《黎曼几何》课程教学资源（英文讲义）Lec11 THE METHOD OF MOVING FRAMES

LECTURE11:THEMETHODOF MOVINGFRAMESIn Riemannian geometry, one frequently encounters with heavy computations(especially for those problems related to curvatures). There are three different meth-ods to do these calculations: the invariant method via global vector fields and tensorfields,the local method via carefully chosen coordinate charts (under the help of Ein-steinsummationconvention),and E.Cartan'smethod of movingframesviacalculusofdifferentialforms.Todaywewillgiveabrief introductiontothemethod ofmov-ing frames where the use of differential forms is emphasized[when compared with tensorfields, differential forms have the advantage that they can be pulled-back via smooth maps, andwe have the powerful tool of exterior derivativel.1.CARTAN'SMETHODOFMOVINGFRAMESI The connection 1-forms for a linear connection in a local frame.Let M be a smoothmanifold and a linear connection on M.We can regardV (acting on vector fields) as a linear mapV : F(TM) → F(TM T*M)So if [ei,... ,em) is a local frame[i.e. for each p e U, ei(p),..,em(p) form a basis ofT,Mj of TM defined on an open set U C M, then one can find a set of one forms[h<i.j<m defined on U so that Vxei = i(X)e, for all X e T(TM), i.e.(1)Vei=ej@.These 's are known as connection 1-forms of with respect to the local frame[ei], which are only locally defined.Moreover, if we choose another local frame [ei,.-.,em] on U, and é, = f'e, onUnu, then ej = (f-1),ex (where f-1 is the inverse of the matrix f = (f?) andthuser@Φ =Vé = V(fiei)= fiVei+e,@dfi=fiex+(f-1)je@df?=é(f-1)f+(f-1)fso we end up with的 =(f-1)f +(f-1),df)on UnU,which canbe writtenin brief as= f-10f +f-1df,(2)

LECTURE 11: THE METHOD OF MOVING FRAMES In Riemannian geometry, one frequently encounters with heavy computations (especially for those problems related to curvatures). There are three different meth￾ods to do these calculations: the invariant method via global vector fields and tensor fields, the local method via carefully chosen coordinate charts (under the help of Ein￾stein summation convention), and E. Cartan’s method of moving frames via calculus of differential forms. Today we will give a brief introduction to the method of mov￾ing frames where the use of differential forms is emphasized[when compared with tensor fields, differential forms have the advantage that they can be pulled-back via smooth maps, and we have the powerful tool of exterior derivative]. 1. Cartan’s method of moving frames ¶ The connection 1-forms for a linear connection in a local frame. Let M be a smooth manifold and ∇ a linear connection on M. We can regard ∇ (acting on vector fields) as a linear map ∇ : Γ∞(TM) → Γ ∞(TM ⊗ T ∗M). So if {e1, · · · , em} is a local frame[i.e. for each p ∈ U, e1(p), · · · , em(p) form a basis of TpM] of TM defined on an open set U ⊂ M, then one can find a set of one forms {θ j i }1≤i,j≤m defined on U so that ∇Xei = θ j i (X)ej for all X ∈ Γ ∞(TM), i.e. (1) ∇ei = ej ⊗ θ j i . These θ j i ’s are known as connection 1-forms of ∇ with respect to the local frame {ei}, which are only locally defined. Moreover, if we choose another local frame {e˜1, · · · , e˜m} on Ue, and ˜ei = f j i ej on U ∩ Ue, then ej = (f −1 ) i j e˜k (where f −1 is the inverse of the matrix f = (f j i )) and thus e˜l ⊗ ˜θ l i = ∇e˜i = ∇(f j i ej ) = f j i ∇ej + ej ⊗ dfj i = f j i ek ⊗ θ k j + (f −1 ) l j e˜l ⊗ dfj i = ˜el ⊗ (f −1 ) l k θ k j f j i + (f −1 ) l jdfj i , so we end up with ˜θ l i = (f −1 ) l k θ k j f j i + (f −1 ) l jdfj i , on U ∩ Ue, which can be written in brief as (2) ˜θ = f −1 θf + f −1 df, 1

2LECTURE11:THEMETHODOFMOVINGFRAMESwhere and are understood as m × m matrices whose entries are 1-forms, whilef and f-1 are invertible m ×m matrices' whose entries are functions (and thus onecan not exchangetheir positions intheproductabove).To develop Riemannian geometry via differential forms only, let's first derivethe dual formula for covariant derivative of differential forms via these connection1-forms.We denote by {w',...,wm) thelocal dual co-frame[ie.w'(e,)=s, for all i,j]of T*M defined on U to the given local frame (ei, ... ,em]. Then we have(Vxw)(e,) = X(w"(e;)) -w'(Vxej) =-w(0,(X)ex) = -0,(X)It follows that the linear connection V acting on one forms, viewed as a mapV: F(T*M) -→F(T*M T*M),can be expressed in terms of the co-frame and the connection 1-forms as(3)Vwi = -wi .I The connection 1-forms: torsion freeness and metric compatibilityNow supposethelinear connectionis torsion free.Thendw"(X,Y) = X(w"(Y)) - Y(w(X)) -w"(X,Y)= X(w(Y)) - Y(w(X)) - w(VxY - VyX)= (Vxw)(Y) - (Vyw")(X)= -w(Y)(X) +w (X),(Y)So the torsion free condition for a linear connection can be written, in terms of thedual co-frameand theconnection 1-forms,as(4)dwi=w-@wj=wi人which can be written in brief as dw = - Aw.Next suppose there is a Riemannian metric g on M, and the connection ismetric compatible.Toencodetheinformationof themetric intoourconsideration,it is reasonable to choose an orthonormal frame (ei, .. ,em) instead of a generalframe.Then0=(Vei,ei)+(ei,Vej)=(ek@f,ej)+(ei,ek@)=+So the metric compatibility of becomes: for any orthonormal frame, the connec-tion 1-forms satisfy +0 = 0,(5)i.e. the matrix of connection 1-forms is anti-symmetric.1So one may regard f as a map from UnU to the general linear group GL(m). If we are in thesetting of Riemannian manifold and we are only using local orthonormal frames, then the groupencountered is O(m)instead.Themethod of moving frameworks in a more general setting,andthere is always such a Lie group behind the theory that plays an important role

2 LECTURE 11: THE METHOD OF MOVING FRAMES where ˜θ and θ are understood as m × m matrices whose entries are 1-forms, while f and f −1 are invertible m × m matrices1 whose entries are functions (and thus one can not exchange their positions in the product above). To develop Riemannian geometry via differential forms only, let’s first derive the dual formula for covariant derivative of differential forms via these connection 1-forms. We denote by {ω 1 , · · · , ωm} the local dual co-frame[i.e. ω i (ej ) = δ i j for all i, j] of T ∗M defined on U to the given local frame {e1, · · · , em}. Then we have (∇Xω i )(ej ) = X(ω i (ej )) − ω i (∇Xej ) = −ω i (θ k j (X)ek) = −θ i j (X). It follows that the linear connection ∇ acting on one forms, viewed as a map ∇ : Γ∞(T ∗M) → Γ ∞(T ∗M ⊗ T ∗M), can be expressed in terms of the co-frame and the connection 1-forms as (3) ∇ω i = −ω j ⊗ θ i j . ¶ The connection 1-forms: torsion freeness and metric compatibility. Now suppose the linear connection ∇ is torsion free. Then dωi (X, Y ) = X(ω i (Y )) − Y (ω i (X)) − ω i ([X, Y ]) = X(ω i (Y )) − Y (ω i (X)) − ω i (∇XY − ∇Y X) = (∇Xω i )(Y ) − (∇Y ω i )(X) = −ω j (Y )θ i j (X) + ω j (X)θ i j (Y ). So the torsion free condition for a linear connection can be written, in terms of the dual co-frame and the connection 1-forms, as (4) dωi = ω j ⊗ θ i j − θ i j ⊗ ω j = ω j ∧ θ i j . which can be written in brief as dω = −θ ∧ ω. Next suppose there is a Riemannian metric g on M, and the connection ∇ is metric compatible. To encode the information of the metric into our consideration, it is reasonable to choose an ✿✿✿✿✿✿✿✿✿✿✿✿ orthonormal✿✿✿✿✿✿✿ frame {e1, · · · , em} instead of a general frame. Then 0 = ⟨∇ei , ej ⟩ + ⟨ei , ∇ej ⟩ = ⟨ek ⊗ θ k i , ej ⟩ + ⟨ei , ek ⊗ θ k j ⟩ = θ j i + θ j i . So the metric compatibility of ∇ becomes: for any orthonormal frame, the connec￾tion 1-forms satisfy (5) θ j i + θ j i = 0, i.e. the matrix of connection 1-forms is anti-symmetric. 1So one may regard f as a map from U ∩ Ue to the general linear group GL(m). If we are in the setting of Riemannian manifold and we are only using local orthonormal frames, then the group encountered is O(m) instead. The method of moving frame works in a more general setting, and there is always such a Lie group behind the theory that plays an important role

3LECTURE11:THEMETHODOFMOVINGFRAMESI Cartan's formulation of Riemannian geometry.It turns out that one can develop Riemannian geometry starting with localframes and connection 1-forms (i.e. via the differential 1-forms w, ') instead of theRiemannian metric g and its Levi-Civita connection[since one can recover the Riemannianmetric g from the local orthonormal co-frame [w'), and then recover the Levi-Civita connection from its connection 1-forms @j]. We start with a simple lemma:Lemma 1.1. Suppose wl, -,w eA'V* (s<m=dim V) are linearly independent.(1) If n',...,nE A'v*and EniAwi=O, then there erist uniquely determinedreal numbers A, (l ≤ i, j <≤ s) with A, = A such that n' = A,w.(2) If s = m, and a collection of linear 1-forms y e A'v* (1<i,j<≤m) satisfyw ^的=0 and 的+=0,then Qj =- 0.Proof. (1) Obviously n' E span[w', -* ,ws]. Write n' = Ajwj. ThenZn Aw-(A, - A)wAwi<jand the conclusion follows.(2) Write j = ai,wk. Then the two conditions becomesajk-akj =0 and djk +dik =0.Thusak=j==-af=ak=a=-aj口and the conclusion follows.Now we state thefundamental theorem of Riemannian geometry [i.e. the existenceand uniqueness of Levi-Civita connection] in the language of connection 1-forms:Theorem 1.2 (E. Cartan). Let wl, ...,wm E 2'(U) be a collection of 1-forms onan open set U C M that are linearly independent at each point. Then there erists aunique collection of 1-forms, , E2'(U)(1≤i,j≤m),so thatdwi =wi ^ and +f =0.[These equations are known as Cartan's structural equations.]Proof. Uniqueness follows from Lemma 1.1 (2). For the existence, one just startwith the Riemannian metric g =w'w (so that the dual frame (ei) of (w') is anorthonormal basis for each point in U) and take , to be the connection 1-forms for the口Levi-Civita connection of thismetric.Remark. How to get from local to global? To glue, one need the connection 1-formsto satisfy the change of frame formula (2) for any orthogonal transformation f

LECTURE 11: THE METHOD OF MOVING FRAMES 3 ¶ Cartan’s formulation of Riemannian geometry. It turns out that one can develop Riemannian geometry starting with local frames and connection 1-forms (i.e. via the differential 1-forms ω i , θi j ) instead of the Riemannian metric g and its Levi-Civita connection[since one can recover the Riemannian metric g from the local orthonormal co-frame {ω i}, and then recover the Levi-Civita connection ∇ from its connection 1-forms θ j i ]. We start with a simple lemma: Lemma 1.1. Suppose ω 1 , · · · , ωs ∈Λ 1V ∗ (s≤m= dim V ) are linearly independent. (1) If η 1 , · · · , ηs ∈ Λ 1V ∗ and Pη i ∧ ω i = 0, then there exist uniquely determined real numbers Ai j (1 ≤ i, j ≤ s) with Ai j = A j i such that η i = Ai jω j . (2) If s = m, and a collection of linear 1-forms θ i j ∈ Λ 1V ∗ (1 ≤ i, j ≤ m) satisfy ω j ∧ θ i j = 0 and θ i j + θ j i = 0, then θ i j = 0. Proof. (1) Obviously η i ∈ span{ω 1 , · · · , ωs}. Write η i = Ai jω j . Then Xη i ∧ ω i = X i<j (A i j − A j i )ω i ∧ ω j and the conclusion follows. (2) Write θ i j = a i jkω k . Then the two conditions becomes a i jk − a i kj = 0 and a i jk + a j ik = 0. Thus a i jk = a i kj = −a k ij = −a k ji = a j ki = a j ik = −a i jk and the conclusion follows. □ Now we state ✿✿✿ the✿✿✿✿✿✿✿✿✿✿✿✿✿✿ fundamental✿✿✿✿✿✿✿✿✿ theorem✿✿ of✿✿✿✿✿✿✿✿✿✿✿✿✿ Riemannian✿✿✿✿✿✿✿✿✿✿✿ geometry [i.e. the existence and uniqueness of Levi-Civita connection] in the language of connection 1-forms: Theorem 1.2 (E. Cartan). Let ω 1 , · · · , ωm ∈ Ω 1 (U) be a collection of 1-forms on an open set U ⊂ M that are linearly independent at each point. Then there exists a unique collection of 1-forms, θ i j ∈ Ω 1 (U) (1 ≤ i, j ≤ m), so that dωi = ω j ∧ θ i j and θ i j + θ j i = 0. [These equations are known as Cartan’s structural equations. ] Proof. Uniqueness follows from Lemma 1.1 (2). For the existence, one just start with the Riemannian metric g = Pω i ⊗ ω i (so that the dual frame {ei} of {ω i} is an orthonormal basis for each point in U) and take θ i j to be the connection 1-forms for the Levi-Civita connection of this metric. □ Remark. How to get from local to global? To glue, one need the connection 1-forms to satisfy the change of frame formula (2) for any orthogonal transformation f

4LECTURE11:THEMETHODOFMOVINGFRAMESThe curvature 2-form.We start with any linear connection on a smooth manifold M. Suppose we aregiven a local co-frame (w'] and the corresponding connection 1-forms j. We mayexpress the curvature using differential forms (in terms of the connection 1-forms)as follows.By definitionR(X,Y)ei=VxVyei-VyVxei-V(x,yjei=Vx((Y)e;) -Vr((X)e;) -([X,Y))e)=X((Y)e;+(Y)的,(X)ex-Y((X))ej-(X),(Y)ek-([X,Y))e=(d)(X,Y)e + ^(X,Y)ej:As a consequence, if we denote R(ek, ei)e, = Ru'ej, then we getde+%^的=Rww=Rut'wkAw.(6)We shall denotenf -Rui'wh Awl,2and call it the curvature 2-form, which can be expressed in terms of 's as(7)2=d+The formula can be taken as definition of curvature (for given connection 1-forms)and is usuallywritten in brief as=where 2 is regarded as an m x m matrix whose entries are 2-forms.Unlike the connection 1-forms, given a linear connection, the curvature 2-formis independent of the choice of co-frame and thus is globally defined. To see this,weuse the frame transformation formula forconnection 1-forms aboveto get=a=(f-)-1(de)f-f(f-1)f+f-1oaof+f-1oadf+f-1ff-10f+f-1dff-1dfInviewof thefactdf-1=-f-1(df)f-1,weget= f-1(do +0)f = f-1nf,which is equivalent to say 2 is independent of the choice of frames.Now suppose (M,g) is a Riemannian manifold. Then we may start with or-thonormal co-frame (w'), and we have Cartan's structural equations, which implies2,=-j.We may also express the curvature 2-form ) using Rijkt := Rm(ei, ej, ek, es) asj = Rufw*Aw' = -RljiwkAwl=RijkwkAwl

4 LECTURE 11: THE METHOD OF MOVING FRAMES ¶ The curvature 2-form. We start with any linear connection on a smooth manifold M. Suppose we are given a local co-frame {ω i} and the corresponding connection 1-forms θ i j . We may express the curvature using differential forms (in terms of the connection 1-forms) as follows. By definition R(X, Y )ei =∇X∇Y ei − ∇Y ∇Xei − ∇[X,Y ]ei =∇X(θ j i (Y )ej ) − ∇Y (θ j i (X)ej ) − θ j i ([X, Y ])ej =X(θ j i (Y ))ej+θ j i (Y )θ k j (X)ek−Y(θ j i (X))ej−θ j i (X)θ k j (Y )ek−θ j i ([X, Y ])ej = (dθj i )(X, Y )ej + θ j k ∧ θ k i (X, Y )ej . As a consequence, if we denote R(ek, el)ei = Rkli j ej , then we get (6) dθj i + θ j k ∧ θ k i = R j kli ω k ⊗ ω l = 1 2 R j kli ω k ∧ ω l . We shall denote Ω j i = 1 2 R j kli ω k ∧ ω l , and call it the curvature 2-form, which can be expressed in terms of θ j i ’s as (7) Ωj i = dθj i + θ j k ∧ θ k i . The formula can be taken as definition of curvature (for given connection 1-forms) and is usually written in brief as Ω = dθ + θ ∧ θ, where Ω is regarded as an m × m matrix whose entries are 2-forms. Unlike the connection 1-forms, given a linear connection, the curvature 2-form is independent of the choice of co-frame and thus is globally defined. To see this, we use the frame transformation formula for connection 1-forms above to get Ω =e d ˜θ + ˜θ ∧ ˜θ =(df −1 ) ∧ θf + f −1 (dθ)f − f −1 θ ∧ df + (df −1 ) ∧ df + f −1 θ ∧ θf + f −1 θ ∧ df + f −1 df ∧ f −1 θf + f −1 df ∧ f −1 df. In view of the fact df −1 = −f −1 (df)f −1 , we get Ω =e f −1 (dθ + θ ∧ θ)f = f −1Ωf, which is equivalent to say Ω is independent of the choice of frames. Now suppose (M, g) is a Riemannian manifold. Then we may start with or￾thonormal co-frame {ω i}, and we have Cartan’s structural equations, which implies Ω j i = −Ω i j . We may also express the curvature 2-form Ωj i using Rijkl := Rm(ei , ej , ek, el) as Ω i j = 1 2 R i klj ω k ∧ ω l = − 1 2 Rkljiω k ∧ ω l = 1 2 Rijklω k ∧ ω l

5LECTURE11:THE METHOD OF MOVINGFRAMESRemark. More generally, one can develop the theory of linear connections on vectorbundles (or principal bundles) via moving frames, as follows. Let E be a rank rvector bundle over M, and [ei, ..,er] a local frame of E. Then one can eitherdefine a linear connectionV: T(E)→T(ET*M)via axioms that we mentioned earlier, or via connection 1-forms (1 ≤ i,j ≤ r)that are locally defined such thatVei=ej.As we calculated above, the matrix transform under change of basis as = f-1of + f-1df.One can further define the curvature 2-form to be2=o+0.2. APPLICATIONS TO RIEMANNIAN GEOMETRYI Calculating curvatures.As the first application, we use moving frames to calculate the curvature ofa Riemannian manifold (M, g). Let [ei, .. ,em) be a local orthonormal frame of(M,g).By definition the sectional curvature of theplane spanned by(e,e,) isK(ei,ej) = Rm(ei,ej,ei,ej) = Rijij = 2;(ej,ei).Theorem 2.1.(M,g) has constant sectional curvature c at p EM if and only iffor any local orthonormal frame feil, at p we have(8)2j=cwiΛwjProof. Suppose (8) holds at p for any orthonormal frame. Let II, be any two di-mensional plane in TpM. Choose an orthonormal basis [ei,e2] of IIp, extend it toan orthonormal frame and denote by w',...,wm the dual co-frame.ThenK(Ilp) = K(e1, e2) = c2(e2,e1) = cw? w'(e2, e1) = c.Conversely suppose (M,g) has constant sectional curvature c at p, then with respectto any orthonormal frame,Rijkl =gg(ei,ej,ek,et) =c(Sindjt-Sjkd)口at p and thus the conclusion follows.Erample. Consider the upper half space Hm with the hyperbolic metric(dr'@drl+...+drm @drm)ghyperbolic(rm

LECTURE 11: THE METHOD OF MOVING FRAMES 5 Remark. More generally, one can develop the theory of linear connections on vector bundles (or principal bundles) via moving frames, as follows. Let E be a rank r vector bundle over M, and {e1, · · · , er} a local frame of E. Then one can either define a linear connection ∇ : Γ∞(E) → Γ ∞(E ⊗ T ∗M) via axioms that we mentioned earlier, or via connection 1-forms θ j i (1 ≤ i, j ≤ r) that are locally defined such that ∇ei = ej ⊗ θ j i . As we calculated above, the matrix θ transform under change of basis as ˜θ = f −1 θf + f −1 df. One can further define the curvature 2-form to be Ω = dθ + θ ∧ θ. 2. Applications to Riemannian geometry ¶ Calculating curvatures. As the first application, we use moving frames to calculate the curvature of a Riemannian manifold (M, g). Let {e1, · · · , em} be a local orthonormal frame of (M, g). By definition the sectional curvature of the plane spanned by {ei , ej} is K(ei , ej ) = Rm(ei , ej , ei , ej ) = Rijij = Ωj i (ej , ei). Theorem 2.1. (M, g) has constant sectional curvature c at p ∈ M if and only if for any local orthonormal frame {ei}, at p we have (8) Ωi j = cωi ∧ ω j . Proof. Suppose (8) holds at p for any orthonormal frame. Let Πp be any two di￾mensional plane in TpM. Choose an orthonormal basis {e1, e2} of Πp, extend it to an orthonormal frame and denote by ω 1 , · · · , ωm the dual co-frame. Then K(Πp) = K(e1, e2) = cΩ 2 1 (e2, e1) = cω2 ∧ ω 1 (e2, e1) = c. Conversely suppose (M, g) has constant sectional curvature c at p, then with respect to any orthonormal frame, Rijkl = c 2 g○∧ g(ei , ej , ek, el) = c(δikδjl − δjkδil) at p and thus the conclusion follows. □ Example. Consider the upper half space Hm with the hyperbolic metric ghyperbolic = 1 (xm) 2 (dx1 ⊗ dx1 + · · · + dxm ⊗ dxm)