LECTURE 6:THE LEVI-CIVITA CONNECTION1.INDUCED LINEAR CONNECTIONS ON TENSORSI Linear connections on the trivial line bundle.LetMbea smoothmanifold,and E avectorbundle over M.As we have seen,a linear connection on E is a bilinear mapV: T(TM)×T(E) →T(E), (X, s) HVxSsuch thatfor anyf eC(M),Vfxs=fVxsandVx(fs) = fVxs+(Xf)s.Again there are numerous choices of linear connections on anyvectorbundle.Consider the simplest vector bundle, the trivial line bundle M ×x C, which willbe regarded as 0,0TM below. Since F(0.TM) = C(M), by definition a linearconnection on this bundle is a bilinear mapV : T(TM) ×C(M)→C(M)that satisfies the two conditions above. Since we are only considering “"directionalderivativeof smooth functions",we have an obvious and perfect candidate, namely(1)V: T(TM) ×C(M)→C(M), (X,f)→Vxf :=Xf,which obviously satisfies the two conditions, and is canonical in the sense that itdepends only on the smooth structure of M.Although we will use the canonical linear connection defined by (1) for smoothfunctions,we should point out that there exist many other interesting linear con-nections. In fact,for any smooth 1-form w E'(M),we haveVx(gf) = X(gf) +gfw(X)= (Xg)f +g(Xf + fw(X)) = (Xg)f +gVxf,which impliesLemma1.1.Forany1-formwonM,Vxf := Xf + fw(X)is a linear connection on 0.oTM.Equivalently, we can write this connection as = d + w.1
LECTURE 6: THE LEVI-CIVITA CONNECTION 1. Induced Linear connections on tensors ¶ Linear connections on the trivial line bundle. Let M be a smooth manifold, and E a vector bundle over M. As we have seen, a linear connection on E is a bilinear map ∇ : Γ∞(TM) × Γ ∞(E) → Γ ∞(E), (X, s) 7→ ∇Xs such that for any f ∈ C ∞(M), ∇fXs = f∇Xs and ∇X(fs) = f∇Xs + (Xf)s. Again there are numerous choices of linear connections on any vector bundle. Consider the simplest vector bundle, the trivial line bundle M × C, which will be regarded as ⊗0,0TM below. Since Γ∞(⊗0,0TM) = C ∞(M), by definition a linear connection on this bundle is a bilinear map ∇ : Γ∞(TM) × C ∞(M) → C ∞(M) that satisfies the two conditions above. Since we are only considering “directional derivative of smooth functions”, we have an obvious and perfect candidate, namely (1) ∇ : Γ∞(TM) × C ∞(M) → C ∞(M), (X, f) 7→ ∇Xf := Xf, which obviously satisfies the two conditions, and is canonical in the sense that it depends only on the smooth structure of M. Although we will use the canonical linear connection ∇ defined by (1) for smooth functions, we should point out that there exist many other interesting linear connections. In fact, for any smooth 1-form ω ∈ Ω 1 (M), we have ∇X(gf) = X(gf) + gfω(X) = (Xg)f + g(Xf + fω(X)) = (Xg)f + g∇Xf, which implies Lemma 1.1. For any 1-form ω on M, ∇ω Xf := Xf + fω(X) is a linear connection on ⊗0,0TM. Equivalently, we can write this connection as ∇ = d + ω. 1
2LECTURE6:THELEVI-CIVITACONNECTIONI The induced linear connection on cotangent bundle.Supposewearegivena linear connectionon 1,oTM=TM.Togetherwiththe canonical linear connection on o.oTM = M × R, next let's try to finda reasonable linear connection on the cotangent bundle T*M. By definition thecovariant derivative we want to construct is a bilinear mapV:r(TM)×(T*M)→T(T*M), (X,w) -Vxwwith given properties. The idea is simple and natural: we need to apply the pairingbetween T*M and TM.Note that the linear connection on TM gives rise to aparallel transport map Po,t : T(o)M → T(t)M, and by taking dual one gets a linearisomorphism(Po,t)* : T(t) M → T(o) M.With this map at hand, it is thus natural to define the covariant derivative to be(Po,t)*w(t) - wp(2)Vxw():= lim0twhere is any curve with (O) = p and (O) = Xp. To get a clear sense of thisformula using on TM instead of using P, let's pair the 1-form Vxw with anyvector field Y, to get(Pot)*w(t)(Yp) -wp(Yp)w(t)(Po,(Yp) -wp(Yp)(Vxw)(Y) = limttt→0We haveW(t)(Po,t(Yp) -wp(Yp) =w(t)(Po,(Yp)) -w(t)(Y(t) +w(t)(Y(t) -wp(Yp)= -w() ( Po,t(Po,t)-1(Y() - Yp) ) + wg()(Y(t) - wp(Yp)So in view of the facts(Po.)-(Y0) -Y= VxYlimt→0tandW(t)(Y(t)) -wp(Yp)limt=0w(Y)((t)) =(O)(w(Y)) = X(w(Y)) = Vx(w(Y)t-→0t.wegetthedesired formula(3)(Vxw)(Y) = Vx(w(Y)) - w(VxY).Note that although it looks like our“definition formula"(2) may depend on thecurve , the formula (3) shows that it is independent of the choice of
2 LECTURE 6: THE LEVI-CIVITA CONNECTION ¶ The induced linear connection on cotangent bundle. Suppose we are given a linear connection ∇ on ⊗1,0TM = TM. Together with the canonical linear connection ∇ on ⊗0,0TM = M × R, next let’s try to find a reasonable linear connection on the cotangent bundle T ∗M. By definition the covariant derivative we want to construct is a bilinear map ∇ : Γ∞(TM) × Γ ∞(T ∗M) → Γ ∞(T ∗M), (X, ω) 7→ ∇Xω with given properties. The idea is simple and natural: we need to apply the pairing between T ∗M and TM. Note that the linear connection ∇ on TM gives rise to a parallel transport map P γ 0,t : Tγ(0)M → Tγ(t)M, and by taking dual one gets a linear isomorphism (P γ 0,t) ∗ : T ∗ γ(t)M → T ∗ γ(0)M. With this map at hand, it is thus natural to define the covariant derivative to be (2) ∇Xω(p) := limt→0 (P γ 0,t) ∗ωγ(t) − ωp t , where γ is any curve with γ(0) = p and ˙γ(0) = Xp. To get a clear sense of this formula using ∇ on TM instead of using P γ , let’s pair the 1-form ∇Xω with any vector field Y , to get (∇Xω)(Y ) = limt→0 (P γ 0,t) ∗ωγ(t)(Yp) − ωp(Yp) t = lim t→0 ωγ(t)(P γ 0,t(Yp)) − ωp(Yp) t We have ωγ(t)(P γ 0,t(Yp)) − ωp(Yp) = ωγ(t)(P γ 0,t(Yp)) − ωγ(t)(Yγ(t)) + ωγ(t)(Yγ(t)) − ωp(Yp) = −ωγ(t) P γ 0,t (P γ 0,t) −1 (Yγ(t)) − Yp + ωγ(t)(Yγ(t)) − ωp(Yp). So in view of the facts lim t→0 (P γ 0,t) −1 (Yγ(t)) − Yp t = ∇XY and lim t→0 ωγ(t)(Yγ(t)) − ωp(Yp) t = d dt|t=0ω(Y )(γ(t)) = ˙γ(0)(ω(Y )) = X(ω(Y )) = ∇X(ω(Y )) we get the desired formula (3) (∇Xω)(Y ) = ∇X(ω(Y )) − ω(∇XY ). Note that although it looks like our “definition formula” (2) may depend on the curve γ, the formula (3) shows that it is independent of the choice of γ
3LECTURE 6:THELEVI-CIVITACONNECTIONI Induced linear connection for tensors.One can continue this process.Let(Pot)(r.s) : 8sT(0)M -→8sT()Mbe the naturally induced linear isomorphism (which equals Po.t on tangent compo-nents, and equals (Po,t)')-1 on cotangent components). Then for any tensor fieldT T(@rsTM), one may naturally define(Po, )(ra)-"Tr() - T,VxT(p) := lim(4)t→0where is any curve with (0) = p and (0) = Xp.After some standard but messy computations as above, one can convert theconceptional definition above to a“computable"formula(VxT)(wi,.. ,Wr,Yi,...,Y) =Vx(T(wi,...,wr,Yi, ..., Y))T(wi,.+, Vxwi,...,wr, Yi,*.,Y.)(5)T(wi,...,wr, Yi,..., VxYj,...,Y.).Erample. Let be a linear connection on M, and g be a Riemannian metric whichis a (O,2)-tensor field on M. Applying the induced linear connection to g we get(Vx9)(Y,Z) = X(Y,Z) - (VxY,Z) - (Y,VxZ)I Parallel tensors.As in the case of vector fields, the linear connection on r,sTM satisfies thethree localityproperties.We may also talk about parallel tensors:Definition 1.2. A tensor field T is called parallel along if V,T = 0, and is calledparallel (in all directions) if VxT = 0 for all X E F(TM).Erample. Under the natural pairing between T,M with T,M, we may view theidentitymap Id:F(TM)→T(TM)as a (1,i)-tensor viaI(w,Y) = w(Y).It is not surprising that I (which comes from the identity map) is parallel:(VxI)(w, Y) = X(w(Y)) - (Vxw)(Y) -w(VxY) = 0.which gives a second explanation of (3)
LECTURE 6: THE LEVI-CIVITA CONNECTION 3 ¶ Induced linear connection for tensors. One can continue this process. Let (P γ 0,t) (r,s) : ⊗ r,sTγ(0)M → ⊗r,sTγ(t)M be the naturally induced linear isomorphism (which equals P γ 0,t on tangent components, and equals ((P γ 0,t) ∗ ) −1 on cotangent components). Then for any tensor field T ∈ Γ ∞(⊗r,sTM), one may naturally define (4) ∇XT(p) := limt→0 (P γ 0,t) (r,s) −1 Tγ(t) − Tp t , where γ is any curve with γ(0) = p and ˙γ(0) = Xp. After some standard but messy computations as above, one can convert the conceptional definition above to a “computable” formula (5) (∇XT)(ω1, · · · , ωr, Y1, · · · , Ys) =∇X(T(ω1, · · · , ωr, Y1, · · · , Ys)) − X i T(ω1, · · · , ∇Xωi , · · · , ωr, Y1, · · · , Ys) − X j T(ω1, · · · , ωr, Y1, · · · , ∇XYj , · · · , Ys). Example. Let ∇ be a linear connection on M, and g be a Riemannian metric which is a (0, 2)-tensor field on M. Applying the induced linear connection to g we get (∇Xg)(Y, Z) = X⟨Y, Z⟩ − ⟨∇XY, Z⟩ − ⟨Y, ∇XZ⟩. ¶ Parallel tensors. As in the case of vector fields, the linear connection ∇ on ⊗r,sTM satisfies the three locality properties. We may also talk about parallel tensors: Definition 1.2. A tensor field T is called parallel along γ if ∇γ˙ T = 0, and is called parallel (in all directions) if ∇XT = 0 for all X ∈ Γ ∞(TM). Example. Under the natural pairing between T ∗ p M with TpM, we may view the identity map Id : Γ∞(TM) → Γ ∞(TM) as a (1, 1)-tensor via I(ω, Y ) = ω(Y ). It is not surprising that I (which comes from the identity map) is parallel: (∇XI)(ω, Y ) = X(ω(Y )) − (∇Xω)(Y ) − ω(∇XY ) = 0, which gives a second explanation of (3)
4LECTURE6:THELEVI-CIVITACONNECTIONI Compatibility of the induced linear connection.Now let M be a smooth manifold, and a linear connection on (the tangentbundle of) M. As we have seen, induces linear connections on all tensor bundles@rsTM over M. It turns out that the induced connections are consistent in thesense that they are compatible with the two natural operations on tensors: thetensor product and the contraction.To see this, let's consider two examples:Erample. For any Y E F(TM) and w E '(M) = F(T*M), applying V to the(1,1)-tensor field Ywwe getVx(Yw)(n, Z) =X(n(Y)w(Z))-(Vxn)(Y)w(Z)-n(Y)w(VxZ)= X(n(Y))w(Z)-(Vxn)(Y)w(Z)+n(Y)X(w(Z))-n(Y)w(VxZ)= ((VxY)@w)(n,Z)+(Y@(Vxw))(n,Z)In other words,Vx(Y@w)=(VxY)w+Y(Vxw)Erample.Here is another way to understand the fact VI =-O: Let Cl be thecontrac-tion map that pairs the first tangent component to the first cotangent component,thenX(w(Y)) = Vx(CI(Y @w)),and by the previous example,CI(Vx(Y @w) = CI(VxY) @w +Y @ Vxw) = w(VxY) + (Vxw)(Y).In other words, the fact "the identity map I being parallel"implies the fact “commutes with Ci" for (1,1)-tensor. Similarly one can show that for an (r,s)-tensor, commutes with all contraction Ci's.Now we can state the compatibility of with the two tensor operations:Theorem 1.3. Given a linear connection on TM, the induced linear connectionV:F(TM) ×F(@"sTM)-→F(@"TM), (X,T) -VxT,ontensorbundles@rsTM above is compatiblewiththetensorproductoperation(6)Vx(TiT2) = (VxT) T2+Ti(VxT2)and commautes with the contractions(7)Ci(VxT) = VxC(T)where1<i<r,l≤j<s, andCj : ("STM) →T(αr-1,s-1TM)is the contraction map that pairs the i-th vector with the j-th covector.IThis fact has another beautiful explanation: For any X, the covariant derivative operator Vxis a derivation on the (graded tensor) algebra of all tensor fields on M!
4 LECTURE 6: THE LEVI-CIVITA CONNECTION ¶ Compatibility of the induced linear connection. Now let M be a smooth manifold, and ∇ a linear connection on (the tangent bundle of) M. As we have seen, ∇ induces linear connections on all tensor bundles ⊗r,sTM over M. It turns out that the induced connections are consistent in the sense that they are compatible with the two natural operations on tensors: the tensor product and the contraction. To see this, let’s consider two examples: Example. For any Y ∈ Γ ∞(TM) and ω ∈ Ω 1 (M) = Γ∞(T ∗M), applying ∇ to the (1, 1)-tensor field Y ⊗ ω we get ∇X(Y ⊗ω)(η, Z) = X(η(Y )ω(Z))−(∇Xη)(Y )ω(Z)−η(Y )ω(∇XZ) = X(η(Y ))ω(Z)−(∇Xη)(Y )ω(Z)+η(Y )X(ω(Z))−η(Y )ω(∇XZ) = ((∇XY )⊗ω)(η, Z)+(Y ⊗(∇Xω))(η, Z). In other words, ∇X(Y ⊗ω) = (∇XY )⊗ω+Y ⊗(∇Xω). Example. Here is another way to understand the fact ∇I = 0: Let C 1 1 be the contraction map that pairs the first tangent component to the first cotangent component, then X(ω(Y )) = ∇X(C 1 1 (Y ⊗ ω)), and by the previous example, C 1 1 (∇X(Y ⊗ ω)) = C 1 1 ((∇XY ) ⊗ ω + Y ⊗ ∇Xω) = ω(∇XY ) + (∇Xω)(Y ). In other words, the fact “the identity map I being parallel” implies the fact “∇ commutes with C 1 1 ” for (1, 1)-tensor. Similarly one can show that for an (r, s)- tensor, ∇ commutes with all contraction C i j ’s. Now we can state the compatibility of ∇ with the two tensor operations: Theorem 1.3. Given a linear connection ∇ on TM, the induced linear connection ∇ : Γ∞(TM) × Γ ∞(⊗ r,sTM) → Γ ∞(⊗ r,sTM), (X, T) 7→ ∇XT, on tensor bundles ⊗r,sTM above is compatible with the tensor product operation 1 (6) ∇X(T1 ⊗ T2) = (∇XT1) ⊗ T2 + T1 ⊗ (∇XT2) and commutes with the contractions (7) C i j (∇XT) = ∇XC i j (T), where 1 ≤ i ≤ r, 1 ≤ j ≤ s, and C i j : Γ∞(⊗ r,sTM) → Γ ∞(⊗ r−1,s−1TM) is the contraction map that pairs the i-th vector with the j-th covector. 1This fact has another beautiful explanation: For any X, the covariant derivative operator ∇X is a derivation on the (graded tensor) algebra of all tensor fields on M!
5LECTURE 6:THELEVI-CIVITA CONNECTIONThe proof is merely a simple but messy computation which we will omit. Instead,we will showhowdo we recover (3)using compatibility conditions (6)and (7):Vx(w(Y))=Vx(CI(Y @w))=CI(Vx(Y @w))=CI(Y@Vxw+VxYw)=(Vxw)(Y)+w(VxY)which is another way to write (3).Moreover,by a tedious messy induction argument in the samephilosophy, onecan even recover (5) by using (6) and (7). In other words,one hasTheorem 1.4.Given any linear connection on the tangent bundleTM, there is aunique linear connection on all tensor fields that coincides with on TM, coincideswith (1) on functions, and satisfies compatibility conditions (6) and (7) above.I The Hessian of a function.Let M be a smooth manifold and a linear connection on M. One may equiv-alentlywritethe induced linearconnections on tensor bundles as mapsV : F(@r$TM) -F(T*M @ (@rsTM)) = T(@"$+ITM)with the understanding thatVT(-.- ,X) = (VxT)(..-))Then one may iterateto getV? : T~(8$TM) →T(@r$+2TM)(or even higher orderpowers) in the understanding that(8)V?T(.-,X,Y) = (VyVT)(...,X) = (VyVxT)(...) -(VVyxT)(...).[Note that V?T(... ,X, Y) + (VyVxT)(-.-) in general.]In particular, if we take r = s =O, i.e. consider functions f e C(M), we getv?f(X,Y) = (Vydf)(X) =YXf - (VyX)fThe bilinear form ? f is known as the Hessian of f with respect to V.I Torsion tensors of a linear connection.For a general linear connection V, the Hessian is not interesting, since it mightbe non-symmetric. A natural question is: when will ?f symmetric? We calculate:V?f(X,Y)- V?f(Y,X)= (VxY)f - (VyX)f - XY f +YXf=(VxY-VyX-[X,YD)Itfollows thatthevector field(9)T(X,Y) = VxY- VyX- [X,Y]measures how far ?f from being symmetric. A direct computation showsT(fX,Y) = T(X,fY) = fT(X,Y)
LECTURE 6: THE LEVI-CIVITA CONNECTION 5 The proof is merely a simple but messy computation which we will omit. Instead, we will show how do we recover (3) using compatibility conditions (6) and (7): ∇X(ω(Y ))=∇X(C 1 1 (Y ⊗ω))=C 1 1 (∇X(Y ⊗ω)) =C 1 1 (Y ⊗∇Xω + ∇XY ⊗ω)= (∇Xω)(Y ) + ω(∇XY ) which is another way to write (3). Moreover, by a tedious messy induction argument in the same philosophy, one can even recover (5) by using (6) and (7). In other words,one has Theorem 1.4. Given any linear connection ∇ on the tangent bundle TM, there is a unique linear connection on all tensor fields that coincides with ∇ on TM, coincides with (1) on functions, and satisfies compatibility conditions (6) and (7) above. ¶ The Hessian of a function. Let M be a smooth manifold and ∇ a linear connection on M. One may equivalently write the induced linear connections on tensor bundles as maps ∇ : Γ∞(⊗ r,sTM) → Γ ∞ T ∗M ⊗ (⊗ r,sTM) = Γ∞(⊗ r,s+1TM) with the understanding that ∇T(· · · , X) = (∇XT)(· · ·). Then one may iterate ∇ to get ∇2 : Γ∞(⊗ r,sTM) → Γ ∞(⊗ r,s+2TM) (or even higher order powers) in the understanding that (8) ∇2T(· · · , X, Y ) = (∇Y ∇T)(· · · , X) = (∇Y ∇XT)(· · ·) − (∇∇Y XT)(· · ·). [Note that ∇2T(· · · , X, Y ) ̸= (∇Y ∇XT)(· · ·) in general.] In particular, if we take r = s = 0, i.e. consider functions f ∈ C ∞(M), we get ∇2 f(X, Y ) = (∇Y df)(X) = Y Xf − (∇Y X)f. The bilinear form ∇2 f is known as the Hessian of f with respect to ∇. ¶ Torsion tensors of a linear connection. For a general linear connection ∇, the Hessian is not interesting, since it might be non-symmetric. A natural question is: when will ∇2 f symmetric? We calculate: ∇2 f(X, Y ) − ∇2 f(Y, X) = (∇XY )f − (∇Y X)f − XY f + Y Xf = (∇XY − ∇Y X − [X, Y ])f It follows that the vector field (9) T (X, Y ) = ∇XY − ∇Y X − [X, Y ] measures how far ∇2 f from being symmetric. A direct computation shows T (fX, Y ) = T (X, fY ) = fT (X, Y )