PROBLEMSET2:CURVATURESDUE:APRIL28,2024Instruction: Please work on at least SiX problems of your interest, and in each problem you aresupposed to work on at least TWO sub-problems.1.[Isometrypreservesgeometry]Let : (M, g) -→ (M,g) be a local isometry. Prove:(a) Given a coordinate system (rl,..., arm) near a point p of M, one define a coordinate system(l,...,am)nearp(p)bylettingi=rop-1.Prove:*gij=gij.(b) Prove: For vector fields X,Y e r(TM), by restricting to neighborhoods of p and (p),one has dp(VxY) = Vde(x)d(Y).(c) State and prove the fact “curvature tensor is invariant under local isometry"(d) State and prove the fact “sectional curvature is invariant under local isometry"(e) Prove that the natural action of the isometry group Iso(sm, ground) on the Grassmannianbundle G2(Tsm) is transitive, and thus (Sm, ground) has constant sectional curvature.2.[Riemannian Geometry of the hyperbolic space:Part 2]Let Hm be the upper half-space in the Rm, ie.Hm=(rl,..,am)eRm [am>0)equipped with the hyperbolic metric1(dar'drl+...+damdam)9Hm:Tm)2(a) Calculate the Christoffel's symbols.(b) Calculate the Riemannian curvature tensor, and show that Hm has constant sectional cur-vature -1.(c) For m = 2 and > 0, find the volume of the vertical stripeS。=[(r,y) I [r|<1,e<y<0].3. [Riemannian Geometry of Product Manifolds].Let (Mi,gi) and (M2,g2) be Riemannian manifolds. Consider the product Riemannian man-ifold (Mi × M2,ig1 +292), where元:Mi×M2→Mbe the canonical projections. For any (p,q) E Mi × M2, we let: M→Mi× M2,p-(p,q)be the embedding of Mi into Mi × M2 as Mi × [g). Similarly one define : M2 → Mi × M2.Denoteby iand RmtheLevi-Civita connection and Riemann curvaturetensor on MA
PROBLEM SET 2: CURVATURES DUE: APRIL 28, 2024 Instruction: Please work on at least SIX problems of your interest, and in each problem you are supposed to work on at least TWO sub-problems. 1. [Isometry preserves geometry] Let φ : (M, g) → (M, f g˜) be a local isometry. Prove: (a) Given a coordinate system (x 1 , · · · , xm) near a point p of M, one define a coordinate system (˜x 1 , · · · , x˜ m) near φ(p) by letting ˜x i = x i ◦ φ −1 . Prove: φ ∗ g˜ij = gij . (b) Prove: For vector fields X, Y ∈ Γ∞(TM), by restricting to neighborhoods of p and φ(p), one has dφ(∇XY ) = ∇e dφ(X)dφ(Y ). (c) State and prove the fact “curvature tensor is invariant under local isometry”. (d) State and prove the fact “sectional curvature is invariant under local isometry”. (e) Prove that the natural action of the isometry group Iso(S m, ground) on the Grassmannian bundle G2(T Sm) is transitive, and thus (S m, ground) has constant sectional curvature. 2. [Riemannian Geometry of the hyperbolic space: Part 2] Let Hm be the upper half-space in the R m, i.e. Hm = {(x 1 , · · · , xm) ∈ R m | x m > 0}, equipped with the hyperbolic metric gHm = 1 (xm) 2 (dx1 ⊗ dx1 + · · · + dxm ⊗ dxm). (a) Calculate the Christoffel’s symbols. (b) Calculate the Riemannian curvature tensor, and show that Hm has constant sectional curvature −1. (c) For m = 2 and ε > 0, find the volume of the vertical stripe Sε = {(x, y) | |x| < 1, ε < y < ∞}. 3. [Riemannian Geometry of Product Manifolds] • Let (M1, g1) and (M2, g2) be Riemannian manifolds. Consider the product Riemannian manifold (M1 × M2, π∗ 1 g1 + π ∗ 2 g2), where πi : M1 × M2 → Mi be the canonical projections. For any (p, q) ∈ M1 × M2, we let ι q 1 : M1 → M1 × M2, p 7→ (p, q) be the embedding of M1 into M1 × M2 as M1 × {q}. Similarly one define ι p 2 : M2 → M1 × M2. Denote by ∇i and Rmi the Levi-Civita connection and Riemann curvature tensor on Mi . 1
2PROBLEM SET 2:CURVATURESDUE:APRIL 28,2024(a) Prove: The Levi-Civita connection V on Mi × M2 satisfiesVx1+x2(Yi + Y2)(p,g) = d(V, Y1)p + d(V,Y2)for any Xi,Yi E T(TM),X2,Y2 er(TM2)(b) Prove: The Riemann curvature tensor of Mi × M2 isRm(X1 + X2, Yi + Y2, Z1 + Z2, Wi + W2) = Rm'(X1, Yi, Z1, Wi) + Rm2(X2, Y2, Z2, W2)for any Xi, Yi1, Zi, Wi Foo(TM1), X2, Y2, Z2, W2 Fo(TM2)(c) Write down similar formulae for the Ricci tensor, the Weyl tensor and the scalar curvatureof MxMintermsofthoseof MandM2.NowconsiderSmxSnendowed withthecanonicalmetric.(d) Prove: All sectional curvatures of sm x sn lie in [0, 1].(e) Find the Ricci and Weyl tensor of sm × sn.(f) When will it be Einstein? When will it be locally conformally flat?(g) Find the scalar curvature of sm × Sn.4. [Polarization formula for Riemann curvature tensor]Denote K(X,Y) = Rm(X,Y,X,Y).(a) Find a formula for Rm(X,Y, X, W) in terms of K.(b) Prove:6Rm(X,Y,Z,W)=K(X + Z,Y +W) - K(Y + Z,X +W)- K(X,Y +W) - K(Z,Y +W)- K(X +Z,Y) - K(X + Z,W)+K(Y,X +W)+ K(Z,X +W)+ K(Y + Z, X) +K(Y + Z,W) +K(X, W)+K(Z,Y) - K(Y, W) - K(Z,X)5. [Curvature-like tensors as curvature tensors](a) Fix r. For 1 ≤i≤j≤m, choose vectors hij E R", and let hji = hij. DefineRijkl=(hik,hjt)-(hi,hjk),where(,)is the standardinnerproduct on Rr.Prove:Rijkie'e @ek@e'is acurvature-liketensor,where e',...,em is a basis of (Rm)*(b) Prove: Fix m, for r large enough, any curvature-like tensor on Rm arises in this way.6. [Riemannian Geometry under Conformal change]Let (M,g) be a Riemannian manifold of dimension m. Let g = e2g be a Riemannian metricthat is conformal to g. In what follows, everything without a “"bar" (e.g. V, Rm, S etc) is for(M,g), and everything with a “bar" (e.g. , Rm, S etc) is for (M,g)(a) Prove: xY = VxY +(X)Y +(Y)X - g(X,Y)V(b) Prove:Rm= e2[Rm- g(Vdp- dpdp+ ldol’g)](c) Prove: S = e-24 [S + 2(m -1)Ap - (m - 2)(m - 1)/Vi)] ,(d)[*] The conformal Laplacian is defined to be the operatorLf =Af+(m-2S4(m-1)Check: For any f ECoo(M),Lf =e-"L(ef)
2 PROBLEM SET 2: CURVATURES DUE: APRIL 28, 2024 (a) Prove: The Levi-Civita connection ∇ on M1 × M2 satisfies ∇X1+X2 (Y1 + Y2)(p, q) = dιq 1 (∇1 X1 Y1)p + dιp 2 (∇2 X2 Y2)q for any X1, Y1 ∈ Γ∞(TM1), X2, Y2 ∈ Γ∞(TM2). (b) Prove: The Riemann curvature tensor of M1 × M2 is Rm(X1 + X2, Y1 + Y2, Z1 + Z2, W1 + W2) = Rm1 (X1, Y1, Z1, W1) + Rm2 (X2, Y2, Z2, W2) for any X1, Y1, Z1, W1 ∈ Γ∞(TM1), X2, Y2, Z2, W2 ∈ Γ∞(TM2). (c) Write down similar formulae for the Ricci tensor, the Weyl tensor and the scalar curvature of M1 × M2 in terms of those of M1 and M2. • Now consider S m × S n endowed with the canonical metric. (d) Prove: All sectional curvatures of S m × S n lie in [0, 1]. (e) Find the Ricci and Weyl tensor of S m × S n . (f) When will it be Einstein? When will it be locally conformally flat? (g) Find the scalar curvature of S m × S n . 4. [Polarization formula for Riemann curvature tensor] Denote Ke(X, Y ) = Rm(X, Y, X, Y ). (a) Find a formula for Rm(X, Y, X, W) in terms of Ke. (b) Prove: 6Rm(X, Y, Z, W) =Ke(X + Z, Y + W) − Ke(Y + Z, X + W) − Ke(X, Y + W) − Ke(Z, Y + W) − Ke(X + Z, Y ) − Ke(X + Z, W) + Ke(Y, X + W) + Ke(Z, X + W) + Ke(Y + Z, X) + Ke(Y + Z, W) + Ke(X, W) + Ke(Z, Y ) − Ke(Y, W) − Ke(Z, X). 5. [Curvature-like tensors as curvature tensors] (a) Fix r. For 1 ≤ i ≤ j ≤ m, choose vectors hij ∈ R r , and let hji = hij . Define Rijkl = ⟨hik, hjl⟩ − ⟨hil, hjk⟩, where ⟨·, ·⟩ is the standard inner product on R r . Prove: Rijkle i⊗e j⊗e k⊗e l is a curvature-like tensor, where e 1 , · · · , em is a basis of (R m) ∗ . (b) Prove: Fix m, for r large enough, any curvature-like tensor on R m arises in this way. 6. [Riemannian Geometry under Conformal change] Let (M, g) be a Riemannian manifold of dimension m. Let ¯g = e 2φg be a Riemannian metric that is conformal to g. In what follows, everything without a “bar” (e.g. ∇, Rm, S etc) is for (M, g), and everything with a “bar” (e.g. ∇, Rm, S etc) is for (M, g¯). (a) Prove: ∇XY = ∇XY + (Xφ)Y + (Y φ)X − g(X, Y )∇φ. (b) Prove: Rm = e 2φ Rm − g○∧ (∇dφ − dφ ⊗ dφ + 1 2 |dφ| 2 g) (c) Prove: S = e −2φ S + 2(m − 1)∆φ − (m − 2)(m − 1)|∇φ| 2 . (d) [*] The conformal Laplacian is defined to be the operator Lf = ∆f + (m − 2)S 4(m − 1) f. Check: For any f ∈ C∞(M), Lf = e − m+2 2 φL(e m−2 2 φ f).
DUE:APRIL 28,2024PROBLEM SET 2:CURVATURES3(e) [*] Deduce the Yamabe equation: If m > 2, and g = um-2 g, then4m-1Au+s.u=s.um7. [Riemannian Geometry of Surfaces]Let M be a two dimensional manifold, i.e. a surface, with Riemannian metric g.(a) [*] Prove: Near each point p, there is a coordinate system so that g is of the formg = E(rl, a2)dal dal + G(rl,r2)da2 @ dr2.(b)Express the sectional curvature of M at each point using the functions E,G above.[Youcanuse eitherdirect computation orthemovingframe.8. [Weyl Tensors](a) Let (M,g) be of constant sectional curvature.Prove: R × M is locally conformally flat.(b) Again suppose (M,g) has constant sectional curvature. Suppose b is a positive smoothfunction on R.Prove:The warped product RXM is locally conformallyflat.(c)Find theWeyl tensorof S2×$?andR?×S?.9. [Riemannian Geometry of Submanifolds: Part 1]Let (M,gm) be a Riemannian manifold, and t : M M be an embedded submanifold, withinduced Riemannian structure gM = t"gm. Using the Riemannian structure of M, for each pointp E M c M, any X, E T,M can be written uniquely asXp=x, +X,where x, e TpM and X, e (TpM)+. In what follows we assume X,Y,z,W e I(TM) aresmooth tangent vector fields on M, and s,C e I(TM+) are smooth normal vector fields on M,i.e. Sp, Sp e (TM)- for all p E M and are smooth.(a) Let be the Levi-Civita connection of (M,g). Explain: VxY is well-defined(b) Let VxY = (VxY)T. Check: V is the Levi-Civita connection on M(c) Set S(X,Y) = (xY). Check: S is Co(M)-linear and symmetric. (S is called the secondfundamental form of M.)(d) Define the shape operator Sg : T,M → T,M byVY ET(TM).(Se(X),Y) = (S(X,Y),s),Prove: Sg is symmetric.(e) Prove: Se(X) = -(x)T(f) Let Rm,Rm be the Riemannian curvature tensor on M,M.Prove: (Gauss equation)Rm(X,Y,Z,W) = Rm(X,Y,Z,W)- (S(X,Z),S(Y,W))+ (S(X,W), S(Y,Z).(g) Use the Gauss equation to prove: (Sm, gsm) has constant curvature 1.(h)[*] Verify the Codazzi equationRm(X,Y,Z,) = -((VxS)(Y, Z),s) + ((VyS)(X, Z),s),where (VxS)(Y,Z) := (Vx(S(Y, Z)) - S(VxY,Z) - S(Y, VxZ)
PROBLEM SET 2: CURVATURES DUE: APRIL 28, 2024 3 (e) [*] Deduce the Yamabe equation: If m > 2, and ¯g = u 4 m−2 g, then 4 m − 1 m − 2 ∆u + S · u = S · u m+2 m−2 . 7. [Riemannian Geometry of Surfaces] Let M be a two dimensional manifold, i.e. a surface, with Riemannian metric g. (a) [*] Prove: Near each point p, there is a coordinate system so that g is of the form g = E(x 1 , x2 )dx1 ⊗ dx1 + G(x 1 , x2 )dx2 ⊗ dx2 . (b) Express the sectional curvature of M at each point using the functions E, G above. [You can use either direct computation or the moving frame.] 8. [Weyl Tensors] (a) Let (M, g) be of constant sectional curvature. Prove: R × M is locally conformally flat. (b) Again suppose (M, g) has constant sectional curvature. Suppose ψ is a positive smooth function on R. Prove: The warped product R ×ψ M is locally conformally flat. (c) Find the Weyl tensor of S 2 × S 2 and R 2 × S 2 . 9. [Riemannian Geometry of Submanifolds: Part 1] Let (M, gM) be a Riemannian manifold, and ι : M ,→ M be an embedded submanifold, with induced Riemannian structure gM = ι ∗ gM. Using the Riemannian structure of M, for each point p ∈ M ⊂ M, any Xp ∈ TpM can be written uniquely as Xp = X T p + X ⊥ p , where X T p ∈ TpM and X ⊥ p ∈ (TpM) ⊥. In what follows we assume X, Y, Z, W ∈ Γ(TM) are smooth tangent vector fields on M, and ξ, ζ ∈ Γ(TM⊥) are smooth ✿✿✿✿✿✿ normal vector fields on M, i.e. ξp, ζp ∈ (TpM) ⊥ for all p ∈ M and are smooth. (a) Let ∇ be the Levi-Civita connection of (M, gM). Explain: ∇XY is well-defined (b) Let ∇XY = (∇XY ) T . Check: ∇ is the Levi-Civita connection on M. (c) Set S(X, Y ) = (∇XY ) ⊥. Check: S is C∞(M)-linear and symmetric. (S is called the second fundamental form of M. ) (d) Define the shape operator Sξ : TpM → TpM by ⟨Sξ(X), Y ⟩ = ⟨S(X, Y ), ξ⟩, ∀ Y ∈ Γ ∞(TM). Prove: Sξ is symmetric. (e) Prove: Sξ(X) = −(∇Xξ) T . (f) Let Rm, Rm be the Riemannian curvature tensor on M, M. Prove: (Gauss equation) Rm(X, Y, Z, W) = Rm(X, Y, Z, W) − ⟨S(X, Z), S(Y, W)⟩ + ⟨S(X, W), S(Y, Z)⟩. (g) Use the Gauss equation to prove: (S m, gSm) has constant curvature 1. (h) [*] Verify the Codazzi equation Rm(X, Y, Z, ξ) = −⟨(∇XS)(Y, Z), ξ⟩ + ⟨(∇Y S)(X, Z), ξ⟩, where (∇XS)(Y, Z) := (∇X(S(Y, Z)))⊥ − S(∇XY, Z) − S(Y, ∇XZ)
4PROBLEMSET2:CURVATURESDUE:APRIL28,2024(i) [*) We will denote = (xs)+ and R(X,Y) = - +V+,s.Verify the Ricci equationRm(X,Y,S,S) = (R-(X,Y)S,S)+<SeSX - ScSeX,Y)(These three equations arethefundamental equations in submanifold geometry.)10.[RiemannianGeometryofLieGroups:Part2]Let G be a Lie group endowed with a bi-invariant Riemannian metric g. Suppose X,Y, Z E gareleft-invariantvectorfieldsonG(a) Prove: 《[X,Y], Z) = ([Y,Z], X)(b) Prove: xX = 0.(c) Prove: VxY = [X,Y].(d) Prove: R(X,Y)Z =-4[X,Y], Z](e) Let X,Y be orthonormal, and II, E TM be the 2-dim plane spanned by Xp,Yp. Prove:K(II,) = I[X,YI?.(f) Prove: G has positive Ricci curvature if the center of G is discrete.1l. [Riemannian Submersions]Let (M,gm) and (M,gM) be Riemannian manifolds, and f : M -→ M a submersion. For anyp EM, wewill call V,=Ker(dfp) CT,M the uertical space at p,and Hp=V CT,M thehorizontal space at p.For any vector Xp e T,M, we will denote X"the vertical component ofX. Note that dfp, when restricted to Hp, is a linear isomorphism. For any X e To(TM), itshorizontal lift is the horizontal vector field X defined by dfp(Xp) = Xf(p). The submersion f issaid to be a Riemannian submersion if for any p e M, dfp : Hp - Tf(p)M is a linear isometry.(a) Prove: For any X,Y E T(TM),=VxY+(,.(b) Prove: For any X,Y,Z,W eI(TM),Rm(X,Y,z,W) =Rm(X,Y,z,W) -《[x,z",[,w])《(,z,[区,)”-[z,)",[区,)")(c) Let II be a plane spanned by orthogonal unit vectors Xp, Yp e TpM and II the plane spannedby X,, Yp. Prove: K(Il) = K(I) +[Xp, Yp]'?12. [Gauss-Bonnet-Chern in dimension 4]Thefamous Gauss-Bonnet-Chernformula saysthatif (M,g)is an orientable closedRiemannianmanifoldofdimensionm=2k,then12 = x(M),2m元m/2(m/2)！Jwhere x(M) is the Euler characteristic of M, and is the following m-form(m-1)32= (-1)°2g(lA0A..An(2)(4)o(m)aESm
4 PROBLEM SET 2: CURVATURES DUE: APRIL 28, 2024 (i) [*] We will denote ∇⊥ Xξ = (∇Xξ) ⊥ and R⊥(X, Y )ξ = −∇⊥ X∇⊥ Y ξ + ∇⊥ Y ∇⊥ Xξ + ∇⊥ [X,Y ] ξ. Verify the Ricci equation Rm(X, Y, ξ, ζ) = ⟨R ⊥(X, Y )ξ, ζ⟩ + ⟨SξSζX − SζSξX, Y ⟩. (These three equations are the fundamental equations in submanifold geometry.) 10. [Riemannian Geometry of Lie Groups: Part 2] Let G be a Lie group endowed with a bi-invariant Riemannian metric g. Suppose X, Y, Z ∈ g are left-invariant vector fields on G. (a) Prove: ⟨[X, Y ], Z⟩ = ⟨[Y, Z], X⟩. (b) Prove: ∇XX = 0. (c) Prove: ∇XY = 1 2 [X, Y ]. (d) Prove: R(X, Y )Z = − 1 4 [[X, Y ], Z]. (e) Let X, Y be orthonormal, and Πp ∈ TpM be the 2-dim plane spanned by Xp, Yp. Prove: K(Πp) = 1 4 ∥[X, Y ]∥ 2 . (f) Prove: G has positive Ricci curvature if the center of G is discrete. 11. [Riemannian Submersions] Let (M, gM) and (M, gM) be Riemannian manifolds, and f : M → M a submersion. For any p ∈ M, we will call Vp = Ker(dfp) ⊂ TpM the vertical space at p, and Hp = V ⊥ p ⊂ TpM the horizontal space at p. For any vector Xp ∈ TpM, we will denote X v the vertical component of X. Note that dfp, when restricted to Hp, is a linear isomorphism. For any X ∈ Γ∞(TM), its horizontal lift is the horizontal vector field X defined by dfp(Xp) = Xf(p) . The submersion f is said to be a Riemannian submersion if for any p ∈ M, dfp : Hp → Tf(p)M is a linear isometry. (a) Prove: For any X, Y ∈ Γ(TM), ∇X Y = ∇XY + 1 2 [X, Y ] v . (b) Prove: For any X, Y, Z, W ∈ Γ(TM), Rm(X, Y ,Z, W) =Rm(X, Y, Z, W) − 1 4 ⟨[X,Z] v , [Y , W] v ⟩ + 1 4 ⟨[Y ,Z] v , [X, W] v − 1 2 ⟨[Z, W] v , [X, Y ] v ⟩. (c) Let Π be a plane spanned by orthogonal unit vectors Xp, Yp ∈ TpM and Π the plane spanned by Xp, Y p. Prove: K(Π) = K(Π) + 3 4 |[Xp, Y p] v | 2 . 12. [Gauss-Bonnet-Chern in dimension 4] The famous Gauss-Bonnet-Chern formula says that if (M, g) is an orientable closed Riemannian manifold of dimension m = 2k, then 1 2mπm/2(m/2)! Z M Ω = χ(M), where χ(M) is the Euler characteristic of M, and Ω is the following m-form Ω = X σ∈Sm (−1)σΩ σ(1) σ(2) ∧ Ω σ(3) σ(4) ∧ · · · ∧ Ω σ(m−1) σ(m)
10PROBLEM SET 2: CURVATURESDUE:APRIL28,2024where Sm is the permutation group of (1,::,m), and 2, is the curvature two form associatedto any orthonormal basis.(a) For m = 2, deduce JM Kda = x(M).(b) [*jFor m = 4, deduce - Jr(IRm)2 - 4|Re? + S)da = x(M).(c) Prove: If (M,g) is an Einstein manifold of dimension 4, then x(M) ≥ 0, and the equalityholds if and only if (M, g) is flat.(d) Prove: If (M,g) is a locally conformally flat manifold of dimension 4, then11(-E)2+s2)d.x(M)= 116元2JM+12°(e) [*J Prove: one can find an orthonormal frame so that R1213 = R1214 = R1223 = R1224 =R1323= R1314 = 0.(f) Prove: If M is a compact orientable Riemannian manifold of dimension 4 which admits ametric of positive sectional curvature, then x(M) > 0
PROBLEM SET 2: CURVATURES DUE: APRIL 28, 2024 5 where Sm is the permutation group of (1, · · · , m), and Ωi j is the curvature two form associated to any orthonormal basis. (a) For m = 2, deduce 1 2π R M Kdx = χ(M). (b) [*]For m = 4, deduce 1 32π2 R M(|Rm| 2 − 4|Rc| 2 + S 2 )dx = χ(M). (c) Prove: If (M, g) is an Einstein manifold of dimension 4, then χ(M) ≥ 0, and the equality holds if and only if (M, g) is flat. (d) Prove: If (M, g) is a locally conformally flat manifold of dimension 4, then χ(M) = 1 16π 2 Z M (−|E| 2 + 1 12 S 2 )dx. (e) [*] Prove: one can find an orthonormal frame so that R1213 = R1214 = R1223 = R1224 = R1323 = R1314 = 0. (f) Prove: If M is a compact orientable Riemannian manifold of dimension 4 which admits a metric of positive sectional curvature, then χ(M) > 0