LECTURE28:BOCHNER'STCHNIQUEAND APPLICATIONSIn studying the relation between the curvatures of a Riemannina manifold and itsgeometry/topology, another very useful method is the so called Bochner technique.1.BOCHNER'SFORMULAI Bochner's formula.WestartwithTheorem 1.1.Let (M,g) be a Riemannian manifold, and X eFo(TM)(1)IfX is symmetric, i.e.(VuX,)=(V,X,u)for all u,uETX,thenA(XI) =VxP +(X, V(diux)+ Re(X, X),(2) IfVX is anti-symmetric, i.e. (VuX,o)=-(V,X,u) for all u,ETrX, thenA(IXIP) = IVXP - Re(X, X),Proof. (1) With Riemannian normal coordinates centered at , we haveVa,O,(r) =0, Vi,j.Recallat one canwrite(2f)r(O,0,) =(0,0,f)() and(f)()=(0,of)()It follows that at &,0,(x, X)=0(VaX, )A(IX}°)=a(VxX,0.)-(Va, VxX, 0)= (VxVa,X,0)-(V(x,a]X,0.)-(R(X,0)X,0.)= (VxVa,X, a.)-(V(x,a)X, a)+Rm(X,0, X,.) ((VxVa,X, a)-(Vx,a.)X,0.)+Rc(X,X)1
LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS In studying the relation between the curvatures of a Riemannina manifold and its geometry/topology, another very useful method is the so called Bochner technique. 1. Bochner’s formula ¶ Bochner’s formula. We start with Theorem 1.1. Let (M, g) be a Riemannian manifold, and X ∈ Γ ∞(TM). (1) If ∇X is symmetric, i.e. ⟨∇uX, v⟩ = ⟨∇vX, u⟩ for all u, v ∈ TxX, then 1 2 ∆(|X| 2 ) = |∇X| 2 + ⟨X, ∇(divX)⟩ + Rc(X, X). (2) If ∇X is anti-symmetric, i.e. ⟨∇uX, v⟩=−⟨∇vX, u⟩ for all u, v ∈ TxX, then 1 2 ∆(|X| 2 ) = |∇X| 2 − Rc(X, X). Proof. (1) With Riemannian normal coordinates centered at x, we have ∇∂i∂j (x) = 0, ∀i, j. Recall at x one can write (∇2 f)x(∂i , ∂j ) = (∂i∂jf)(x) and (∆f)(x)=X(∂i∂if)(x). It follows that at x, 1 2 ∆(|X| 2 )= 1 2 X i ∂i∂i⟨X, X⟩= X i ∂i⟨∇∂iX, X⟩ ⋆ = X i ∂i⟨∇XX, ∂i⟩ = X i ⟨∇∂i∇XX, ∂i⟩ = X i ⟨∇X∇∂iX, ∂i⟩−⟨∇[X,∂i]X, ∂i⟩−⟨R(X, ∂i)X, ∂i⟩ = X i ⟨∇X∇∂iX, ∂i⟩−⟨∇[X,∂i]X, ∂i⟩+Rm(X, ∂i , X, ∂i) = X i ⟨∇X∇∂iX, ∂i⟩−⟨∇[X,∂i]X, ∂i⟩ +Rc(X, X). 1
2LECTURE28:BOCHNER'STCHNIQUEANDAPPLICATIONSNote that if we write X = Xioi, thenTr(Vx) =(Va,X,0) =a(X, α) =o,X= divXitfollows(VxVa,X,0.) =x(Va,X,a.) = X(divX) = (X, Vdivx)On the other hand, since[Oi, X] =Va,-VxO, =Va,X(V(x,a)X,a.) =(Vva,xX,a)(Va,X, Va,X)= /VX/2.So the conclusion follows.(2) If X is anti-symmetric, then there will be a negative sign at the right handside of the two , and we will get O after the . So the conclusion follows.口I Bochner's formula for smooth functions.In particular,if u E Co(M),then X = vuis smooth and vX =?u issymmetric.Morever,divX=divVu=u.ItfollowsTheorem 1.2. For any u ECo(M),(/Vul2) = /V?uj + (Vu, V(u) + Rc(Vu, Vu),2Sometimes one need to replace theHessian term /v?uj2bya simpler one.Notethat by Cauchy-Schwartz inequality,forany A= (ai),[AI =lag]P ≥ ≥=(au) =(TrA)2.YIijAs a result, we get /V?u}? ≥ (u)? and thusCorollary 1.3. For any u E Co(M),(IVul) ≥ =(Au)* + (Vu, V(Au) + Rc(Vu, Vu),.2Remark. In particular, if Ric≥ (m-1)k, then for any uE Co(M)(IVul) ≥(u)2 + (Vu, V(u) + (m -1)/Vu2.(*)
2 LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS Note that if we write X = Xi∂i , then Tr(∇X) = X i ⟨∇∂iX, ∂i⟩ ♣= X i ∂i⟨X, ∂i⟩ = X i ∂iX i = divX. it follows X i ⟨∇X∇∂iX, ∂i⟩ = X i X⟨∇∂iX, ∂i⟩ = X(divX) = ⟨X, ∇divX⟩. On the other hand, since [∂i , X] = ∇∂i∂ − ∇X∂i = ∇∂iX, − X i ⟨∇[X,∂i]X, ∂i⟩ = X i ⟨∇∇∂iXX, ∂i⟩ ⋆ = X i ⟨∇∂iX, ∇∂iX⟩ = |∇X| 2 . So the conclusion follows. (2) If ∇X is anti-symmetric, then there will be a negative sign at the right hand side of the two ⋆ =, and we will get 0 after the ♣=. So the conclusion follows. □ ¶ Bochner’s formula for smooth functions. In particular, if u ∈ C ∞(M), then X = ∇u is smooth and ∇X = ∇2u is symmetric. Morever, divX = div∇u = ∆u. It follows Theorem 1.2. For any u ∈ C ∞(M), 1 2 ∆(|∇u| 2 ) = |∇2u| 2 + ⟨∇u, ∇(∆u)⟩ + Rc(∇u, ∇u). Sometimes one need to replace the Hessian term |∇2u| 2 by a simpler one. Note that by Cauchy-Schwartz inequality, for any A = (aij ), |A| 2 = X ij |aij | 2 ≥ X I a 2 ii ≥ 1 m ( X i aii) 2 = 1 m (TrA) 2 . As a result, we get |∇2u| 2 ≥ 1 m (∆u) 2 and thus Corollary 1.3. For any u ∈ C ∞(M), 1 2 ∆(|∇u| 2 ) ≥ 1 m (∆u) 2 + ⟨∇u, ∇(∆u)⟩ + Rc(∇u, ∇u). Remark. In particular, if Ric ≥ (m − 1)k, then for any u ∈ C ∞(M), (*) 1 2 ∆(|∇u| 2 ) ≥ 1 m (∆u) 2 + ⟨∇u, ∇(∆u)⟩ + (m − 1)k|∇u| 2
3LECTURE28:BOCHNER'STCHNIQUEANDAPPLICATIONSConversely, if this inequality holds for any u E Co(M), then we must have Ric ≥(m - 1)k. To see this, given any ro E M and any Xo e Tr. M, we take u E Coo(M)so that Vu(ro) = Xo and (V2u)ro = cld. Then by Bochner formula and (*),A(/VulP) - (Vu, V(Au)) ≥ 二(Au)? + (m - 1)K|Xo/2,/V?u|?+Rc(Xo,X)=rOn the other hand, our choice of u implies /v?u/? = (Au), so we getRic≥(m-1)kThe condition (*) is used in discrete geometric analysis (on graphs one can define and △ but not curvature tensor) as a definition of “Ric ≥ (m - 1)k".I Bochner formula for closed 1-forms.Recall from Lecture2that given any smooth 1-form w E 2'(M), the musicalisomorphism produce a smooth vector field X - #w. It is not hard to check[X|= [w], /VX]2 = [Vw2.Now suppose w E 2'(M) is a closed 1-form. Then locally w is exact, i.e. locally ofthe form w=du.As a result,X=tw =Vu, and vX is symmetricl.So wemayapply part (1) of Theorem 1.1 to get()=/Vw/?+(,divw)+Rc(ww)nToproceed let'sdo somelocal computation.Supposew=w;dr.Then#w=wid,divw-awi,div#w=adwidji.3and thus《wdi)owiijOn the other hand, as we have seen in Lecture 4, for any smooth function fdiv(f#w)=fdiv#w+(Vf,#w).Now we assume Mis compact. After integration we getdiv(f#w)dVg =(fdivw + (Vf, #w)dVg=(fdivw+(df,w))dVg0=Itfollowsthat(df,w)L2 = / (df,w)dVg = /f(-div#w)dVg=<f,-div#w)L2.So if we define Sw=-divw.Then :2'(M)-→Co(M)is theL?-dual of d.Vf EC(M),w E2'(M)(df,w)L2 =(f,8w)L2,lIn fact one can show that vx is symmetric if and only if w = bx is closed
LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS 3 Conversely, if this inequality holds for any u ∈ C ∞(M), then we must have Ric ≥ (m − 1)k. To see this, given any x0 ∈ M and any X0 ∈ Tx0M, we take u ∈ C ∞(M) so that ∇u(x0) = X0 and (∇2u)x0 = cId. Then by Bochner formula and (*), |∇2u| 2 + Rc(X0, X0) = 1 2 ∆(|∇u| 2 ) − ⟨∇u, ∇(∆u)⟩ ≥ 1 m (∆u) 2 + (m − 1)k|X0| 2 , On the other hand, our choice of u implies |∇2u| 2 = 1 m (∆u) 2 , so we get Ric ≥ (m − 1)k. The condition (*) is used in ✿✿✿✿✿✿✿✿ discrete✿✿✿✿✿✿✿✿✿✿✿ geometric ✿✿✿✿✿✿✿✿✿ analysis (on graphs one can define ∇ and ∆ but not curvature tensor) as a definition of “Ric ≥ (m − 1)k”. ¶ Bochner formula for closed 1-forms. Recall from Lecture 2 that given any smooth 1-form ω ∈ Ω 1 (M), the musical isomorphism produce a smooth vector field X = ♯ω. It is not hard to check |X| = |ω|, |∇X| 2 = |∇ω| 2 . Now suppose ω ∈ Ω 1 (M) is a ✿✿✿✿✿✿ closed 1-form. Then locally ω is exact, i.e. locally of the form ω = du. As a result, X = ♯ω = ∇u, and ∇X is symmetric1 . So we may apply part (1) of Theorem 1.1 to get 1 2 ∆(|ω| 2 ) = |∇ω| 2 + ⟨♯ω, ∇div♯ω⟩ + Rc(♯ω, ♯ω). To proceed let’s do some local computation. Suppose ω = ωidxi . Then ♯ω = X i ωi∂i , div♯ω = X i ∂iωi , ∇div♯ω = X i,j ∂j∂iωi∂j . and thus ⟨♯ω, ∇div♯ω⟩ = X i,j ωj∂j∂iωi . On the other hand, as we have seen in Lecture 4, for any smooth function f, div(f ♯ω) = fdiv♯ω + ⟨∇f, ♯ω⟩. Now we assume M✿✿✿✿✿ is ✿✿✿✿✿✿✿✿✿ compact. After integration we get 0 = Z M div(f ♯ω)dVg = Z M (fdiv♯ω + ⟨∇f, ♯ω⟩)dVg = Z M (fdiv♯ω + ⟨df, ω⟩)dVg. It follows that ⟨df, ω⟩L2 = Z M ⟨df, ω⟩dVg = Z M f(−div♯ω)dVg = ⟨f, −div♯ω⟩L2 . So if we define δω = −div♯ω. Then δ : Ω1 (M) → C ∞(M) is the L 2 -dual of d, ⟨df, ω⟩L2 = ⟨f, δω⟩L2 , ∀f ∈ C ∞(M), ω ∈ Ω 1 (M). 1 In fact one can show that ∇X is symmetric if and only if ω = ♭X is closed
4LECTURE28:BOCHNER'STCHNIOUE AND APPLICATIONSFor any closed 1-form w we define w = d&w. Then locallyAw= d(-0wi) =-aowiiijand thus(w,Aw) = -wjo,owi=-(#w, Vdiv#w).ijSo we end withTheorem 1.4 (Bochner's formula for closed 1-form).Let (M,g) be a compact Rie-mannian manifold, then for any closed 1-form w E 2'(M),A(lol) = ulP - (a, Aw) + Re(tu, u),.I Harmonic k-form.More generally, one can define : 2*(M) -→ k-1(M) so that(w,dn)L2 = (8w,n)L2, Vw E2*(M),nE2k-1(M)and define the Hodge Laplacian on all smooth k-forms to beA:=d8+8d:(M)-2M)One can check that whenk =O this definition coincides with the Laplace-Beltramioperator on smooth functions (and thus differed with =tr2by a negativesign). A differential form w k(M) is called a harmonic k-form if w = 0. Inview of the fact(w,w)12=(,d8w)12+ (w,8dw)2 = (dw,dw)12+(8w,8)L2and the definition of ,we haveProposition 1.5.w E2k(M)isharmonic if and onlyif dw=0 and ow=0.Now suppose w is a harmonic 1-form on compact Riemannian manifold (M,g).Then w is closed, and thus by Theorem 1.4,MA(lol) = //V+//Rc(#w, #w)0=So we getTheorem 1.6 (Bochner). Let (M,g) be a compact Riemannian manifold, then(1)SupposeRic≥0.If△w=0,thenVw=0.(2)SupposeRic≥O,and Ric>0 at onepoint.Ifw=0,thenw=0.According to the famous Hodge theory, the space of harmonic k-forms is iso-morphic to the de Rham cohomology group Har(M). So we concludeCorollary 1.7. Let (M, g) be a closed oriented Riemannian manifold, Ric ≥ O, andRic > 0 at one point. Then bi(M) = 0
4 LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS For any closed 1-form ω we define ∆ω = dδω. Then locally ∆ω = d(− X i ∂iωi) = − X i,j ∂j∂iωi and thus ⟨ω, ∆ω⟩ = − X i,j ωj∂j∂iωi = −⟨♯ω, ∇div♯ω⟩. So we end with Theorem 1.4 (Bochner’s formula for closed 1-form). Let (M, g) be a compact Riemannian manifold, then for any closed 1-form ω ∈ Ω 1 (M), 1 2 ∆(|ω| 2 ) = |∇ω| 2 − ⟨ω, ∆ω⟩ + Rc(♯ω, ♯ω). ¶ Harmonic k-form. More generally, one can define δ : Ωk (M) → Ω k−1 (M) so that ⟨ω, dη⟩L2 = ⟨δω, η⟩L2 , ∀ω ∈ Ω k (M), η ∈ Ω k−1 (M), and define the Hodge Laplacian on all smooth k-forms to be ∆ := dδ + δd : Ωk (M) → Ω k (M). One can check that when k = 0 this definition coincides with the Laplace-Beltrami operator ∆ on smooth functions (and thus differed with ∆ = tr∇2 by a ✿✿✿✿✿✿✿✿ negative sign). A differential form ω ∈ Ω k (M) is called a harmonic k-form if ∆ω = 0. In view of the fact ⟨ω, ∆ω⟩L2 = ⟨ω, dδω⟩L2 + ⟨ω, δdω⟩L2 = ⟨dω, dω⟩L2 + ⟨δω, δω⟩L2 and the definition of ∆, we have Proposition 1.5. ω ∈ Ω k (M) is harmonic if and only if dω = 0 and δω = 0. Now suppose ω is a harmonic 1-form on compact Riemannian manifold (M, g). Then ω is closed, and thus by Theorem 1.4, 0 = Z M 1 2 ∆(|ω| 2 ) = Z M |∇ω| 2 + Z M Rc(♯ω, ♯ω). So we get Theorem 1.6 (Bochner). Let (M, g) be a compact Riemannian manifold, then (1) Suppose Ric ≥ 0. If ∆ω = 0, then ∇ω = 0. (2) Suppose Ric ≥ 0, and Ric > 0 at one point. If ∆ω = 0, then ω = 0. According to the famous Hodge theory, the space of harmonic k-forms is isomorphic to the de Rham cohomology group Hk dR(M). So we conclude Corollary 1.7. Let (M, g) be a closed oriented Riemannian manifold, Ric ≥ 0, and Ric > 0 at one point. Then b1(M) = 0
LECTURE28:BOCHNER'STCHNIOUE ANDAPPLICATIONS5Remarks. There is a Bochner-Weitzenbock formula that generalize the Bochner for-mula aboveto k-forms using which one canprove:If(M,g)isa closed Riemannianmanifold with nomnegative curvature.operator, then all harmonic forms of order1 ≤ k ≤ m - 1 on M are parallel.I Bochner formula for Killing forms.Now let's turn to part (2) in Theorem 1.1.As we have seen in PSet 1, vX isanti-symmetric if and only if X is a Killing field on (M,g). So we getCorollary 1.8. For any Killing vector field on M,A(IXI) = /VXI2 - Rc(X, X).As a result,Theorem 1.9 (Bochner, 1946).AnyKilling vector field of a compactRiemannianmanifold with negative Ricci curvature must be zero.Since the space of all Killing vector fields on (M, g) is the Lie algebra of theisometry group Iso(M,g)[which is a Lie group,and the isometry group of any com-pact Riemannianmanifold is compact, we concludethattheisometrygroup of anycompactRiemannian manifold with negative Ricci curvaturemustbea finitegroup.2.CHEEGER-GROMOLL SPLITTING THEOREMTCheeger-Gromoll splittingtheorem.Let (M,g) be a complete non-compact connected Riemannian manifold.Recallthat a line in M is a normal geodesic : R → M so thatd(%(a),(b)) = [a - bl, Va,b e R.Unlike the case of rays,it is possible that there is no ray in a complete non-compactRiemannian manifold.For example, a cylinder Rx Sl admitsmany lines, whiletheparaboloid z= r2+y?admits no line at all.As another application of Bochner formula, we prove the following structuretheoremfor Riemannian manifoldswithpositiveRiccicurvature that admit lines:Theorem 2.1 (Cheeger-Gromoll, 1971).Let (M.g)be a complete non-compact Riemannian manifold withRic≥O.Suppose there erists a line in M.Then (M,g)is isometric toR×N,where N is an (m-I)-dimensional completeRiemannianmanifold with Ric≥0.The idea is to construct a function on M which behaves like the functionf(r,r)=r on N × R, so thatthe level sets of f gives the desired componentN.Sowhat is the speciality of thefunctionf(r,r)=r?It is smooth,with gradientf = o, which has length 1, and has Hessian ? f = o (so that Vf is parallel). Itis in the proof of "? f = o" that we need Bochner's formula
LECTURE 28: BOCHNER’S TCHNIQUE AND APPLICATIONS 5 Remarks. There is a Bochner-Weitzenb¨ock formula that generalize the Bochner formula above to k-forms using which one can prove: If (M, g) is a closed Riemannian manifold with ✿✿✿✿✿✿✿✿✿✿✿✿ nonnegative✿✿✿✿✿✿✿✿✿✿✿ curvature ✿✿✿✿✿✿✿✿✿✿ operator, then all harmonic forms of order 1 ≤ k ≤ m − 1 on M are parallel. ¶ Bochner formula for Killing forms. Now let’s turn to part (2) in Theorem 1.1. As we have seen in PSet 1, ∇X is anti-symmetric if and only if X is a Killing field on (M, g). So we get Corollary 1.8. For any Killing vector field on M, 1 2 ∆(|X| 2 ) = |∇X| 2 − Rc(X, X). As a result, Theorem 1.9 (Bochner, 1946). Any Killing vector field of a compact Riemannian manifold with negative Ricci curvature must be zero. Since the space of all Killing vector fields on (M, g) is the Lie algebra of the isometry group Iso(M, g)[which is a Lie group], and the isometry group of any compact Riemannian manifold is compact, we conclude that the isometry group of any compact Riemannian manifold with negative Ricci curvature must be a finite group. 2. Cheeger-Gromoll splitting theorem ¶ Cheeger-Gromoll splitting theorem. Let (M, g) be a complete non-compact connected Riemannian manifold. Recall that a ✿✿✿✿ line in M is a normal geodesic γ : R → M so that d(γ(a), γ(b)) = |a − b|, ∀a, b ∈ R. Unlike the case of rays, it is possible that there is no ray in a complete non-compact Riemannian manifold. For example, a cylinder R × S 1 admits many lines, while the paraboloid z = x 2 + y 2 admits no line at all. As another application of Bochner formula, we prove the following structure theorem for Riemannian manifolds with positive Ricci curvature that admit lines: Theorem 2.1 (Cheeger-Gromoll, 1971). Let (M, g) be a complete non-compact Riemannian manifold with Ric ≥ 0. Suppose there exists a line in M. Then (M, g) is isometric to R × N, where N is an (m − 1)-dimensional complete Riemannian manifold with Ric ≥ 0. The idea is to construct a function on M which behaves like the function f(x, r) = r on N × R, so that the level sets of f gives the desired component N. So what is the speciality of the function f(x, r) = r? It is smooth, with gradient ∇f = ∂r which has length 1, and has Hessian ∇2 f = 0 (so that ∇f is parallel). It is in the proof of “∇2 f = 0” that we need Bochner’s formula