LECTURE27:THE SPHERETHEOREM1. CRITICAL POINT THEORY OF DISTANCE FUNCTIONSI A glimpse into Morse theory.As we have mentioned in Lecture 20, Morse theory is a basic tool in differentialtopology relates the topology of M to the critical points of a Morse.function on M,and the theory has manyapplications in Riemanniangeometry.On major theme in Morse theory is to study the change of topology of sub-levelsets M. = {r L f(r) < a) as a varies. Two crucial facts in Morse theory areTheorem A (Isotopy lemma).Suppose f E Co(M),f-1([a,bl)is compact, andf-1([a,b])nCrit(f)=0. Then Ma is diffeomorphic [and isa deformation retract] to Mb.Idea of proof.“Push" Me down to Malong trajectories of (which are of con-stant speed and are perpendicular to each level set f = c).The topology is not口changed during this procedure. [See my notes on smooth manifolds for detail]One can showthat on any smooth manifold,therearelots of“good Morsefunctions" [i.e, the critical points are disjoint, non-degenerate and take dfferent values].Theorem B. Suppose f E Co(M), p is a non-degenerate critical point of f,f-1([c-e,c+el) is compact, and f-i([c-e,c+el)nCrit(f)=(p). Then Me+e ishomotopy equivalent to "Mc- with a A-cell attached", where is the inder of p.As a result, one can detect the homotopy type of M from a good Morse function.A useful theorem in differential topology that can be used to produce a sphere isTheorem C (Brown). If M is a compact manifold, M = U,u U2, and Ui, U2 arebothhomeomorphictoRm,then M ishomeomorphictoSm.As a consequence, one hasTheorem D (Reeb). If M is compact, f e Co(M) is a Morse function that hasonly two critical points, then M is homeomorphic to Sm.Proof. The two critical points have to be the maximum/minimum of f., Take aclose to the minimal value of f and b closed to the maximal value of f, so thatboth f-1((-oo,a)) and f-1(b, +oo)) are homeomorphic to Rm. Take b betweenb and the maximal value of f. By Theorem A f-1(-oo,a) is homeomorphic tof-1((-oo,b)). Since M = f-1((-oo,b))U f-1(b, +oo)),by Brown's theorem, M is口homeomorphic to Sm.Note that in this proof we avoided the use of Theorem B.1
LECTURE 27: THE SPHERE THEOREM 1. Critical Point Theory of Distance Functions ¶ A glimpse into Morse theory. As we have mentioned in Lecture 20, Morse theory is a basic tool in differential topology relates the topology of M to the critical points of a Morse ✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿ function on M, and the theory has many applications in Riemannian geometry. On major theme in Morse theory is to study the change of topology of sub-level sets Ma = {x | f(x) ≤ a} as a varies. Two crucial facts in Morse theory are Theorem A (Isotopy lemma). Suppose f ∈ C ∞(M), f −1 ([a, b]) is compact, and f −1 ([a, b])∩Crit(f)=∅. Then Ma is diffeomorphic [and is a deformation retract] to Mb. Idea of proof. “Push” Mb down to Ma along trajectories of ∇f |∇f| (which are of constant speed and are perpendicular to each level set f = c). The topology is not changed during this procedure. [See my notes on smooth manifolds for detail] One can show that on any smooth manifold, there are lots of “good Morse functions” [i.e., the critical points are disjoint, non-degenerate and take different values]. Theorem B. Suppose f ∈ C ∞(M), p is a non-degenerate critical point of f, f −1 ([c − ε, c + ε]) is compact, and f −1 ([c − ε, c + ε]) ∩ Crit(f) = {p}. Then Mc+ε is homotopy equivalent to “Mc−ε with a λ-cell attached”, where λ is the index of p. As a result, one can detect the homotopy type of M from a good Morse function. A useful theorem in differential topology that can be used to produce a sphere is Theorem C (Brown). If M is a compact manifold, M = U1 ∪ U2, and U1, U2 are both homeomorphic to R m, then M is homeomorphic to S m. As a consequence, one has Theorem D (Reeb). If M is compact, f ∈ C ∞(M) is a Morse function that has only two critical points, then M is homeomorphic to S m. Proof. The two critical points have to be the maximum/minimum of f. Take a close to the minimal value of f and b closed to the maximal value of f, so that both f −1 ((−∞, a)) and f −1 ((b, +∞)) are homeomorphic to R m. Take b 0 between b and the maximal value of f. By Theorem A f −1 ((−∞, a)) is homeomorphic to f −1 ((−∞, b0 )). Since M = f −1 ((−∞, b0 )) ∪ f −1 ((b, +∞)),by Brown’s theorem, M is homeomorphic to S m. Note that in this proof we avoided the use of Theorem B. 1
2LECTURE27:THESPHERETHEOREMTCritical points of the distancefunction.Now let (M,g) be a Riemannian manifold, and p E M be a point. In somesensethe distancefunctions d,'s arethemost natural functions that are defined onM.Although d,g Co(M), Grove and Shiohama succeeded in developing a Morsetheoryfor d,in1977whichplayed an importantrolein studying thetopology ofRiemannian manifolds. To get an idea let's examine the behavior of d, on (M, g):. As we have seen, the distance function d, is smooth at any q Cut(p)u (p),with (Vdp)q = (d(p,q), where is the unique minimizing normal geodesicfrom p to q. In particular, /Vdpl = 1 at any q & Cut(p) U (p). As a result,these points are not critical points of dp.The singularity of dp at the point p is not too bad, since it is the onlyminimum of dp, and the change of topology near p is well-understood. Thispoint can be regarded as a "trivial critical point" of dp. So we are more interested in those points q e Cut(p). They are candidatesof critical points for dp. To get a better idea, let's take a closer look at theexample S' xR: given any p = (ei, zo) E S'xR, Cut(p) = [(e-in, z) / z E R).When will the topology of Ma =(q| d(p,g)< a) change as a varies?Obviously- the topology of Ma will not change for a < [no critical points there], the topology of Ma will change when a pass the value , i.e. pass thecutpoint p=(e-ie,zo),-the topology of Ma will not change for a >, although there are twonon-smooth points of d, for each such a.Why the topology for Mawill not change for a > ? Because althoughthere are two minimizing geodesics meeting at one point q Cut(p) withd,(q) = a, their directions at q lie in the same open half space. As a result,there is one direction that one can“flow-out" Ma to Mb (b > a),and thatisadirectionwhoseangleswithbothgeodesicsareobtuse.Whysuch“flow-out" argument fails for a = π, ie. at p = (e-i, zo)? Because for the twogeodesics meeting at p, one can't find such a direction whose angles withbothgeodesics are obtuse!Wearethus led tothefollowing definition:Definition 1.1. A point q + p is called a critical point of d, [or a critical point of p] iffor all X.e T.M, there exists a minimizing geodesic from g =(o) to p so that(%(0), Xg) ≥ 0,i.e. the angle α between (O) and X, is no moore than The set of all such critical points of of dp will be denoted as CP(p). Note thatif g is not a critical point of dp, then the tangent vector of all minimizing geodesicfrom q to p lie in an open half space of T,M
2 LECTURE 27: THE SPHERE THEOREM ¶ Critical points of the distance function. Now let (M, g) be a Riemannian manifold, and p ∈ M be a point. In some sense the distance functions dp’s are the most natural functions that are defined on M. Although dp 6∈ C ∞(M), Grove and Shiohama succeeded in developing a Morse theory for dp in 1977 which played an important role in studying the topology of Riemannian manifolds. To get an idea let’s examine the behavior of dp on (M, g): • As we have seen, the distance function dp is smooth at any q 6∈ Cut(p)∪ {p}, with (∇dp)q = ˙γ(d(p, q)), where γ is the unique minimizing normal geodesic from p to q. In particular, |∇dp| = 1 at any q 6∈ Cut(p) ∪ {p}. As a result, these points are not critical points of dp. • The singularity of dp at the point p is not too bad, since it is the only minimum of dp, and the change of topology near p is well-understood. This point can be regarded as a “trivial critical point” of dp. • So we are more interested in those points q ∈ Cut(p). They are candidates of critical points for dp. To get a better idea, let’s take a closer look at the example S 1×R: given any p = (e iθ, z0) ∈ S 1×R, Cut(p) = {(e −iθ, z) | z ∈ R}. When will the topology of Ma = {q | d(p, q) < a} change as a varies? Obviously – the topology of Ma will not change for a < π [no critical points there], – the topology of Ma will change when a pass the value π, i.e. pass the cut point ˜p = (e −iθ, z0), – the topology of Ma will not change for a > π, although there are two non-smooth points of dp for each such a. Why the topology for Ma will not change for a > π? Because although there are two minimizing geodesics meeting at one point q ∈ Cut(p) with dp(q) = a, their directions at q lie in the same open half space. As a result, there is one direction that one can “flow-out” Ma to Mb (b > a), and that is a direction whose angles with both geodesics are obtuse. Why such “flowout” argument fails for a = π, i.e. at ˜p = (e −iθ, z0)? Because for the two geodesics meeting at ˜p, one can’t find such a direction whose angles with both geodesics are obtuse! We are thus led to the following definition: Definition 1.1. A point q 6= p is called a critical point of dp [or a critical point of p] if for all Xq ∈ TqM, there exists a minimizing geodesic γ from q = γ(0) to p so that hγ˙(0), Xqi ≥ 0, i.e. the angle α between ˙γ(0) and Xq is no moore than π 2 . The set of all such critical points of of dp will be denoted as CP(p). Note that if q is not a critical point of dp, then the tangent vector of all minimizing geodesic from q to p lie in an open half space of TqM
3LECTURE27:THESPHERETHEOREMExamples of critical pointsof thedistance function.Erample.Here are some immediate examples:.M = S? the standard sphere: the only critical point of p is its antipodal p. M = Sl × Ri the cylinder: the only critical point of (eie, z) is (e-i, z).. M - s1 × s1 the fat torus with fundamental domain a square centered at p:the critical points are the two midpoints of the sides and the corner point..If is a closed geodesic of length 21 so that both lo,j and ly,2 are minimal,then () is a critical point of (0).Recall that for any point g Cut(p), there is a unique minimizing geodesicjoining p to q. So exp, is injective on an open ball Bp(O,r) C T,M if B(p,r) CMCut(p).Moreover,for most points in Cut(p),thereexists at least twominimizingnormal geodesic top(c.f.PSet 3).Soweconcludeandinj(M,g) = inf dist(p, Cut(p).inj,(M,g) = dist(p, Cut(p)PEMProposition 1.2. If q E Cut(p) is not conjugate to p andd(p,q) = d(p,Cut(p),then there are eractly two minimizing normal geodesic and from p to g, ando(U)=-(l). In particular, q is a critical point of p.Proof. We have seen in Theorem 1.7 in Lecture 21 that there are at least two mini-mizing normal geodesic ,g from p to q. We shall prove () = -(). Suppose not,then there exists X, eTqM with[Xg =1, such that(Xg, ())<0and(Xg,o(1) <0.Since q is not conjugate to p along , there exists U l(O) such that expp lu is adiffeomorphism.LetE(s) = (expp lu)-1 expg(sXg).Then (t) = exp,(s(s) is a geodesic variation of . By the first variation formula,d%la=0L(%) = (Xg,() <0. So for s > 0 small enough, L(%) < L() = l.Similarly one can construct a geodesic variations(t) = expp(_n(s), where n(s) = (exPplv)-1 expg(sXg)of the minimizing geodesic so that L(s) < L(o) = I for s > 0 small enough. Notethat for each s, both s and o, are geodesics from p to exp.(sXg). Moreover, fors > 0 small enough,Is := d(p, exPg(sXg) ≤ L(s) ≤ 1.So exP, is NOT injective on BP(o, ), which contradicts with the fact that exPp口is a diffeomorphism on Bp(o, I), since I = d(p, Cut(p))
LECTURE 27: THE SPHERE THEOREM 3 ¶ Examples of critical points of the distance function. Example. Here are some immediate examples: • M = S 2 the standard sphere: the only critical point of p is its antipodal ¯p. • M = S 1 × R 1 the cylinder: the only critical point of (e iθ, z) is (e −iθ, z). • M = S 1 ×S 1 the flat torus with fundamental domain a square centered at p: the critical points are the two midpoints of the sides and the corner point. • If γ is a closed geodesic of length 2l so that both γ|[0,l] and γ|[l,2l] are minimal, then γ(l) is a critical point of γ(0). Recall that for any point q 6∈ Cut(p), there is a unique minimizing geodesic joining p to q. So expp is injective on an open ball Bp (0, r) ⊂ TpM if B(p, r) ⊂ M\Cut(p). Moreover, for most points in Cut(p), there exists at least two minimizing normal geodesic to p (c.f. PSet 3). So we conclude injp (M, g) = dist(p, Cut(p)) and inj(M, g) = inf p∈M dist(p, Cut(p)). Proposition 1.2. If q ∈ Cut(p) is not conjugate to p and d(p, q) = d(p, Cut(p)), then there are exactly two minimizing normal geodesic γ and σ from p to q, and σ˙(l) = −γ˙(l). In particular, q is a critical point of p. Proof. We have seen in Theorem 1.7 in Lecture 21 that there are at least two minimizing normal geodesic γ, σ from p to q. We shall prove ˙σ(l) = −γ˙(l). Suppose not, then there exists Xq ∈ TqM with |Xq| = 1, such that hXq, γ˙(l)i < 0 and hXq, σ˙(l)i < 0. Since q is not conjugate to p along γ, there exists U 3 lγ˙(0) such that expp |U is a diffeomorphism. Let ξ(s) = (expp |U ) −1 expq (sXq). Then γs(t) = expp ( t l ξ(s)) is a geodesic variation of γ. By the first variation formula, d ds|s=0L(γs) = hXq, γ˙(l)i < 0. So for s > 0 small enough, L(γs) < L(γ) = l. Similarly one can construct a geodesic variation σs(t) = expp ( s l η(s)), where η(s) = (expp |V ) −1 expq (sXq) of the minimizing geodesic σ so that L(σs) < L(σ) = l for s > 0 small enough. Note that for each s, both γs and σs are geodesics from p to expq (sXq). Moreover, for s > 0 small enough, ls := d(p, expq (sXq) ≤ L(γs) ≤ l. So expp is NOT injective on Bp (0, ls+l 2 ), which contradicts with the fact that expp is a diffeomorphism on Bp (0, l), since l = d(p, Cut(p)).
4LECTURE27:THESPHERETHEOREMI The isotopy lemma for dp.As in the usual Morse theory the following fact is crucial in all applications.Theorem 1.3(TheIsotopyLemma).Suppose (M,g)is complete, b>a>0,andd,'([a, b])nCP(d)=0. Then Ma is diffeomorphic [and is a deformation retract) to Mb.Proof. For any point q CP(p), then there exists X. E TM so that for any mini-mizing geodesic from q to p, the angle2(Xg, (0)<.2Extend the vector X. to a vector field X defined on a neighborhood U. of g so thatfor any q eU, and any minimizing geodesic from q to p,2(x(@.() 1Take a locally finite covering (Uq,/ of B(p, b)/B(p, a) using such neighborhoods, anda smooth partition of unity (pi) subordinate to this covering. Let X =piX9iClearly X is a smooth non-vanishing vector field on B(p,b)B(p,a), since(X(q), (0)) =>pi(X(q), (0)) > 0, VqE B(p,b) / B(p, a),We normalize X sothat x(@)/ = 1 at each g,and then repeat the proof ofTheorem A.More precisely, for any e B(p,b)/B(p,a)we let i be the integralcurve of Xpassing j,and forany i(t)eB(p,b)/B(p,a)welettbe theminimizinggeodesic from o'(t) to p.Then by thefirst variation formula,d是L(%) = -(X(o(0),(0),(dp(αi(t)) =dtdtFix ti < t2 so that i([ti,tal) c B(p,t2) / B(p,ti). By compactness, 3e > 0 so that-(X(α9(t), t(0)) ≤-cos(-E)<02for all t E[ti,t2]. It follows(dp(gi(t)dt ≤-(t2 - ti) cos(dp(α(t2)) - dp(α9(ti)) =-E)<0dtSo as t increases, d, is strictly decreasing along the integral curves oi(t) of X inside口B(p,ta)B(p,ti) as. So the flow of X gives the desired diffeomorphism.Since the topology changes after the“farthest point", we getCorollary 1.4.Let (M,g)be a compact Riemannian manifold,p E M, and q isafarthest point from p, then q is a critical point of p.In particular, if d(p, g) = diam(M, g), then for any X, e T,M, there is a minimalgeodesic from p = (0) to q so that ((0), Xp) ≥ 0
4 LECTURE 27: THE SPHERE THEOREM ¶ The isotopy lemma for dp. As in the usual Morse theory the following fact is crucial in all applications. Theorem 1.3 (The Isotopy Lemma). Suppose (M, g) is complete, b > a > 0, and d −1 p ([a, b])∩CP(d)=∅. Then Ma is diffeomorphic [and is a deformation retract] to Mb. Proof. For any point q 6∈ CP(p), then there exists Xq ∈ TqM so that for any minimizing geodesic γ from q to p, the angle ∠(Xq, γ˙(0)) < π 2 . Extend the vector Xq to a vector field Xq defined on a neighborhood Uq of q so that for any ¯q ∈ Uq and any minimizing geodesic ¯γ from ¯q to p, ∠(X q (¯q), γ¯˙(0)) < π 2 . Take a locally finite covering {Uqi } of B(p, b)\B(p, a) using such neighborhoods, and a smooth partition of unity {ρi} subordinate to this covering. Let X = PρiXqi . Clearly X is a smooth non-vanishing vector field on B(p, b) \ B(p, a), since hX(¯q), γ¯˙(0)i = XρihX qi (¯q), γ¯˙(0)i > 0, ∀q¯ ∈ B(p, b) \ B(p, a). We normalize X so that |X(¯q)| = 1 at each ¯q, and then repeat the proof of Theorem A. More precisely, for any ¯q ∈ B(p, b) \ B(p, a) we let σ q¯ be the integral curve of X passing ¯q, and for any σ q¯ (t) ∈ B(p, b)\B(p, a) we let ¯γt be the minimizing geodesic from σ q¯ (t) to p. Then by the first variation formula, d dt(dp(σ q¯ (t)) = d dtL(¯γt) = −hX(σ q¯ (t)), γ¯˙ t(0)i. Fix t1 < t2 so that σ q¯ ([t1, t2]) ⊂ B(p, t2) \ B(p, t1). By compactness, ∃ε > 0 so that −hX(σ q¯ (t)), γ¯˙ t(0)i ≤ − cos(π 2 − ε) < 0 for all t ∈ [t1, t2]. It follows dp(σ q¯ (t2)) − dp(σ q¯ (t1)) = Z t2 t1 d dt(dp(σ q¯ (t)))dt ≤ −(t2 − t1) cos(π 2 − ε) < 0. So as t increases, dp is strictly decreasing along the integral curves σ q¯ (t) of X inside B(p, t2) \ B(p, t1) as. So the flow of X gives the desired diffeomorphism. Since the topology changes after the “farthest point”, we get Corollary 1.4. Let (M, g) be a compact Riemannian manifold, p ∈ M, and q is a farthest point from p, then q is a critical point of p. In particular, if d(p, q) = diam(M, g), then for any Xp ∈ TpM, there is a minimal geodesic γ from p = γ(0) to q so that hγ˙(0), Xpi ≥ 0.
LECTURE27:THESPHERETHEOREM5I The Reeb theorem for d.Although there is no Morse lemma and there is no index for the critical pointsof a distance function, near the trivial critical point p of d, the sub-level set is stillan m-ball. Similar phenomena happens near a“"non-degenerate (=discrete)farthestpoint". So it is not amazing that we still have the following analogue to the Reebtheorem fordp:Corollary 1.5.Let (M,g)be a compact Riemannian manifold and p E M.If dphas only one nontrivial critical point q p, then M is homeomorphic to Sm.Proof. According to Corollary 1.4, q has to be the only farthest point of p. Takeri small so that both B(p,ri) and B(q,ri) are homeomorphic to Rm.Take r2 (r1,d(p, q)) large so that B(p, r2) U B(q,ri) = M. Then d, has no critical point inB(p,r2))B(p, ri). By the isotopy lemma, B(p, r2) is homeomorphic to B(p, ri), andthus is homeomorphic to Rm. By Brown's theorem, M is homeomorphic to Sm.2.SOME SPHERE THEOREMSI The diameter sphere theorem of Grove-Shiohama.As a first application of the critical point theory of distance function, we shallprove thefollowing diameter spheretheorem:Theorem 2.1 (Grove-Shiohama). Let (M,g) be a complete connected Riemannianmanifoldwith1K71and diam(M,g)≥π,then M is homeomorphicto Sm.Proof. Since M is compact, there exists k > so that K ≥ k. By Bonnet-Meyer,diam(M,g)≤ . In view of Cheng's maximal diameter theorem, we may assumediam(M,g) = 1 <VKLet p, q E M so that d(p, q) = I = diam(M, g). By Corollary 1.4, q is a critical pointof p. By Corollary 1.5, it is enough to prove that p has no other critical points.Suppose to the contrary, + q is a critical point of p. Denote l' = d(p, q) and"=d(g,q).Let be aminimizing normal geodesic from q=(o) to q =(t").By definition of critical points, there exists a minimizing normal geodesic fromq = o(0) to p = (l') so thatα= 2(一(0),0(0)≤2:ApplytheToponogov comparison theorem (triangle version),we conclude thatthereis a geodesic triangle in Sm()whose sides havelengths l,,",so that the opposite
LECTURE 27: THE SPHERE THEOREM 5 ¶ The Reeb theorem for dp. Although there is no Morse lemma and there is no index for the critical points of a distance function, near the trivial critical point p of dp the sub-level set is still an m-ball. Similar phenomena happens near a “non-degenerate (=discrete) farthest point”. So it is not amazing that we still have the following analogue to the Reeb theorem for dp: Corollary 1.5. Let (M, g) be a compact Riemannian manifold and p ∈ M. If dp has only one nontrivial critical point q 6= p, then M is homeomorphic to S m. Proof. According to Corollary 1.4, q has to be the ✿✿✿✿ only farthest point of p. Take r1 small so that both B(p, r1) and B(q, r1) are homeomorphic to R m. Take r2 ∈ (r1, d(p, q)) large so that B(p, r2) ∪ B(q, r1) = M. Then dp has no critical point in B(p, r2)\B(p, r1). By the isotopy lemma, B(p, r2) is homeomorphic to B(p, r1), and thus is homeomorphic to R m. By Brown’s theorem, M is homeomorphic to S m. 2. Some sphere theorems ¶ The diameter sphere theorem of Grove-Shiohama. As a first application of the critical point theory of distance function, we shall prove the following diameter sphere theorem: Theorem 2.1 (Grove-Shiohama). Let (M, g) be a complete connected Riemannian manifold with K > 1 4 and diam(M, g) ≥ π, then M is homeomorphic to S m. Proof. Since M is compact, there exists k > 1 4 so that K ≥ k. By Bonnet-Meyer, diam(M, g) ≤ √π k . In view of Cheng’s maximal diameter theorem, we may assume diam(M, g) = l < π √ k Let p, q ∈ M so that d(p, q) = l = diam(M, g). By Corollary 1.4, q is a critical point of p. By Corollary 1.5, it is enough to prove that p has no other critical points. Suppose to the contrary, ¯q 6= q is a critical point of p. Denote l 0 = d(p, q¯) and l 00 = d(q, q¯). Let γ be a minimizing normal geodesic from q = γ(0) to ¯q = γ(l 00). By definition of critical points, there exists a minimizing normal geodesic σ from q¯ = σ(0) to p = σ(l 0 ) so that α = ∠(−γ˙(l), σ˙(0)) ≤ π 2 . Apply the Toponogov comparison theorem (triangle version), we conclude that there is a geodesic triangle in S m( √ 1 k ) whose sides have lengths l, l0 , l00, so that the opposite