《黎曼几何》课程教学资源（英文讲义）Lec24 THE GLOBAL HESSIAN AND TOPONOGOV COMPARISON

LECTURE24:THE GLOBALHESSIANANDTOPONOGOVCOMPARISONLast time we proved various local comparison theorems that holds away fromcut locus. Today weturn to global comparisonthatholds onM.1.THEHESSIANCOMPARISONTHEOREM:GLOBALFORMI Hessian of distance for Mm.Let Mm be a space form, i.e. a complete connected Riemannian manifold withconstant sectional curvature k. Let : [o,ll → Mm be a normal geodesic in Mmfrom p to q Cut(p) U (p), then for any X, e ((i)+, the Jacobi field V along with V(O) = 0 and V(U) = X issnk(t)X,(t)V(t) =snk(1)where Xq(t) is the parallel vector field along with X(l) = Xq, and we used( sin(Vkt)k>0cos(Vkt),k>0,Vksnk(t) =k=0and cnk(t) = sns(t) =k = 0,1N1sinh(V-kt)cosh(V-kt), k < 0.k<0V-kAs a result, foranyY eT,M,cnk(l)(V2dp)a(Xq, Ya) = (Vs()V,Ya) =(Xq, Yq)snk(l)So the Hessian of dp at the point g, with respect to an orthonormal basis ei() =(),e2(U), ..-,em(), is00/0...cnx()00sna()(V2dp)g::.-cnk()00snx(0)The Hessian matrix is almost a constant matrix, with the only exception thatthere is a zero for the top-left entry. We will carefully choose a function f so thatV2(f odp) is a constant matrix. For this purpose we calculate, for any Xq, Yq e TqM,V?(fodp)(Xg,Ya)=(VxV(fodp),Yq)=(Vx,(f(dp)Vdp),Ya)=f"(dp)(VxVdp,Ya)+f"(dp)(Vdp,Xa)(Vdp,Ya)=f'(dp)2dp(Xg,Y)+f"(dp)<(0),Xa)((l),Yg)

LECTURE 24: THE GLOBAL HESSIAN AND TOPONOGOV COMPARISON Last time we proved various local comparison theorems that holds away from cut locus. Today we turn to global comparison that holds on M. 1. The Hessian comparison theorem: global form ¶ Hessian of distance for Mm κ . Let Mm κ be a space form, i.e. a complete connected Riemannian manifold with constant sectional curvature κ. Let γ : [0, l] → Mm κ be a normal geodesic in Mm κ from p to q ̸∈ Cut(p) ∪ {p}, then for any Xq ∈ ( ˙γ(l))⊥, the Jacobi field V along γ with V (0) = 0 and V (l) = Xq is V (t) = snk(t) snk(l) Xq(t), where Xq(t) is the parallel vector field along γ with Xq(l) = Xq, and we used snk(t) =    sin(√ kt) √ k , k > 0 t, k = 0 sinh(√ −kt) √ −k , k < 0 and cnk(t) = sn′ k (t) =    cos(√ kt), k > 0, 1, k = 0, cosh(√ −kt), k < 0. As a result, for any Yq ∈ TqM, (∇2 dp)q(Xq, Yq) = ⟨∇γ˙ (l)V, Yq⟩ = cnk(l) snk(l) ⟨Xq, Yq⟩. So the Hessian of dp at the point q, with respect to an orthonormal basis e1(l) = γ˙(l), e2(l), · · · , em(l), is (∇2 dp)q =   0 0 · · · 0 0 cnκ(l) snκ(l) · · · 0 . . . . . . . . . . . . 0 0 · · · cnκ(l) snκ(l)   . The Hessian matrix is almost a constant matrix, with the only exception that there is a zero for the top-left entry. We will carefully choose a function f so that ∇2 (f ◦dp) is a constant matrix. For this purpose we calculate, for any Xq, Yq ∈ TqM, ∇2 (f ◦dp)(Xq,Yq)=⟨∇Xq∇(f ◦dp), Yq⟩=⟨∇Xq (f ′ (dp)∇dp), Yq⟩ =f ′ (dp)⟨∇Xq∇dp, Yq⟩+f ′′(dp)⟨∇dp,Xq⟩⟨∇dp,Yq⟩ =f ′ (dp)∇2 dp(Xq,Yq)+f ′′(dp)⟨γ˙(l),Xq⟩⟨γ˙(l),Yq⟩, 1

2LECTURE24:THEGLOBALHESSIANANDTOPONOGOV COMPARISONwhere is the minimizing geodesic from p to q. As a result, for ?(f o dp) to be aconstant matrix under the given basis, we should choose f so that'(t) cnx(t)= f"(t).snr(t)So the simplest solution is to take f to satisfy f'(t) = snk(t), i.e. take f to be=cos(V/kr)if k> 0,Amd,(r) =if k= 0,snr(t)dt =-cosh(V-kr)if k<0.Itfollows(md o dp)g = cns(1)Id.I The cosine law in Mm.It is easy to check cnx(r) = 1 - k md(r). So if we let p(t) = md o dp o (t),where is a geodesic in Mm away from Cut(p) U (p), thenp"(t) = cnk 0 d, o(t) = 1 -kmd(d, o(t)) = 1 - kp(t).As an application, we may derive the cosine law in Mm.Consider a geodesictriangle △ABC in Mm with side lengths a,b,c and angles A, B,C, where for k > 0we assume a,b,c < /Vk. Let 1 : [0,a] → Mm be the normal geodesic with(0) = B, (a) = C and let 2 : [0, b] -→ Mm be the normal geodesic with 2(0) =C, 2(b) = A.Now take p = B and = 2, i.e. consider (t) = md, o dg o 2(t). Thenp(0) = md,(a),p(b) =md,(c)and(0) = sn(a)(%1(a), 2(0)) = -sn(a) cos C.So if k = 0, we getα212p(t)=号-acos Ct+22and thus p(b) = md (c) becomes c2 = a? + b? - 2ab cos CFor K≠o,weget(t) = I+csn:()+ Ccn(t),The initial conditions (0) = mds(a) and '(O) = -sns(a)cosC implies ci Icnk(a). So the equation p(b) = mdx(c) becomes the fol--snk(a)cosC,c2lowingcosinelawinMm,cn(c)=cn(a)cnk(b)+ksnk(a)snk(b)cosC.Asadirectcorollary,wegetProposition 1.1. In Mm, take an angle α with side lengths l and l2 fired. Letf(a) be the distance between the end points. Then f(a) is increasing in α

2 LECTURE 24: THE GLOBAL HESSIAN AND TOPONOGOV COMPARISON where γ is the minimizing geodesic from p to q. As a result, for ∇2 (f ◦ dp) to be a constant matrix under the given basis, we should choose f so that f ′ (t) cnκ(t) snκ(t) = f ′′(t). So the simplest solution is to take f to satisfy f ′ (t) = snk(t), i.e. take f to be mdκ(r) = Z r 0 snκ(t)dt =    1−cos(√ κr) κ , if κ > 0, r 2 2 , if κ = 0, 1−cosh(√ −κr) κ , if k < 0. It follows ∇2 (mdκ ◦ dp)q = cnk(l)Id. ¶ The cosine law in Mm κ . It is easy to check cnκ(r) = 1 − κ mdκ(r). So if we let φ(t) = mdκ ◦ dp ◦ γ(t), where γ is a geodesic in Mm κ away from Cut(p) ∪ {p}, then φ ′′(t) = cnκ ◦ dp ◦ γ(t) = 1 − κ mdκ(dp ◦ γ(t)) = 1 − κφ(t). As an application, we may derive the cosine law in Mm κ . Consider a geodesic triangle △ABC in Mm κ with side lengths a, b, c and angles A, B, C, where for κ > 0 we assume a, b, c < π/√ κ. Let γ1 : [0, a] → Mm κ be the normal geodesic with γ1(0) = B, γ1(a) = C and let γ2 : [0, b] → Mm κ be the normal geodesic with γ2(0) = C, γ2(b) = A. Now take p = B and γ = γ2, i.e. consider φ(t) = mdκ ◦ dB ◦ γ2(t). Then φ(0) = mdκ(a), φ(b) = mdκ(c) and φ ′ (0) = snκ(a)⟨γ˙ 1(a), γ˙ 2(0)⟩ = −snκ(a) cos C. So if κ = 0, we get φ(t) = a 2 2 − a cos Ct + 1 2 t 2 and thus φ(b) = mdκ(c) becomes c 2 = a 2 + b 2 − 2ab cos C. For κ ̸= 0, we get φ(t) = 1 κ + c1snκ(t) + c2cnκ(t). The initial conditions φ(0) = mdκ(a) and φ ′ (0) = −snκ(a) cos C implies c1 = −snκ(a) cos C, c2 = − 1 κ cnκ(a). So the equation φ(b) = mdκ(c) becomes the fol￾lowing cosine law in Mm κ , cnκ(c) = cnκ(a)cnκ(b) + κsnκ(a)snκ(b) cos C. As a direct corollary, we get Proposition 1.1. In Mm κ , take an angle α with side lengths l1 and l2 fixed. Let f(α) be the distance between the end points. Then f(α) is increasing in α

LECTURE24:THEGLOBALHESSIANANDTOPONOGOV COMPARISON3I Compare in the barrier sense.It turns out that the Hessian and Laplacian comparison theorems holds globallyon the whole of M in several weak sense: in thebarrier sense,in theviscosity senseand in the distribution sense'. Here we only discuss the first one, since the conditionis often easier to check.The notion of barrier sense wasfirst introduced by Calabiin1958Definition 1.2. Let f be a continuous function defined on (M,g).(1) If g e C2(U) is defined in a neighborhood U of p, andf(p) = g(p),and f(q)≤g(q), Vq EUthen we call g an upper barrier function of f at p.(2) If for any e > o, there is an upper barrier function ge of f at p, such thatAge(p) ≤c+e, then we say△f(p)≤c in the barrier sense.(3) If for any normal geodesic with o(0) = q, one has (f o o)"(0) ≤ c in thebarrier sense, then we say(2f)(q)≤c.Idinthebarriersense.Erample. Note that by taking M = (a, b), we get a definition of “f"(to) ≤ c inthe barrier sense" for continuous function f : (a,b) → R. For example, considerf(r) = -rl. Then g = 0 is a upper barrier function of f at = 0. As a result,f"(O)≤0 in the barrier sense.As observed by Calabi, one can easily construct upper barrier functions for thedistance function:Erample. If is a minimizing geodesic from p to q, then for 0 < n < d(p, q), thefunction rn(r)=n +d(r,(n)) is an upper barrier function for d, at qThe proof of the following lemma will be left as an exercise.Lemma 1.3. Let f:(a,b)-→R be continuous.(1) If f is C2, then f"(to) ≤c in the barrier sense if and only if f"(to) ≤c inthe usual sense.(2) If f takes its minimum at to, and f"(to)< c in the barrier sense, then c≥0.(3) If f"<O in the barrier sense, then f is concave.We also mention thefollowingHopf strong maximal principle withoutproof.Theorem 1.4 (Hopf-Calabi strong maximum principle). Let 2 M be a con-nected open set. Suppose f 0 in M in the barrier sense, and f has an interiorminimum, then f is constant on 2.'It can be proven that if f ≤ g holds in the barrier sense, then it also holds in the viscositysense and in the distribution sense

LECTURE 24: THE GLOBAL HESSIAN AND TOPONOGOV COMPARISON 3 ¶ Compare in the barrier sense. It turns out that the Hessian and Laplacian comparison theorems holds globally on the whole of M in several weak sense: in the barrier sense, in the viscosity sense and in the distribution sense1 . Here we only discuss the first one, since the condition is often easier to check. The notion of barrier sense was first introduced by Calabi in 1958: Definition 1.2. Let f be a continuous function defined on (M, g). (1) If g ∈ C 2 (U) is defined in a neighborhood U of p, and f(p) = g(p), and f(q) ≤ g(q), ∀q ∈ U, then we call g an upper barrier function of f at p. (2) If for any ε > 0, there is an upper barrier function gε of f at p, such that ∆gε(p) ≤ c + ε, then we say ∆f(p) ≤ c in the barrier sense. (3) If for any normal geodesic σ with σ(0) = q, one has (f ◦ σ) ′′(0) ≤ c in the barrier sense, then we say (∇2 f)(q) ≤ c · Id in the barrier sense. Example. Note that by taking M = (a, b), we get a definition of “f ′′(t0) ≤ c in the barrier sense” for continuous function f : (a, b) → R. For example, consider f(x) = −|x|. Then g = 0 is a upper barrier function of f at x = 0. As a result, f ′′(0) ≤ 0 in the barrier sense. As observed by Calabi, one can easily construct upper barrier functions for the distance function: Example. If γ is a minimizing geodesic from p to q, then for 0 < η < d(p, q), the function rη(x) = η + d(x, γ(η)) is an upper barrier function for dp at q. The proof of the following lemma will be left as an exercise. Lemma 1.3. Let f : (a, b) → R be continuous. (1) If f is C 2 , then f ′′(t0) ≤ c in the barrier sense if and only if f ′′(t0) ≤ c in the usual sense. (2) If f takes its minimum at t0, and f ′′(t0) ≤ c in the barrier sense, then c ≥ 0. (3) If f ′′ ≤ 0 in the barrier sense, then f is concave. We also mention the following Hopf strong maximal principle without proof. Theorem 1.4 (Hopf-Calabi strong maximum principle). Let Ω ⊂ M be a con￾nected open set. Suppose ∆f ≤ 0 in M in the barrier sense, and f has an interior minimum, then f is constant on Ω. 1 It can be proven that if ∆f ≤ g holds in the barrier sense, then it also holds in the viscosity sense and in the distribution sense

4LECTURE24:THEGLOBALHESSIANANDTOPONOGOV COMPARISONI The global Hessian comparison theorem.Now weare ready to state and proveTheorem 1.5 (The global Hessian comparison theorem). Let (M, g) be a Riemann-ian manifold with sectional curvature K ≥ k. Then for any p E M,V(mds o dp) ≤cnk o dp Id in the barrier sense.Proof.According to the(local)Hessiancomparison theorem that weproved lasttime, the theorem holds at smooth points q of dp.We first prove the conclusion atthe point p:Forany normal geodesic with o(O)=p, we havemdko doo(t) =md(It)) =mds(t)Thus (mdo d,o )"(0)= md(0) = cn(0).It remains to prove the conclusion for a non-smooth point q + p of dp. So we let : [o,i] -→ M be a minimizing normal geodesic from p to q, and let be a normalgeodesic with o(0) = q. Note that by Bonnet-Myers theorem, I ≤ if k > 0.For 0 < n < d(p, q) small, the functionrn(r)=n +d(r,(n))is an upper barrier function of dp at q.Apply Hessian comparison to rn we get(md,orno0)"(0)=mk,()(rn 0 0)"(0) + md'(1)<(1), 0(0)?=sns(1)(V2dg(m)g(o(0), 0(0)) + cn(1)<%(0),(0)2≤sn() m(二%)10-(0)P + cn:()(), (0)2snr(l-nsn()en(n)-cn()sn()o+(0)P+cn()(lα+(0)P+((),(0))snr(l-n)sn (m) /0+(0)P + cn ().snr(l -n)So if ≤ 0 or if l < when > 0, given any > 0, for n is small enough, we have(md 0 rn o)"(0) ≤ cnk(l) +e,whichimplies?(md o d) ≤ cnk o d,in the barrier sense.For k > 0 and I = , we will prove (M,g) is isomorphic to Sm, in which case we口may take α such that (O) = (l), and the desired conclusion follows

4 LECTURE 24: THE GLOBAL HESSIAN AND TOPONOGOV COMPARISON ¶ The global Hessian comparison theorem. Now we are ready to state and prove Theorem 1.5 (The global Hessian comparison theorem). Let (M, g) be a Riemann￾ian manifold with sectional curvature K ≥ κ. Then for any p ∈ M, ∇2 (mdκ ◦ dp) ≤ cnk ◦ dp · Id in the barrier sense. Proof. According to the (local) Hessian comparison theorem that we proved last time, the theorem holds at smooth points q of dp. We first prove the conclusion at the point p: For any normal geodesic σ with σ(0) = p, we have mdk ◦ dp ◦ σ(t) = mdk(|t|) = mdk(t). Thus (mdk ◦ dp ◦ σ) ′′(0) = md′′ k (0) = cnk(0). It remains to prove the conclusion for a non-smooth point q ̸= p of dp. So we let γ : [0, l] → M be a minimizing normal geodesic from p to q, and let σ be a normal geodesic with σ(0) = q. Note that by Bonnet-Myers theorem, l ≤ √π κ if κ > 0. For 0 < η < d(p, q) small, the function rη(x) = η + d(x, γ(η)) is an upper barrier function of dp at q. Apply Hessian comparison to rη we get (mdκ◦rη ◦σ) ′′(0)=mk′ κ (l)(rη ◦ σ) ′′(0) + md′′ κ (l)⟨γ˙(l), σ˙(0)⟩ 2 =snκ(l)(∇2 dγ(η))q( ˙σ(0), σ˙(0)) + cnκ(l)⟨γ˙(l), σ˙(0)⟩ 2 ≤snκ(l) cnκ(l − η) snκ(l − η) |σ˙ ⊥(0)| 2 + cnκ(l)⟨γ˙(l), σ˙(0)⟩ 2 = snκ(l)cnκ(l−η)−cnκ(l)snκ(l−η) snκ(l−η) |σ˙ ⊥(0)| 2+cnκ(l)(|σ˙ ⊥(0)| 2+⟨γ˙(l), σ˙(0)⟩ 2 ) = snκ(η) snκ(l − η) |σ˙ ⊥(0)| 2 + cnκ(l). So if κ ≤ 0 or if l < √π κ when κ > 0, given any ε > 0, for η is small enough, we have (mdκ ◦ rη ◦ σ) ′′(0) ≤ cnκ(l) + ε, which implies ∇2 (mdκ ◦ dp) ≤ cnk ◦ dp in the barrier sense. For κ > 0 and l = √π κ , we will prove (M, g) is isomorphic to S m κ , in which case we may take σ such that ˙σ(0) = ˙γ(l), and the desired conclusion follows. □

LECTURE24:THEGLOBALHESSIANANDTOPONOGOV COMPARISON52.THE TOPONOGOV COMPARISON THEOREMThe purpose of this section is to prove a very useful global comparison theorem,due to Toponogov in 1959. It quantifies the assertion (c.f. PSet 4) that a pair ofgeodesics emanating from a point p spread apart more slowly in a region of highcurvature than they would in a region of low curvature.I Geodesic triangles and hinges.Definition 2.1. Let (M, g) be complete.(1) A geodesic triangle △ABC consists of three points A, B,C in M (which arecalled the ertices) and three minimizing normal geodesics (which are calledthe sides)AB,Bc,CA joining each two of them.Ifonly two sides, say AB and Ac, are minimizing, whilethethird is a normalgeodesic [which need not be minimizing] that satisfies the triangle inequalityL(BC) ≤ L(AB) + L(AC),then we will call ABC a generalized geodesic triangle(2) A geodesic hinge /BAC consists of a point A in M (which is again calledthe verter) and two minimizing normal geodesics AB, Ac (called the sides)emanating from A, with end points B and C in M.If one side is minimizing, while the other side is a normal geodesic[which neednot be minimizingl, we call ZBAC a generalized geodesic hinge.In what follows when we say hinge or triangle, we always mean generalizedgeodesic hinge or generalized geodesic triangle.Remark. In the definition of generalized geodesic hinge, we required that at leastone curve is minimizing.Otherwise the Toponogov comparison theorem below mayfail: If one take two geodesics of length in Mk+e, then the other endpoints of thecomparing hinge in M,will meet.Lemma2.2. Let (M,g)be a completeRiemannian manifold of dimension m whosesectional curvature K≥ k. Then(1)Let/BAC be a generalized geodesic hinge inM.If k>O we furtherassumethat all the sides of LABC have lengths no more than Then there isa generalized geodesic hinge BAC in Mm with same angle and the samecorresponding side lengths. [We will call it a comparing hinge.](2)Let ABC be a generalized geodesic triangle in M. If k > 0 we furtherassume that all the sides of ABC have lengths no more than k-Thenthere is a triangle ABC in Mm whose corresponding sides have the samelength asABC.[We will call it a comparing triangle.]

LECTURE 24: THE GLOBAL HESSIAN AND TOPONOGOV COMPARISON 5 2. The Toponogov Comparison Theorem The purpose of this section is to prove a very useful global comparison theorem, due to Toponogov in 1959. It quantifies the assertion (c.f. PSet 4) that a pair of geodesics emanating from a point p spread apart more slowly in a region of high curvature than they would in a region of low curvature. ¶ Geodesic triangles and hinges. Definition 2.1. Let (M, g) be complete. (1) A geodesic triangle △ABC consists of three points A, B, C in M (which are called the vertices) and three minimizing normal geodesics (which are called the sides) γAB, γBC, γCA joining each two of them. If only two sides, say γAB and γAC, are minimizing, while the third is a normal geodesic [which need not be minimizing] that satisfies the triangle inequality L(γBC) ≤ L(γAB) + L(γAC), then we will call △ABC a generalized geodesic triangle. (2) A geodesic hinge ∠BAC consists of a point A in M (which is again called the vertex ) and two minimizing normal geodesics γAB, γAC (called the sides) emanating from A, with end points B and C in M. If one side is minimizing, while the other side is a normal geodesic[which need not be minimizing], we call ∠BAC a generalized geodesic hinge. In what follows when we say hinge or triangle, we always mean generalized geodesic hinge or generalized geodesic triangle. Remark. In the definition of generalized geodesic hinge, we required that at least one curve is minimizing. Otherwise the Toponogov comparison theorem below may fail: If one take two geodesics of length √π κ in Mκ+ε, then the other endpoints of the comparing hinge in Mκ will meet. Lemma 2.2. Let (M, g) be a complete Riemannian manifold of dimension m whose sectional curvature K ≥ k. Then (1) Let ∠BAC be a generalized geodesic hinge in M. If κ > 0 we further assume that all the sides of ∠ABC have lengths no more than √π κ . Then there is a generalized geodesic hinge ∠BeAeCe in Mm k with same angle and the same corresponding side lengths. [We will call it a comparing hinge.] (2) Let △ABC be a generalized geodesic triangle in M. If κ > 0 we further assume that all the sides of △ABC have lengths no more than √π κ . Then there is a triangle △AeBeCe in Mm k whose corresponding sides have the same length as △ABC. [We will call it a comparing triangle.]