Y (hx, Zi) Z(hX, Y i) = hy,[x, Z]i + hx, [Y,Z]i + hz, [Y,X]i +2hZ, r X Y i which yields the Koszul formula. We will not prove existence here. The reader should consult the standard texts for a proof.

23 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 617 Remark: In a chart, (U, '), if we k g k (g ij ) then it can be shown that the Christo el symbols are given by k ij = 1 nx g kl (@ i g jl j g l g ij ), 2 l=1 where (g kl )istheinverseofthematrix(g kl ). It can be shown that a connection is torsion-free i k ij = k 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 submanifold.

24 618 CHAPTER 12. CONNECTIONS ON MANIFOLDS Proposition Let M be any Riemannian manifold and let N be any submanifold of M equipped with the induced metric. If r M and r N are the Levi-Civita connections on M and N, respectively, induced by the metric on M, then for any two vector fields, X and Y in X(M) with X(p),Y(p) 2 T p N, for all p 2 N, we have r N XY =(r M X Y ) k, where (r M X Y )k (p) is the orthogonal projection of r M X Y (p) onto T pn, for every p 2 N. In particular, if is a curve on a surface, M R 3,then avectorfield,x(t), along is parallel i X 0 (t) isnormal to the tangent plane, T (t) M. If r is a metric connection, then we can say more about the parallel transport along a curve. Recall from Section 12.2, Definition 12.3, that a vector field, X,alongacurve,,isparalleli DX =0. dt

25 12.3. CONNECTIONS COMPATIBLE WITH A METRIC 619 Proposition Given any Riemannian manifold, M, and any metric connection, r, on M, for every curve, :[a, b]! M, on M, if X and Y are two vector fields along, then d DX dt hx(t),y(t)i = dt,y + X, DY. dt Using Proposition we get Proposition Given any Riemannian manifold, M, and any metric connection, r, on M, for every curve, :[a, b]! M, on M, if X and Y are two vector fields along that are parallel, then hx, Y i = C, for some constant, C. In particular, kx(t)k is constant. Furthermore, the linear isomorphism, P : T (a)! T (b), is an isometry. In particular, Proposition shows that the holonomy group, Hol p (r), based at p, isasubgroupofo(n).

26 620 CHAPTER 12. CONNECTIONS ON MANIFOLDS