A Study on Sphere Theorems and the Curvature on Exotic Spheres LEUNG, Wai Sing A Thesis Submitted in Partial Fulfilment of the Requirements for the Degree of Master of Philosophy in Mathematics The Chinese University of Hong Kong August 2010 § ！ ”參 Thesis Committee Professor CHOU Kai Seng (Chair) Professor LUK Hing Sun (Thesis Supervisor) Professor TAM Luen Fai (Committee Member) Professor YAU Shing-Toung Stephen (External Examiner) 1 Sphere theorems and Exotic spheres 1 Abstract This thesis studies the sphere theorems in differential geometry and exotic spheres. In chapter 1, we examine in detail those simply connected complete Riemannian manifolds for which the sectional curvature is bounded below by 1/4 and bounded above by 1. Then in chapter 2 we present a proof of a differentiable sphere theo- rem that there exists S > 1/4, which is independent of the dimension, such that if the lower bound of the curvature is S and the upper bound is 1, then the manifold is diffeomorphic to a standard sphere. In chapter 4, we present the construction of an exotic sphere with nonnegative sectional curvature due to Gromoll and Meyer, using O'Neill's formula for Riemannian submersions which is proved in chapter 3. Perturbations of the Gromoll-Meyer sphere by Cheeger's method yield exotic spheres with almost positive sectional curvature. We conclude the thesis by stating some recent results about exotic spheres with positive curvature and some related open problems. Sphere theorems and Exotic spheres 2 摘要 此論文討論微分幾何中的球定理及異種球面。在第一課，我們會測試單連通 完備及截面曲率介乎1/4到1之間的黎曼流形。在第二課我們會給微分球定理 證明，存在一正數J > 1/4,並與維度無關，使得如果其單連通完備及截面曲 率介乎到1之間，那麼該流形會與正常球面微分同胚。在第四課，我們會根 據Gromoll及Meyer，利用在黎曼浸沒上的0’Nem’s formula在構作一個非負截 面曲率的異種球面。而用Cheeger方法在Gromoll及Meyer球面作擾動會得到幾 乎正截面曲率的異種球面。最後，我們會以一些近期關於正截面曲率異種球面 的結果及其有關的未解決的問題作結。 Sphere theorems and Exotic spheres 3 ACKNOWLEDGMENTS I wish to express my gratitude to Prof H.S. Luk for his supervision. His stim- ulating guidance and encouragement has helped me a lot throughout the whole period of my postgraduate study and on the preparation of this thesis. I also want to think everyone who are interested in my thesis. Leung Wai Sing The Chinese University of Hong Kong July, 2010 Contents 0.1 Introduction . 6 1 Sphere Theorems 8 1.1 Rauch-Berger-Klingenberg Sphere Theorem 8 1.2 Maximal Diameter Theorem 15 1.3 Minimal Diameter Theorem 17 2 A Differentiable Sphere Theorem 27 2.1 Definitions 27 2.2 Preliminary results not related to curvature 28 2.3 Preliminary result related to the curvature 33 2.4 Differentiable Sphere Theorem 35 3 The fundamental equations of Riemannian submer- sions 43 3.1 Definitions 43 3.2 The fundamental tensors T and A 44 3.3 Covariant derivatives of T and A 47 3.4 Fundamental equations and O'Neill's formulas .... 49 4 Sphere theorems and Exotic spheres 5 4 A study on exotic spheres 52 4.1 Construction of Milnor sphere 52 4.2 Construction of GM-sphere (S^) 53 4.3 The curvature of E^ 54 4.4 Some recent results and open questions 59 Bibliography 61 Sphere theorems and Exotic spheres 6 0.1 Introduction The purpose of this paper is to study sphere theorems and some results related to exotic spheres. In 1951, Rauch[Ra]showed that a simply connected manifolds with the sectional curvature KM satisfying 1 > KM > ^ ~ 0.74 is homeomorphic to a sphere. In 1960, Berger and Klingenberg proved it for 5 = 1/4 by studying the injectivity radius. In chapter 1, we will give the proof of Rauch-Berger- Klingenberg Sphere Theorem and the Berger，s Diameter Sphere Theorems. In chapter 2, we will present the proof of a differentiable sphere theorem. An early differentiable sphere theorem was proved by Gromoll and Calabi in 1966. However, they needed a different lower bound 6n for Km in each dimension n and lim^-^oo ^n = 1- Then it was shown by Shiohama that a lower bound 5 for KM can be chosen independent of n. The actual value of 5 was improved successively by Sufimoto, Shiohama, Karcher(1971)[SSK] and E.Ruh (1971)[Ru]. The estimate is 6 ^ 0.80. Surprisingly, Simon Brendle and Richard Schoen utilized Ricci flow to prove in 2007[BS] that S = 1/4. Thus, those manifolds which satisfies the hy- pothesis of the sphere theorem are indeed diffeomorphic to the standard sphere. Moreover, the proof of Brendle and Schoen uses only a weaker assumption of pointwise pinching rather than the above global pinching. In diaper 2 we will give a proof of the theorem due to Sufimoto, Shiohama, Karcher and Ruh. The first exotic sphere was constructed by John Milnor. Using O'Neill's for- mula, Gromoll and Meyer constructed the first example of an exotic sphere E^ {GM-sphere) with a metric of non-negative sectional curvature as the base space of a Riemannian submersion on Sp{2). By means of Cheeger's perturbation on the E^, Peter Petersen and Frederick Wilhelm proved that it admits a metric with positive sectional curvature. In chapter 3 and 4，we will first prove in detail the O'Neill's formulas and present the construction of E?, then we briefly present some ideas Petersen and Wilhelm to give a proof that E"^ admits a metric with almost positive sectional Sphere theorems and Exotic spheres 7 curvature. We conclude this thesis by stating some recent results and some open problems, including a problem related to complex geometry and CR-geometry. Chapter 1 Sphere Theorems In this chapter, we will study smooth manifolds with the curvature 1 > KM > 1/4. Such manifolds are called 1 /‘pinched. We will show that a l/A-pinched manifold which is simply connected is homeomorphic to a sphere. Moreover, if a Riemannian manifold admits a metric such that 1 > KM > 1/4 which is not homeomorphic to a sphere, then it is isometric to a symmetric space. The following proofs are from [CE]Ch.6 1.1 Rauch-Berger-Klingenberg Sphere Theorem Theorem 1.1 If M is complete, simply connected n-dimensional Riemannian manifold with 1 > Km > H > 1/4, where H is a constant, then M is homeomor- phic to the n-sphere. Remark: If M can be written as the union of two imbedded balls joined along their common boundaries, then we can match the imbedded balls onto the upper half and lower half of the sphere. To construct the balls we have 3 lemmas: Lemma 1.1 Let M be any compact manifold. Fix p G M, and choose q G M such that p{p, q) is maximal. Then, for any v G Mp, there exists a maximal 8 Sphere theorems and Exotic spheres 9 geodesic 7 from q to p such that Z(7'(0), v) < 7^/2. Proof: Let a be the geodesic from q such that (7,(0) 二 v. Pick a sequence a —> 0, Qi = a(ei) — q, and 7e. be a normal geodesic from qi to p, which is minimal. Then there are two cases: Case I: If there exists a subsequence h such that •乂) < f then i^a(。)(7“(0)) G Mq has a limit point, say v. Take 7 be a normal geodesic from q such that y(0) = v. Then Z(7,(0),力 < 7r/2 and P(g,7(0) = lim p{qjnej{lj)) = lim Ij = I where I = p{p, q), and Ij = p{p,^{ej{0))). Case II: If there exists some to > 0 such that for any minimal geodesic from a{t) to p, we have at) > 7r/2, then consider a variation {c^} of c^ goes from p to a{t — s). Then by 1st variation formula, 知c』=〈7>,〉<0 办 5 = 0 Therefore, as t — 0, p{p, <j(t)) is strictly decreasing to p{p, q). But this contradicts the fact that q = (j(0) is at maximal distance from p. • Sphere theorems and Exotic spheres 10 p 〜 yq Figure la Lemma 1.2 If KM > H > 0 and suppose d{M) > 7r/2y/H, then for p,q above, B^/2Mp) U B叫H� q) 二 M Proof: Given any x G M, let 7i, 72 be a minimal geodesic from x to p and from p to q respectively. If x G M ~ 丑(P) then L[7i] > 7r/2y^H. And we claim pM{oc,q) = AM(7I(0),72(Z2)) < By Toponogov's Theorem [To PM(7I(0)，72(^2)) < so it suffices to check this is the case the M is a sphere of curvature H. Let a = ^(—71(^1)572(0))? then (五,节，a) is a hinge on Si/^H. On the other hand, it is easy to show that ps^^^^(^(0)5^(^2)) is a increasing function w.r.t to a, so by the Lemma 1.1, we can assume a = 7r/2. Let 0 denote the segment of ^ from ^(k) to ^{tx/^H), T denote the segment of —五 from 五(0) to ^{Ti/y/H), then r and 9 meet at the antipodal point of ^(0). and — k)，-A命—h)) = n/2. Sphere theorems and Exotic spheres H I 1 乂 I I 乂 l%io)J antipotal point of p Figure lb Since L[6'], L[T] < 7r/2->/i/, it follows that And hence PM{X, q) < < •批 口 Remark Tsukamoto[Ts] proves the lemma using Rauch comparison theorem Ra]instead of Toponogov's theorem. Lemma 1.3 Suppose KM > > 1/4 and the injectivity radius of M= z(M) > TT, and q be at maximal distance, then for any normal geodesic 7 from p with L[7] < n/y/H^ there exists a unique point r G 7 such that Proof: {Existence) Since i{M) > tt > �)= tfoi 0 < t < TT.