《黎曼几何》课程教学资源（英文讲义）Lec05 THE LINEAR CONNECTION.pdf

LECTURE 5:THELINEAR CONNECTIONToday we will introduce a new structure on (the tangent bundle of) a smoothmanifold:the linear connection structure. Roughly speaking, a linear connection isa structure that can be viewed as an abstraction of the concept“parallel"(which isof course one of the most important concepts in geometry): geometrically it startsbygiving us a way to connect or identify different tangent spaces over nearby points.When endowed with a linear connection structure on its tangent bundle, a manifoldwill look infinitesimally like Euclidean space not just smoothly, but as an affinespace. In particular, with a linear connection structure, one can.transport vectors in a“parallel way"along a curve,?differentiate vectorfields as if they werefunctions on themanifold with valuesin a fixed vector space (so that “parallel vector fields" have derivative zero).It turns out that the structure of defining“parallel transport" on a smooth manifoldis equivalent to the structure of defining “covariant derivative".1. LINEAR CONNECTIONS: MANY FACESThere are many different ways to define a linear connection on the tangentbundle.Here wemainlyfocus on two of them which aremost useful for us.I Parallel transports.From its geometric origin, linear connection was used to characterize parallelism.We start by axiomize the structure of parallelism on smooth manifolds.By staringattheparallelismstructureintheEuclideanspace,itisnothardtoseethatthefirststep should be “identifying tangent spaces in Euclidean space Rm by translations".On a smooth manifold, instead of transporting/translating a vector along a line in aparallel fashion, we will have to transport a vector along a curve since we no longerhave the concept of straight lines in general.How to transport a vector in a parallel way along a curve? If we transport avector from one point to another point along different paths, do we get the sameresult (i.e. path-dependence)? One may start with two simple examples:.For the standard Euclidean space: the simplest way is to identify all T,Rmwith Rm via the global coordinate system, and thus transport a vector fromone point to another“"without changing its direction"..For the round sphere:According to the hairy ball theorem, there is no way tofind a global coordinate system on s2.However,we can still try to transport(in some reasonable way)the north-pointing vector at a point on the equatorto the north pole. You may try to transport it along the longitude directly1

LECTURE 5: THE LINEAR CONNECTION Today we will introduce a new structure on (the tangent bundle of) a smooth manifold: the linear connection structure. Roughly speaking, a linear connection is a structure that can be viewed as an abstraction of the concept “parallel” (which is of course one of the most important concepts in geometry): geometrically it starts by giving us a way to connect or identify different tangent spaces over nearby points. When endowed with a linear connection structure on its tangent bundle, a manifold will look infinitesimally like Euclidean space not just smoothly, but as an affine space. In particular, with a linear connection structure, one can • transport vectors in a “parallel way” along a curve, • differentiate vector fields as if they were functions on the manifold with values in a fixed vector space (so that “parallel vector fields” have derivative zero). It turns out that the structure of defining “parallel transport” on a smooth manifold is equivalent to the structure of defining “covariant derivative”. 1. Linear connections: many faces There are many different ways to define a linear connection on the tangent bundle. Here we mainly focus on two of them which are most useful for us. ¶ Parallel transports. From its geometric origin, linear connection was used to characterize parallelism. We start by axiomize the structure of parallelism on smooth manifolds. By staring at the parallelism structure in the Euclidean space, it is not hard to see that the first step should be “identifying tangent spaces in Euclidean space R m by translations”. On a smooth manifold, instead of transporting/translating a vector along a ✿✿✿✿ line in a parallel fashion, we will have to transport a vector along a ✿✿✿✿✿✿ curve since we no longer have the concept of straight lines in general. How to transport a vector in a parallel way along a curve? If we transport a vector from one point to another point along different paths, do we get the same result (i.e. path-dependence)? One may start with two simple examples: • For the standard Euclidean space: the simplest way is to identify all TxR m with R m via the global coordinate system, and thus transport a vector from one point to another “without changing its direction”. • For the round sphere: According to the hairy ball theorem, there is no way to find a global coordinate system on S 2 . However, we can still try to transport (in some reasonable way) the north-pointing vector at a point on the equator to the north pole. You may try to transport it along the longitude directly 1

2LECTURE5:THELINEARCONNECTIONto the north pole, or first transport it along the equator to the opposite pointandthentransporttheresultingvectoralongthelongitudetothenorthpole.Although we have not specified what do we mean by“parallel transport",by looking at this example with the most natural way of transport, you mayconvince yourself that “parallel transport" depends on the path.Of course in these two examples, we really used some kind of Riemannian metricstructure to get a geometrically reasonable parallel transport. In fact, the theoryof connection/covariant derivative as developed by Ricci, Levi-Civita at the turnof 2Othcenturydidrequirethepresence of aRiemannianstructure,but itwassoonrealizedbymanyothermathematicians(includingE.Cartan,Schouten andWeyl) that such a structure could be defined abstractly without the presence of aRiemannian metric.The following definition is complicated (you don't need to memorize this definitionsince we will mainly use the definition of linear connection via covariant derivative below) butquite natural:Definition 1.1. A parallel transport structure on a smooth manifold M is a family[P : is a piecewise smooth curve on M]that assigns to each piecewise smooth curve : [a,b] → M a linear isomorphismP : T(a)M → T(b)Mso that(1)Supposethestartpointof2istheendpointof1,thenP2=P2Pmwhere12is the curveformed by“first1,then2"(2) Denote by - “the curve in the opposite direction", then P- = (P)-1.(3) P depends smoothly on in the following sense: Suppose U is an opensubset in M, X is a smooth vector field on U, u() is a smooth family ofcurves (i.e. I(u,t) =u(t) is a smooth map from Ux[0, 1] to M) in M with parameterin U such that u(O) = u, then the map U -→ TM given by u → P(X(u))is smooth.(4)If1and2are in firstorderosculating,i.e.1(0) = 2(0),1(0) = 2(0),then for any Xo E Tn(0),dadlt=0Poi(Xo) =lt=oPo(Xo)dtwhere Pti,t represents the parallel transport along the curve ([ti,tal) (sothat Po(Xo) is a curve in TM starting at the point (i(O), Xo).).Given such a parallel transport structure on M, one may define the directionalderivative of a vector field Y along a vector field X. To illustrate,we start with the

2 LECTURE 5: THE LINEAR CONNECTION to the north pole, or first transport it along the equator to the opposite point and then transport the resulting vector along the longitude to the north pole. Although we have not specified what do we mean by “parallel transport”, by looking at this example with the most natural way of transport, you may convince yourself that “parallel transport” depends on the path. Of course in these two examples, we really used some kind of Riemannian metric structure to get a geometrically reasonable parallel transport. In fact, the theory of connection/covariant derivative as developed by Ricci, Levi-Civita at the turn of 20th century did require the presence of a Riemannian structure, but it was soon realized by many other mathematicians (including E. Cartan, Schouten and Weyl) that such a structure could be defined abstractly without the presence of a Riemannian metric. The following definition is complicated (you don’t need to memorize this definition since we will mainly use the definition of linear connection via covariant derivative below) but quite natural: Definition 1.1. A parallel transport structure on a smooth manifold M is a family {P γ : γ is a piecewise smooth curve on M} that assigns to each piecewise smooth curve γ : [a, b] → M a linear isomorphism P γ : Tγ(a)M → Tγ(b)M so that (1) Suppose the start point of γ2 is the endpoint of γ1, then P γ1γ2 = P γ2 ◦ P γ1 , where γ1γ2 is the curve formed by “first γ1, then γ2”. (2) Denote by −γ “the curve γ in the opposite direction”, then P −γ = (P γ ) −1 . (3) P γ depends smoothly on γ in the following sense: Suppose U is an open subset in M, X is a smooth vector field on U, γu(·) is a smooth family of curves (i.e. Γ(u, t) = γu(t) is a smooth map from U×[0, 1] to M) in M with parameter in U such that γu(0) = u, then the map U → TM given by u 7→ P γu (X(u)) is smooth. (4) If γ1 and γ2 are in first order osculating, i.e. γ1(0) = γ2(0), γ˙ 1(0) = ˙γ2(0), then for any X0 ∈ Tγ1(0), d dt|t=0P γ1 0,t(X0) = d dt|t=0P γ2 0,t(X0), where P γ t1,t2 represents the parallel transport along the curve γ([t1, t2]) (so that P γ1 0,t(X0) is a curve in TM starting at the point (γ1(0), X0).). Given such a parallel transport structure on M, one may define the directional derivative of a vector field Y along a vector field X. To illustrate, we start with the

3LECTURE5:THELINEARCONNECTIONEuclidean case.Let X and Y betwo smooth vector fields on IRm.Then ata pointr the directional derivative of Y along X is the limitY(r+tX)-Y(r)(1)(DxY)(r) = limt0twhere both r and X are viewed as vectors in Rm. On a smooth manifold M withX,Y e T(TM), it makes no sense to talk about Y(p +t) since p+tX is nolonger a point on the manifold. However, in the Euclidean case (t) = +tX issimply a curve starting at r in the direction X. So on M, one may replace + tXby a curve (t) on M with (0) = p and (O) = Xp, and consider the limitY((t) - Y(p)limtt-0which, however, is still not well-defined since the vectors Y((t) and Y(p) belongto different vector spaces. It is at this step that we may use our parallel transportstructure to identify two different tangent spaces! As a result, we may define thedirectional derivative we are looking for to be(Pto,t)-1(Y((t)) - Y((to)DxY(p) = limt-to→towhere is a smooth curve on M such that (to) = p and (to) = X, e T,MIn particular, different choices of parallel transport structure may give us differentdirectional derivatives.Remark. There is an even abstract way to define the connection/parallel transportstructure via suitable choices of "horizontal subspaces at each point"It has theadvantage that it can be extended to define connections on general fiber bundles(One may search the word “Ehresmann connection" in internet to get more details.)I Linear connections as directional derivatives.So by attaching to a smooth manifold an extra (and complicated)parallel trans-port structure,one can define directional derivative of a vector field alonganothervectorfield.Itturnsoutthat lifewill bemucheasierif,instead ofaxiomizingtheparallism structure, we start by axiomizing the directional derivative (since it has amuch simpler algebraic structure).To illustrate the structure behind directional derivative, again we start with theEuclidean case, ie. the formula (1). Let X - X'o, and Y = Yia,. Then a simplecomputationyieldsDxY = X'o,(yi)o,In order to generalize this to vector fields on manifolds, one may take a closer lookof the dependence of DxY with X and Y. It is not hard to see. Fixing Y, the mapDY :T(TR)→F(TRm), X -DxY

LECTURE 5: THE LINEAR CONNECTION 3 Euclidean case. Let X and Y be two smooth vector fields on R m. Then at a point x the directional derivative of Y along X is the limit (1) (DXY )(x) = limt→0 Y (x + tX) − Y (x) t , where both x and X are viewed as vectors in R m. On a smooth manifold M with X, Y ∈ Γ ∞(TM), it makes no sense to talk about Y (p + tX) since p + tX is no longer a point on the manifold. However, in the Euclidean case γ(t) = x + tX is simply a curve starting at x in the direction X. So on M, one may replace x + tX by a curve γ(t) on M with γ(0) = p and ˙γ(0) = Xp, and consider the limit lim t→0 Y (γ(t)) − Y (p) t which, however, is still not well-defined since the vectors Y (γ(t)) and Y (p) belong to different vector spaces. It is at this step that we may use our parallel transport structure to identify two different tangent spaces! As a result, we may define the directional derivative we are looking for to be DXY (p) = limt→t0 (P γ t0,t) −1 (Y (γ(t))) − Y (γ(t0)) t − t0 , where γ is a smooth curve on M such that γ(t0) = p and ˙γ(t0) = Xp ∈ TpM. In particular, different choices of parallel transport structure may give us different directional derivatives. Remark. There is an even abstract way to define the connection/parallel transport structure via suitable choices of “horizontal subspaces at each point”. It has the advantage that it can be extended to define connections on general fiber bundles. (One may search the word “Ehresmann connection” in internet to get more details.) ¶ Linear connections as directional derivatives. So by attaching to a smooth manifold an extra (and complicated) parallel transport structure, one can define directional derivative of a vector field along another vector field. It turns out that life will be much easier if, instead of axiomizing the parallism structure, we start by axiomizing the directional derivative (since it has a much simpler algebraic structure). To illustrate the structure behind directional derivative, again we start with the Euclidean case, i.e. the formula (1). Let X = Xi∂i and Y = Y j∂j . Then a simple computation yields DXY = X i ∂i(Y j )∂j . In order to generalize this to vector fields on manifolds, one may take a closer look of the dependence of DXY with X and Y . It is not hard to see • Fixing Y , the map DY : Γ∞(TR m) → Γ ∞(TR m), X 7→ DXY

4 LECTURE 5: THE LINEAR CONNECTION is C ∞(M)-linear with respect to X, i.e. (2) DfXY = fDXY, ∀f ∈ C ∞(R m). [In other words, DY can be viewed as a (1, 1)-tensor via (DY )(ω, X) := ω(∇XY ).] • Fixing X, the map (3) DX : Γ∞(TR m)Y → Γ ∞(TR m), Y 7→ DXY satisfies the Leibniz rule DX(fY ) = fDXY + (Xf)Y ∀f ∈ C ∞(R m). Now let M be any smooth manifold (so no Riemannian structure is assumed). For any smooth vector fields X, Y ∈ Γ ∞(TM), we would like to study the “directional derivative” of Y along the direction of X. Here are two natural candidates: • The Lie derivative LXY = [X, Y ] is not good for this purpose, since it is not C ∞(M)-linear (tensorial) in X. • One may embed M into some Euclidean space R N as we did in Lecture 1: For any X, Y ∈ Γ ∞(TM), one can extend them to vector fields X, e Ye on R N , and then define DXY to be some kind of “projection” of DXe Ye onto the tangent space of M. One can check that what we get satisfies the two conditions (2) and (3) that we want. So, with only smooth structure at hand, there are lots of different ways (e.g. arising from different embeddings) to define the directional derivative of Y with respect to X. Note that both (2) and (3) are natural in defining the “directional derivative” of Y with respect to X: X represents the direction that we want to take “derivative”, so one should have pointwise linearity; Y represents the vector field to be differentiated, so one should have a Leibniz law. With the ✿✿✿✿✿✿✿✿ purpose ✿✿ as✿✿ a✿✿✿✿✿✿✿✿✿ guiding, we define Definition 1.2. A linear connection ∇ on a smooth manifold M is a bilinear map ∇ : Γ∞(TM) × Γ ∞(TM) → Γ ∞(TM), (X, Y ) 7→ ∇XY such that for any X, Y ∈ Γ ∞(TM) and any f ∈ C ∞(M), (1) ∇fXY = f∇XY , (2) ∇X(fY ) = f∇XY + (Xf)Y. The vector field ∇XY is called the covariant derivative of Y along X. Example. Let M = R m. Then the usual directional derivative ∇XY = ∇Xi∂i (Y j ∂j ) = X i ∂i(Y j )∂j . is a linear connection. More generally, for any choice of m3 functions γ k ij ∈ C ∞(R m), ∇XY := X i ∂i(Y j )∂j + X iY j γ k ij∂k. defines a linear connection on R m

LECTURE 5: THE LINEAR CONNECTION 5 One can regard a linear connection as a map ∇ : Γ∞(TM) → Γ ∞(T ∗M ⊗ TM), Y 7→ ∇Y in the understanding that ∇Y (X, ω) := ω(∇XY ). Then we automatically have C ∞(M)-linearity on X, and the condition (2) of a linear connection becomes ∇(fY ) = df ⊗ Y + f∇Y. More generally, given any vector bundle E over M, one can define a linear connection (or a covariant derivative) over E to be a linear map ∇ : Γ∞(E) → Γ ∞(T ∗M ⊗ E) such that ∇(fs) = df ⊗ s + f∇s, ∀f ∈ C ∞(M), s ∈ Γ ∞(E). Remark. In general, if E, F are two vector bundles over a smooth manifold M, then a linear map P : Γ∞(E) → Γ ∞(F) is a differential operator if supp(P u) ⊂ supp(u) for all section u ∈ Γ ∞(E). So a connection in E is a first order differential operator from section of E to sections of T ∗M ⊗ E (which has an extra property that “its principal symbol is the identity map in T ∗M ⊗ E”). 2. Basic properties of linear connections ¶ Locality of linear connections. Now let ∇ be a linear connection on a smooth manifold M. We shall prove that ∇XY depends only on local information of X and Y . Since ∇XY is tensorial in X and is a “derivative” in Y , one immediately gets Proposition 2.1 (Locality I). For any open subset U ⊂ M, if X|U = X˜|U and Y |U = Y˜ |U , then ∇XY |U = ∇X˜Y˜ |U . Proposition 2.2 (Locality II). If X(p) = X˜(p), then ∇XY (p) = ∇X˜Y (p). As a consequence, for any vector v ∈ TpM and any vector field Y ∈ Γ ∞(TM), one can define ∇vY (p), the “directional derivative” of Y at p along the direction v, to be the vector ∇XY (p), where X is any vector field such that X(p) = v. On the other hand side, it is not hard to construct vector fields X, Y, Y such that Y (p) = Y (p) but ∇XY (p) ̸= ∇XY (p). However, we have Proposition 2.3 (Locality III). Let γ : (−ε, ε) → M be a smooth curve on M with γ(0) = p and γ˙(0) = v. Suppose X, Y, Y are vector fields on M such that X(p) = v and Y (γ(t)) = Y (γ(t)), −ε < t < ε. Then ∇XY (p) = ∇XY (p)