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
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) 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
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 small enough, L(γs) 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)).
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)) 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 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.
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
6 LECTURE 27: THE SPHERE THEOREM angle of l is ˜α ≤ π 2 . Since π ≤ l cos(√ kl) = cos(√ kl0 ) cos(√ kl00) + sin(√ kl0 ) sin(√ kl00) cos(˜α) ≥ cos(√ kl0 ) cos(√ kl00), which implies that exactly one of l 0 and l 00 is strictly greater than π 2 √ k , and the other is strictly smaller than π 2 √ k . Without loss of generality, assume 0 cos(√ kl0 ). In other words, l 0 so that if 1 4 − ε(m) ≤ K ≤ 1, then M is either homeomorphic to S m or diffeomorphic to one of the CROSSes: CPm/2 , HPm/4 , CaP 2 . (2) (Abresch-Meyer, 1994) If m is odd, then there exists ε > 0 so that if 1 4 − ε ≤ K ≤ 1, then M is homeomorphic to S m. Remark. The other half of Hopf’s question was proved by Brendle and Schoen: Theorem 2.3 (Differential sphere theorem, Brendle-Schoen 2009). Let (M, g) be complete and simply connected, such that any p ∈ M, 0 < supΠp K(Πp) < 4 infΠp K(πp), then M is diffeomorphic to S m. Remark (Sphere theorem in lower dimensions).
LECTURE 27: THE SPHERE THEOREM 7 (1) For m = 2: let M be an oriented compact surface with K > 0, then by the Gauss-Bonnet formula M is ✿✿✿✿✿✿✿✿✿✿✿✿✿✿ diffeomorphic to S 2 . (2) For m = 3, by introducing the method of Ricci ✿✿✿✿✿✿✿✿✿✿ flow, R.Hamilton proved in 1982 that if (M, g) is a 3 dimensional compact Riemannian manifold with Ric > 0, then (M, g) is ✿✿✿✿✿✿✿✿✿✿✿✿✿✿ diffeomorphic to S 3 . (3) For m = 4 there is a very interesting ✿✿✿✿✿✿✿✿✿✿✿ conformal ✿✿✿✿✿✿✿ sphere✿✿✿✿✿✿✿✿✿ theorem: Theorem 2.4 (Chang-Gursky-Yang 2003). Let (M, g) be a compact 4-manifold whose Yamabe invariant is positive. Suppose R M |W| 2dv < 16π 2χ(M), then M is diffeomorphic to S 4 or RP4 . In view of Grove-Shiohama’s diameter sphere theorem, to prove topological sphere theorem it is enough to prove Theorem 2.5 (Klingenberg injectivity radius estimate). Let (M, g) be a complete simply connected Riemannian manifold with 1/4 < K ≤ 1, then inj(M, g) ≥ π. In what follows we will prove Klingenberg’s injectivity radius estimate for m even. The odd case is more involved. ¶ Klingenberg lemma. We need Lemma 2.6 (Klingenberg lemma). Let (M, g) be a compact Riemannian manifold whose sectional curvature satisfies K ≤ C for some constant C. Then either inj(M, g) ≥ π √ C or there exists a closed geodesic γ in M whose length is minimum among all closed geodesics, such that inj(M, g) = 1 2 L(γ). Proof. Take p ∈ M and q ∈ Cut(p) so that dist(p, q) = inj(M, g). If q is conjugate to p along some minimizing geodesic, then by Corollary 2.2 in Lecture 23, inj(M, g) = dist(p, q) ≥ π √ C . If q is not conjugate to p, then by Proposition 1.2, there exists two minimizing normal geodesics σ, τ joining p to q so that ˙σ(l) = −τ˙(l), where l = dist(p, q). Since p is also a cut point of q, and by definition p realize the distance from q to Cut(q). It follows that ˙σ(0) = −τ˙(0). So σ and τ together form a closed geodesic. If we denote this closed geodesic by γ, then inj(M, g) = 1 2 L(γ). Finally we prove γ has minimal length among all closed geodesics: Otherwise if there is another closed geodesic γ 0 with length L(γ 0 ) < L(γ), and let p 0 , q 0 be
8 LECTURE 27: THE SPHERE THEOREM two “antipodal” points on γ 0 , i.e. dist(p 0 , q0 ) = 1 2 L(γ 0 ), then by definition there is a point q 00 on γ 0 which lies in Cut(p 0 ), and dist(p 0 , q00) ≤ 1 2 L(γ 0 ) < inj(M, g). Contradiction. ¶ Proof of Klingenberg injectivity radius estimate, m even. [In what follows we only need to assume M is orientable (which implies M is simply connected by Sygne’s theorem).] By Bonnet-Myers’ theorem, M is compact. So there exists p ∈ M and q ∈ Cut(p) so that dist(p, q) = inj(M, g) =: l. Suppose the theorem fails, i.e. l < π. Then by Corollary 2.2 in lecture 23, q is not conjugate to p. So according to Klingenberg lemma, there exists a closed normal geodesic γ in M passing p = γ(0) and q = γ(l) whose length is L(γ) = 2l < 2π. Since M is of even-dimension and is oriented, by repeating the proof of Synge’s theorem, we can find a vector field X(t) parallel along γ with X(2l) = X(0) = Xp ∈ γ˙(0)⊥, so that the variation of γ with variation field X satisfies d 2 ds2 s=0 E(γs) = − Z R( ˙γ, X, γ, X˙ )dt < 0. In other words, L(γs) < L(γ) for all small s 6= 0. Denote ps = γs(0) and let qs = γs(ls) be the point on γs which is farthest to ps. Then dist(ps, qs) < l = inj(M, g), so there exists a unique normal minimizing geodesic σs joint qs = σs(0) to ps. Since lims→0 qs = q, there exists a sequence si → 0 so that ˙σsi (0) converges to a unit vector Yq ∈ TqM. By continuity, σ(t) = expq (tYq) is a minimizing normal geodesic connecting q to p. In what follows we will show ˙σ(0) ⊥ γ˙(l), so that σ is not one of the two parts of γ. As a consequence, we get three minimizing geodesic from q to p. This contradicts with Proposition 1.2. It remains to prove ˙σ(0) ⊥ γ˙(l). We let σs,t be the minimizing normal geodesic from ps = γs(0) to γs(t) for γs(t) close to qs = γs(ls). Then σs,t is a variation of σs = σs,ls . By the choice of qs, L(σs,t) ≤ L(σs). So according to the first variation formula, 0 = d dt t=ls E(σs,t) = −hσ˙ s(0), γ˙ s(ls)i. It follows that ˙σs(0) ⊥ γ˙ s(ls). Passing to the subsequence si and taking limit, we get ˙σ(0) = Yq ⊥ γ˙(l). Remark. By checking the prove above one can see that for the case m = dim M even, it’s enough to assume that (M, g) is oriented and satisfies the weaker curvature condition that there exists ε such that 0 < ε < K ≤ 1.