《黎曼几何》课程教学资源（英文讲义）PSet3 GEODESICS

PROBLEMSET3:GEODESICSDUE:MAY16,2024Instruction:Please work on at least Five problems of your interest, and in eachproblem you aresupposed to work on at least TWO sub-problems.1. [Examples of geodesics](a) Consider (S,ground).(i) Show that the“equator"(0) = (cos , sin , 0) is a geodesic.(i) Show that the “meridian"(z) = (V1-z2, o, z) is not a geodesic, then find a correctparametrization so that it becomes a geodesic(b) Describe the relations between the geodesics on the product Riemannian manifold (Mi x×Mk,g1X.×gk)withthegeodesicson(Mi,gi)'s(c) Consider the hyperbolic space H? = (z, y) I y > 0) endowed with the hyperbolic metricg=(dadr+dydy).(i) Prove: The maps (r,y) =(-r,y) and o(r,y) =(p,+) are isometries.(i) Deduce that the upper unit semicircle and the positive y-axis are geodesics.(i) Prove: The maps (r,y) -→ (r + a,y) and (r,y) -→ (br,by) are isometries for any aand any b > 0.(iv) Figure out all geodesics on H?. [Write down the correct parametrization.](v) Is (H2,g) complete?(d) Describe geodesics on Lie groups (endowed with a bi-invariant metric).2.[Torsion freeconnectionv.s.geodesics]Let M be a smooth manifold and let v,i be two linear connections on M. For any vectorfields X,Y ET(TM), defineA(X,Y)=VxY- VY.(a) Prove: A is a (0,2)-tensor.(b) Prove:(i) and ' have the same torsion if and only ifA(X,Y) = A(Y,X), VX,Y ET(TM),(i) and V' have the same geodesics if and only ifA(X,Y) =-A(Y,X), VX,Y EF(TM)(ii) For any linear connection V, there is a unique torsion free connection ' with thesame geodesics.(c) Suppose g is a Riemannian metric on M, and suppose V is a g-compatible linear connection.(i) Prove: ' is g-compatible if and only ifVX,Y,Z EF(TM).g(A(X,Y),Z) =-g(Y,A(X,Z)),(i) Give another proof of the uniqueness part of the fundamental theorem in Riemanniangeometry.1

PROBLEM SET 3: GEODESICS DUE: MAY 16, 2024 Instruction: Please work on at least Five problems of your interest, and in each problem you are supposed to work on at least TWO sub-problems. 1. [Examples of geodesics] (a) Consider (S 2 , ground). (i) Show that the “equator” γ(θ) = (cos θ,sin θ, 0) is a geodesic. (ii) Show that the “meridian” γ(z) = (√ 1 − z 2, 0, z) is not a geodesic, then find a correct parametrization so that it becomes a geodesic. (b) Describe the relations between the geodesics on the product Riemannian manifold (M1 × · · · × Mk, g1 × · · · × gk) with the geodesics on (Mi , gi)’s. (c) Consider the hyperbolic space H2 = {(x, y) | y > 0} endowed with the hyperbolic metric g = 1 y 2 (dx ⊗ dx + dy ⊗ dy). (i) Prove: The maps φ(x, y) = (−x, y) and ϕ(x, y) = ( x x2+y 2 , y x2+y 2 ) are isometries. (ii) Deduce that the upper unit semicircle and the positive y-axis are geodesics. (iii) Prove: The maps (x, y) 7→ (x + a, y) and (x, y) 7→ (bx, by) are isometries for any a and any b > 0. (iv) Figure out all geodesics on H2 . [Write down the correct parametrization.] (v) Is (H2 , g) complete? (d) Describe geodesics on Lie groups (endowed with a bi-invariant metric). 2. [Torsion free connection v.s. geodesics] Let M be a smooth manifold and let ∇, ∇′ be two linear connections on M. For any vector fields X, Y ∈ Γ(TM), define A(X, Y ) = ∇XY − ∇′ XY. (a) Prove: A is a (0,2)-tensor. (b) Prove: (i) ∇ and ∇′ have the same torsion if and only if A(X, Y ) = A(Y, X), ∀X, Y ∈ Γ ∞(TM). (ii) ∇ and ∇′ have the same geodesics if and only if A(X, Y ) = −A(Y, X), ∀X, Y ∈ Γ ∞(TM). (iii) For any linear connection ∇, there is a unique torsion free connection ∇′ with the same geodesics. (c) Suppose g is a Riemannian metric on M, and suppose ∇ is a g-compatible linear connection. (i) Prove: ∇′ is g-compatible if and only if g(A(X, Y ), Z) = −g(Y, A(X, Z)), ∀X, Y, Z ∈ Γ ∞(TM). (ii) Give another proof of the ✿✿✿✿✿✿✿✿✿✿ uniqueness✿✿✿✿ part of the fundamental theorem in Riemannian geometry. 1

2PROBLEM SET 3:GEODESICSDUE:MAY16,20243. [Examples of the exponential maps]For each of the following, write down the exponential mapexpp:(a) M = Rm with standard go, and p = 0 the origin.(b) M =H? (the Poincare upper plane), and p = (0, 1).(c) M = s? with the standard round metric, and p = the north pole(d) M = s1 × R with the standard product metric, and p = ((1, 0), 0)(e) M be the standard paraboloid defined by z=r2+y?in R3, and p = (0,0,0)(f) M = G (equipped with bi-invariant Riemannian metric) a matrix Lie group, and p = I.4. [Riemannian geometry of surfaces of revolution]Consider the surfaceof revolutionS(r,0) = (y(r)coso,y(r) sin0, z(r)),a 0 so that for any p E M and X, E T,M, there isa normal geodesic : [0, eo] → M with (0) = p, (O) = X,". Prove: (M,g) is complete.(b) Suppose (M, g) is complete and g is another Riemannian metric on M with g ≥ g. Prove:(M,g) is complete.(c) Let (M, g) be a Riemannian manifold and there exists a proper[jie. the pre-images of compactsets are compact) Lipschitz function f : M -→ R. Prove: (M,g) is complete.(d) Prove: (M,g) is complete if and only if for some fixed r, the closed geodesic ball B,(p) iscompactfor anyp EM.(e)A Riemannian homogeneous space is a Riemannian manifold such that its isometry groupacts transitively on it, i.e. for any p,q e M there exists an isometry of (M,g) so that(p) = q. Prove: Any Riemannian homogeneous space is complete

2 PROBLEM SET 3: GEODESICS DUE: MAY 16, 2024 3. [Examples of the exponential maps] For each of the following, write down the exponential map expp : (a) M = R m with standard g0, and p = 0 the origin. (b) M = H2 (the Poincare upper plane), and p = (0, 1). (c) M = S 2 with the standard round metric, and p = the north pole. (d) M = S 1 × R with the standard product metric, and p = ((1, 0), 0). (e) M be the standard paraboloid defined by z = x 2 + y 2 in R 3 , and p = (0, 0, 0). (f) M = G (equipped with bi-invariant Riemannian metric) a matrix Lie group, and p = I. 4. [Riemannian geometry of surfaces of revolution] Consider the surface of revolution S(x, θ) = (y(x) cos θ, y(x) sin θ, z(x)), a 0 so that for any p ∈ M and Xp ∈ TpM, there is a normal geodesic γ : [0, ε0] → M with γ(0) = p, γ˙(0) = Xp”. Prove: (M, g) is complete. (b) Suppose (M, g) is complete and ˜g is another Riemannian metric on M with ˜g ≥ g. Prove: (M, g˜) is complete. (c) Let (M, g) be a Riemannian manifold and there exists a proper[i.e. the pre-images of compact sets are compact] Lipschitz function f : M → R. Prove: (M, g) is complete. (d) Prove: (M, g) is complete if and only if for some fixed r, the closed geodesic ball Br(p) is compact for any p ∈ M. (e) A Riemannian homogeneous space is a Riemannian manifold such that its isometry group acts transitively on it, i.e. for any p, q ∈ M there exists an isometry φ of (M, g) so that φ(p) = q. Prove: Any Riemannian homogeneous space is complete

DUE:MAY16,2024PROBLEMSET3:GEODESICS37.[Completeness of Riemannian metrics II:Existence](a) Let's prove the existence of a complete Riemannian metric on any smooth manifold M. Infact,we shall provethat given anyRiemannian metric g,thereexists a completeRiemannianmetricqthatisconformaltog.WLOG.supposeMisnon-compactand gisincomplete(i) Let r(p) = sup[r|Br(p) is compact). Prove: 0 ):(iv) Define a new metric g' on M by g' = w?g. Denote by B;(p) the closed geodesic ballof radius r around p with respect to the new metric g'. Prove: B() c Bp(“P) forall p.(v) Concludethat (M,g)is complete.(b)Next let's prove that ifeveryRiemannian.metric.on.M is.complete.thenM is.compactEquivalently,wewant to construct a incomplete Riemannian metric on anynon-compactsmooth manifold. Again we shall prove a stronger result: for any (M,g) with M non-compact,there is a Riemannian metric gconformal to gwhich is incomplete.WLOG,assume g is complete.(i) Fix p E M and let w be a smooth function on M so that w(g) > dist(p,g). (Theexistence is the same as part (a)(ii) above.) Let g' = e-2wg. Prove: Under g' thenewdistancedist'(p,q)<1 for all q.(ii)Concludethat(M,g)is incomplete.8. [Rays in complete noncompact Riemannian manifolds]Let (M,g)be a complete noncompact Riemannian manifold.. A normal geodesic : [0, oo) -→ M is called a ray if dist((a),(b)) =a-b| for any a,b ≥ 0.. A normal geodesic : (-oo, oo) -→ M is called a geodesic line if dist((a), (b) = [a - b)for any a,beR(a) Prove: From any point p e M there exists a ray so that (O) = p.(b) Wesay M is discontinuous at infinity if thereexists a compact subset K in M so that MKcontains at least two non-compact connected components. Prove: If (M,g) is complete andM is discontinuous at infinity, then M contains a geodesic line(c)ConstructacompletenoncompactRiemannianmanifoldonwhichthereisnogeodesicline.(d) Now let 1,2 be two geodesic rays on M.We say 1 and 2 are asymptotic if there existsC e R so that dist(i(t), 2(t)) < C for all t ≥ 0. This defines an equivalent relation onthe set of geodesic rays. We will denote the set of equivalent classes by M(co), and call itthe sphere at infinity of M. Figure out M(oo) for M - Rn, sl x RI and H?, each equippedwiththestandard Riemannianmetric.9. [Existence of variation]Let V(t) a piecewise smooth vector field along a smooth curve : [a, b] -→ M.(a) Generalize the concept of variation to allow piecewise variation field.(b)Prove:Thereexistsa variation of whosevariationfield isV.(c) Prove: If V(a) = 0, V(b) = O, then the variation could be chosen to be proper

PROBLEM SET 3: GEODESICS DUE: MAY 16, 2024 3 7. [Completeness of Riemannian metrics II: Existence] (a) Let’s prove ✿✿✿ the✿✿✿✿✿✿✿✿✿✿ existence ✿✿ of✿✿ a✿✿✿✿✿✿✿✿✿ complete✿✿✿✿✿✿✿✿✿✿✿✿ Riemannian✿✿✿✿✿✿✿ metric✿✿✿ on✿✿✿✿ any✿✿✿✿✿✿✿✿ smooth✿✿✿✿✿✿✿✿✿ manifold✿✿✿M. In fact, we shall prove that given any Riemannian metric g, there exists a complete Riemannian metric g ′ that is conformal to g. WLOG, suppose M is non-compact and g is incomplete. (i) Let r(p) = sup{r|Br(p) is compact}. Prove: 0 1 r(p) . (iv) Define a new metric g ′ on M by g ′ = ω 2 g. Denote by B′ r (p) the closed geodesic ball of radius r around p with respect to the new metric g ′ . Prove: B′ p ( 1 3 ) ⊂ Bp( r(p) 2 ) for all p. (v) Conclude that (M, g′ ) is complete. (b) Next let’s prove that ✿ if✿✿✿✿✿✿ every✿✿✿✿✿✿✿✿✿✿✿✿✿ Riemannian ✿✿✿✿✿✿✿ metric ✿✿✿ on✿✿✿M✿✿ is✿✿✿✿✿✿✿✿✿✿ complete,✿✿✿✿✿✿ then ✿✿✿ M ✿✿ is✿✿✿✿✿✿✿✿✿ compact. Equivalently, we want to construct a incomplete Riemannian metric on any non-compact smooth manifold. Again we shall prove a stronger result: for any (M, g) with M non￾compact, there is a Riemannian metric g ′ conformal to g which is incomplete. WLOG, assume g is complete. (i) Fix p ∈ M and let ω be a smooth function on M so that ω(q) > dist(p, q). (The existence is the same as part (a)(iii) above.) Let g ′ = e −2ω g. Prove: Under g ′ the new distance dist′ (p, q) < 1 for all q. (ii) Conclude that (M, g′ ) is incomplete. 8. [Rays in complete noncompact Riemannian manifolds] Let (M, g) be a complete noncompact Riemannian manifold. • A normal geodesic γ : [0, ∞) → M is called a ray if dist(γ(a), γ(b)) = |a−b| for any a, b ≥ 0. • A normal geodesic γ : (−∞, ∞) → M is called a geodesic line if dist(γ(a), γ(b)) = |a − b| for any a, b ∈ R. (a) Prove: From any point p ∈ M there exists a ray so that γ(0) = p. (b) We say M is discontinuous at infinity if there exists a compact subset K in M so that M \K contains at least two non-compact connected components. Prove: If (M, g) is complete and M is discontinuous at infinity, then M contains a geodesic line. (c) Construct a complete noncompact Riemannian manifold on which there is no geodesic line. (d) Now let γ1, γ2 be two geodesic rays on M. We say γ1 and γ2 are asymptotic if there exists C ∈ R so that dist(γ1(t), γ2(t)) < C for all t ≥ 0. This defines an equivalent relation on the set of geodesic rays. We will denote the set of equivalent classes by M(∞), and call it the sphere at infinity of M. Figure out M(∞) for M = R n , S1 × R 1 and H2 , each equipped with the standard Riemannian metric. 9. [Existence of variation] Let V (t) a piecewise smooth vector field along a smooth curve γ : [a, b] → M. (a) Generalize the concept of variation to allow piecewise variation field. (b) Prove: There exists a variation of γ whose variation field is V . (c) Prove: If V (a) = 0, V (b) = 0, then the variation could be chosen to be proper

4PROBLEM SET 3:GEODESICSDUE:MAY16,202410. [Second variation of length]Let:[a,b]t→M beasmooth curveandf : [a,b] ×(-,e)u× (-5,0)→Mbe a smooth variation of with two parameters v, w. Denote %,w(t) = f(t, u, w) and0df(t, , w) = dfte,m)(%), fe(, )= d(t)(%),fa(t, )=d()(%as usual and leto0V(t) = df(.0.0)(%)W(t) = df(c.0.0(ube the corresponding variation fields of the two parameter directions.(a) Prove:82(Va/atfuVa/atfu)+(R(t,fu)ft,fe)+(Va/ata/awfu,ft)L(w,w) =wooof. )(/of a(b) Let be a normal geodesic, provea2L(%,u) =((V,V,V,W)+(R(%, W)), V)-(%, V)(%, W))dt+(Vwfu,)1awou(00)(c) Let V+, W+ be the orthogonal component of V,W with respect to , i.e.V+=V-(V,), W+=W-(W,)Show that82L(,w)=/ (V,Vl, V,W-)+(R(%,W+)),Vl)dt+(Vwfu,)1ao(0.0)

4 PROBLEM SET 3: GEODESICS DUE: MAY 16, 2024 10. [Second variation of length] Let γ : [a, b]t → M be a smooth curve and f : [a, b] × (−ε, ε)v × (−δ, δ)w → M be a smooth variation of γ with two parameters v, w. Denote γv,w(t) = f(t, v, w) and ft(t, v, w) = df(t,v,w) ( ∂ ∂t), fv(t, v, w) = df(t,v,w) ( ∂ ∂v ), fw(t, v, w) = df(t,v,w) ( ∂ ∂w) as usual and let V (t) = df(t,0,0)( ∂ ∂v ), W(t) = df(t,0,0)( ∂ ∂w) be the corresponding variation fields of the two parameter directions. (a) Prove: ∂ 2 ∂w∂vL(γv,w) = Z b a 1 |ft |  ⟨∇e ∂/∂tfv, ∇e ∂/∂tfw⟩ + ⟨R(ft , fw)ft , fv⟩ + ⟨∇e ∂/∂t∇e ∂/∂wfv, ft⟩ − 1 |ft | 2 ⟨∇e ∂/∂tfv, ft⟩⟨∇e ∂/∂tfw, ft⟩  dt (b) Let γ be a ✿✿✿✿✿✿ normal geodesic, prove ∂ 2 ∂w∂v     (0,0) L(γv,w)=Z b a ￾ ⟨∇γ˙ V, ∇γ˙ W⟩+⟨R( ˙γ, W) ˙γ, V ⟩−γ˙⟨γ, V ˙ ⟩γ˙⟨γ, W˙ ⟩
dt+⟨∇e W fv, γ˙⟩|b a . (c) Let V ⊥, W⊥ be the orthogonal component of V, W with respect to ˙γ, i.e. V ⊥ = V − ⟨V, γ˙⟩γ, W ˙ ⊥ = W − ⟨W, γ˙⟩γ. ˙ Show that ∂ 2 ∂w∂v     (0,0) L(γv,w)=Z b a ￾ ⟨∇γ˙ V ⊥, ∇γ˙ W⊥⟩+⟨R( ˙γ, W⊥) ˙γ, V ⊥⟩
dt+⟨∇e W fv, γ˙⟩|b a