DOCSLIB.ORG

A Study on Sphere Theorems and the Curvature on Exotic Spheres

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Study on Sphere Theorems and the Curvature on Exotic Spheres 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.
Load more

Recommended publications
  • Marcel Berger Remembered
    Marcel Berger Remembered Claude LeBrun, Editor and Translator Gérard Besson EDITOR’S NOTE. Claude LeBrun has kindly assembled this memorial for Marcel Berger. This month also Marcel Berger,1 one of the world’s leading differential marks the conference “Riemannian Geometry Past, geometers and a corresponding member of the French Present and Future: An Homage to Marcel Berger” at Academy of Sciences for half a century, passed away on IHÉS. October 15, 2016, at the age of eighty-nine. Marcel Berger’s contributions to geometry were both broad and deep. The classification of Riemannian holo- nomy groups provided by his thesis has had a lasting impact on areas ranging from theoretical physics to alge- braic geometry. His 1960 proof that a complete oriented even-dimensional manifold with strictly quarter-pinched positive curvature must be a topological sphere is the Claude LeBrun is professor of mathematics at Stony Brook Univer- sity. His email address is [email protected]. Gérard Besson is CNRS Research Director at Institut Fourier, Uni- For permission to reprint this article, please contact: versité Grenoble, France. His email address is g.besson@ujf [email protected]. -grenoble.fr. DOI: http://dx.doi.org/10.1090/noti1605 1Pronounced bare-ZHAY. December 2017 Notices of the AMS 1285 research director by the CNRS. He served as president of the French Mathematical Society (SMF) during the period of 1979–80 and in this capacity helped oversee the foundation of the International Center for Mathematical Workshops (CIRM) in Luminy. In 1985 he then became director of the Institute for Higher Scientific Studies (IHÉS) in Bures-sur-Yvette, a position in which he served until 1993, when he was succeeded by his former student Jean-Pierre Bourguignon.
    [Show full text]
  • Hamilton's Ricci Flow
    The University of Melbourne, Department of Mathematics and Statistics Hamilton's Ricci Flow Nick Sheridan Supervisor: Associate Professor Craig Hodgson Second Reader: Professor Hyam Rubinstein Honours Thesis, November 2006. Abstract The aim of this project is to introduce the basics of Hamilton's Ricci Flow. The Ricci flow is a pde for evolving the metric tensor in a Riemannian manifold to make it \rounder", in the hope that one may draw topological conclusions from the existence of such \round" metrics. Indeed, the Ricci flow has recently been used to prove two very deep theorems in topology, namely the Geometrization and Poincar´eConjectures. We begin with a brief survey of the differential geometry that is needed in the Ricci flow, then proceed to introduce its basic properties and the basic techniques used to understand it, for example, proving existence and uniqueness and bounds on derivatives of curvature under the Ricci flow using the maximum principle. We use these results to prove the \original" Ricci flow theorem { the 1982 theorem of Richard Hamilton that closed 3-manifolds which admit metrics of strictly positive Ricci curvature are diffeomorphic to quotients of the round 3-sphere by finite groups of isometries acting freely. We conclude with a qualitative discussion of the ideas behind the proof of the Geometrization Conjecture using the Ricci flow. Most of the project is based on the book by Chow and Knopf [6], the notes by Peter Topping [28] (which have recently been made into a book, see [29]), the papers of Richard Hamilton (in particular [9]) and the lecture course on Geometric Evolution Equations presented by Ben Andrews at the 2006 ICE-EM Graduate School held at the University of Queensland.
    [Show full text]
  • Book Reviews
    BOOK REVIEWS BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 11, Number 1, July 1984 ©1984 American Mathematical Society 0273-0979/84 $1.00 + $.25 per page Riemannian geometry, by Wilhelm Klingenberg, de Gruyter Studies in Mathematics, Vol. 1, Walter de Gruyter & Co., Berlin, 1982, x + 396 pp., $49.00. ISBN 3-1100-8373-5 The basic concepts of Riemannian geometry have become useful in a surprising variety of mathematical subjects. The terminology of manifolds, bundles, Riemannian metrics, and connections has become a lingua franca over much of partial differential equations, mathematical physics, and algebraic geometry, among other fields. This increasingly widespread use of the terminology and methods of geometry has tended to obscure the fact that Riemannian geometry as such is a subject with a quite precisely focused program, namely, to determine how the topology of a manifold is influenced by the local properties of its metric structure. These local properties are usually formulated in terms of curvature; this formulation is justified by the theorem of E. Cartan that the curvature tensor and its covariant derivatives of all orders at a point determine the formal Taylor expansion of the metric at that point. The program of obtaining global topological information from local metric information actually applies only to complete Riemannian manifolds, i.e., those that are complete as metric spaces when the distance between points is defined to be the infimum of the Riemannian-metric arc length of curves joining the two points. This restriction to complete manifolds, long a standard one in the subject, has been given explicit justification by a result of M.
    [Show full text]
  • Curvature Cones and the Ricci Flow
    Institut Fourier — Université de Grenoble I Actes du séminaire de Théorie spectrale et géométrie Thomas RICHARD Curvature cones and the Ricci flow. Volume 31 (2012-2014), p. 197-220. <http://tsg.cedram.org/item?id=TSG_2012-2014__31__197_0> © Institut Fourier, 2012-2014, tous droits réservés. L’accès aux articles du Séminaire de théorie spectrale et géométrie (http://tsg.cedram.org/), implique l’accord avec les conditions générales d’utilisation (http://tsg.cedram.org/legal/). cedram Article mis en ligne dans le cadre du Centre de diffusion des revues académiques de mathématiques http://www.cedram.org/ Séminaire de théorie spectrale et géométrie Grenoble Volume 31 (2012-2014) 197-220 CURVATURE CONES AND THE RICCI FLOW. Thomas Richard Abstract. — This survey reviews some facts about nonnegativity conditions on the curvature tensor of a Riemannian manifold which are preserved by the action of the Ricci flow. The text focuses on two main points. First we describe the known examples of preserved curvature conditions and how they have been used to derive geometric results, in particular sphere theorems. We then describe some recent results which give restrictions on general preserved conditions. The paper ends with some open questions on these matters. The Ricci flow is the following evolution equation: ( ∂g = −2 Ric (0.1) ∂t g(t) g(t) = g0 where (g(t))t∈[0,T ) is a one-parameter family of smooth Riemannian metrics on a fixed manifold M, and g0 is a given smooth Riemannian metric on M. It was introduced by R. Hamilton in 1982 ([11]), where it was used to study the topology of compact 3-manifolds with positive Ricci curvature.
    [Show full text]
  • Examples of Riemannian Manifolds with Non-Negative Sectional Curvature
    EXAMPLES OF RIEMANNIAN MANIFOLDS WITH NON-NEGATIVE SECTIONAL CURVATURE WOLFGANG ZILLER Manifolds with non-negative sectional curvature have been of interest since the beginning of global Riemannian geometry, as illustrated by the theorems of Bonnet-Myers, Synge, and the sphere theorem. Some of the oldest conjectures in global Riemannian geometry, as for example the Hopf conjecture on S2 £ S2, also ¯t into this subject. For non-negatively curved manifolds, there are a number of obstruction theorems known, see Section 1 below and the survey by Burkhard Wilking in this volume. It is somewhat surprising that the only further obstructions to positive curvature are given by the classical Bonnet-Myers and Synge theorems on the fundamental group. Although there are many examples with non-negative curvature, they all come from two basic constructions, apart from taking products. One is taking an isometric quotient of a compact Lie group equipped with a biinvariant metric and another a gluing procedure due to Cheeger and recently signi¯cantly generalized by Grove-Ziller. The latter examples include a rich class of manifolds, and give rise to non-negative curvature on many exotic 7-spheres. On the other hand, known manifolds with positive sectional curvature are very rare, and are all given by quotients of compact Lie groups, and, apart from the classical rank one symmetric spaces, only exist in dimension below 25. Due to this lack of knowledge, it is therefore of importance to discuss and understand known examples and ¯nd new ones. In this survey we will concentrate on the description of known examples, although the last section also contains suggestions where to look for new ones.
    [Show full text]
  • Arxiv:1710.03952V3 [Math.MG] 11 Jul 2021 Example Lp-Unit Spheres)
    THE ISOPERIMETRIC INEQUALITY ON COMPACT RANK ONE SYMMETRIC SPACES AND BEYOND YASHAR MEMARIAN Abstract. Klartag's needle decomposition technique enables one to obtain strong isoperimetric inequalities on Riemannian manifolds other than the clas- sical known examples. As a result, in this paper, we obtain sharp isoperimetric inequalities for compact rank one symmetric spaces (CROSS). Namely, for the n real projective space RP , we demonstrate that the isoperimetric regions are k n given by either the geodesic balls or tubes around some RP ⊂ RP . For the n complex projective space CP , the isoperimetric regions are given by either k n the geodesic balls or tubes around some CP ⊂ CP . And for the quaternionic projective space, the isoperimetric regions are given by either the geodesic balls k n or tubes around some HP ⊂ HP . 1. Introduction Isoperimetric problems are some of the oldest problems in geometry. Given a space, one looks for domains of a given volume with the least boundary area. It is known that in model spaces (i.e. Euclidean spaces, spheres and hyperbolic spaces), for every given number v, the intrinsic balls with volume v have the least surface area among every domain with the same volume. When we leave the world of model spaces, or when we are dealing with geometric spaces with boundary, the solution to isoperimetric problems has provided real difficulties. Of course, when the number v is small enough, for spaces which locally look like Euclidean spaces (for instance manifolds), one expects that the solution of isoperimetric problems still would be the metric balls. However, for larger v and shapes with non-constant curvature or with non-smooth boundaries, the isoperimetric problem in general is very hard to solve.
    [Show full text]
  • Comparison Theorems in Riemannian Geometry
    COMPARISON THEOREMS IN RIEMANNIAN GEOMETRY JEFF CHEEGER DAVID G. EBIN AMS CHELSEA PUBLISHING !MERICAN -ATHEMATICAL 3OCIETY s 0ROVIDENCE 2HODE )SLAND COMPARISON THEOREMS IN RIEMANNIAN GEOMETRY David G. Ebin and Jeff Cheeger G.EbinandJeff David Photo courtesy of Ann Billingsley COMPARISON THEOREMS IN RIEMANNIAN GEOMETRY JEFF CHEEGER DAVID G. EBIN AMS CHELSEA PUBLISHING American Mathematical Society • Providence, Rhode Island M THE ATI A CA M L ΤΡΗΤΟΣ ΜΗ N ΕΙΣΙΤΩ S A O C C I I R E E T ΑΓΕΩΜΕ Y M A F O 8 U 88 NDED 1 2000 Mathematics Subject Classification. Primary 53C20; Secondary 58E10. For additional information and updates on this book, visit www.ams.org/bookpages/chel-365 Library of Congress Cataloging-in-Publication Data Cheeger, Jeff. Comparison theorems in riemannian geometry / Jeff Cheeger, David G. Ebin. p. cm. — (AMS Chelsea Publishing) Originally published: Amsterdam : North-Holland Pub. Co. ; New York : American Elsevier Pub. Co., 1975, in series: North-Holland mathematical library ; v. 9. Includes bibliographical references and index. ISBN 978-0-8218-4417-5 (alk. paper) 1. Geometry, Riemannian. 2. Riemannian manifolds. I. Ebin, D. G. II. American Mathe- matical Society. III. Title. QA649 .C47 2008 516.373—dc22 2007052113 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy a chapter for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society.
    [Show full text]
  • Comparison Geometry for Ricci Curvature
    Comparison Geometry for Ricci Curvature Xianzhe Dai Guofang Wei 1 1Partially supported by NSF grant DMS-08 2 A Ricci curvature bound is weaker than a sectional curvature bound but stronger than a scalar curvature bound. Ricci curvature is also special that it occurs in the Einstein equation and in the Ricci flow. Comparison geometry plays a very important role in the study of manifolds with lower Ricci curva- ture bound, especially the Laplacian and the Bishop-Gromov volume compar- isons. Many important tools and results for manifolds with Ricci curvature lower bound follow from or use these comparisons, e.g. Meyers' theorem, Cheeger- Gromoll's splitting theorem, Abresch-Gromoll's excess estimate, Cheng-Yau's gradient estimate, Milnor's result on fundamental group. We will present the Laplacian and the Bishop-Gromov volume comparison theorems in the first lec- ture, then discuss their generalizations to integral Ricci curvature, Bakry-Emery Ricci tensor and Ricci flow in the rest of lectures. Contents 1 Basic Tools for Ricci Curvature 5 1.1 Bochner's formula . 5 1.2 Mean Curvature and Local Laplacian Comparison . 7 1.3 Global Laplacian Comparison . 10 1.4 Volume Comparison . 12 1.4.1 Volume of Riemannian Manifold . 12 1.4.2 Comparison of Volume Elements . 15 1.4.3 Volume Comparison . 16 1.5 The Jacobian Determinant of the Exponential Map . 20 1.6 Characterizations of Ricci Curvature Lower Bound . 22 1.7 Characterization of Warped Product . 24 2 Geometry of Manifolds with Ricci Curvature Lower Bound 27 2.1 Cheeger-Gromoll's Splitting Theorem .
    [Show full text]
  • Injectivity Radius Estimates and Sphere Theorems
    Comparison Geometry MSRI Publications Volume 30, 1997 Injectivity Radius Estimates and Sphere Theorems UWE ABRESCH AND WOLFGANG T. MEYER Abstract. We survey results about the injectivity radius and sphere the- orems, from the early versions of the topological sphere theorem to the 1 authors' most recent pinching below- 4 theorems, explaining at each stage the new ideas involved. Introduction Injectivity radius estimates and sphere theorems have always been a central theme in global differential geometry. Many tools and concepts that are now fundamental for comparison geometry have been developed in this context. This survey of results of this type reaches from the early versions of the 1 topological sphere theorem to the most recent pinching below- 4 theorems. Our main concern is to explain the new ideas that enter at each stage. We do not cover the differentiable sphere theorem and sphere theorems based on Ricci curvature. In Sections 1–3 we give an account of the entire development from the first sphere theorem of H. E. Rauch to M. Berger’s rigidity theorem and his pinching 1 below-4 theorem. Many of the main results depend on subtle injectivity radius estimates for compact, simply connected manifolds. In Section 4 we present our recent injectivity radius estimate for odd-dimen- n 1 sional manifolds M with a pinching constant below 4 that is independent of n [Abresch and Meyer 1994]. With this estimate the restriction to even-dimensional 1 manifolds can be removed from the hypotheses of Berger’s pinching below- 4 theorem. Additional work is required in order to get a sphere theorem for odd-dimen- n 1 sional manifolds M with a pinching constant < 4 independent of n.
    [Show full text]
  • The Differentiable Sphere Theorem for Manifolds with Positive Ricci Curvature
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 3, March 2012, Pages 1011–1021 S 0002-9939(2011)10952-3 Article electronically published on July 21, 2011 THE DIFFERENTIABLE SPHERE THEOREM FOR MANIFOLDS WITH POSITIVE RICCI CURVATURE HONG-WEI XU AND JUAN-RU GU (Communicated by Jianguo Cao) Abstract. We prove that if M n is a compact Riemannian n-manifold and if − Ricmin > (n 1)τnKmax,whereKmax(x):=maxπ⊂TxM K(π), Ricmin(x):= · · minu∈UxM Ric(u), K( )andRic( ) are the sectional curvature and Ricci cur- − 6 vature of M respectively, and τn =1 5(n−1) ,thenM is diffeomorphic to a spherical space form. In particular, if M is a compact simply connected manifold with K ≤ 1andRicM > (n − 1)τn,thenM is diffeomorphic to the standard n-sphere Sn. We also extend the differentiable sphere theorem above to submanifolds in Riemannian manifolds with codimension p. 1. Introduction and main results In 1989, Shen [14] obtained the following important topological sphere theorem for manifolds of positive Ricci curvature (see also [16]). Theorem A. Let M n be an n-dimensional complete and simply connected mani- fold. If the sectional curvature satisfies KM ≤ 1, and the Ricci curvature satisfies RicM ≥ (n − 1)δn,where 5 − 3 8 8(n−1) for even n, δn = 5 − 3 8 4(n−1) for odd n, then M is homeomorphic to the n-sphere Sn. A natural question related to Shen’s sphere theorems is as follows. Question. Can one prove a differentiable sphere theorem for manifolds satisfying similar pinching condition? In 1966, Calabi (unpublished) and Gromoll [8] first investigated the differen- tiable pinching problem for positive pinched compact manifolds.
    [Show full text]
  • Ricci Flow and Manifolds with Positive Curvature
    Ricci flow and manifolds with positive curvature Lei Ni Dedicated to Nolan Wallach on his 70th birthday Abstract This is an expository article based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. We discuss various notions of positivity and their relations with the study of the Ricci flow, including a proof of the assertion, due to Wolfson and the author, that the Ricci flow preserves the positivity of the complex sectional curvature. We discuss the examples of Wallach of the manifolds with positive pinched sectional curvature and the behavior of Ricci flow on some examples. Finally we discuss the recent joint work with Wilking on the manifolds with pinched flag curvature and some open problems. Keywords: Positivity of the Curvature • Ricci flow • Flag curvature pinching Mathematics Subject Classification: 53C44 1 Introduction This article is based on the author’s lecture delivered at the conference Lie Theory and Its Applications in March 2011, UCSD. Gauss curvature was defined for surfaces in three-dimensional Euclidean space R3 by the determinant of the second fundamental form of the embedding with respect to the first fundamental form, namely the induced metric. The Theorem The research of the author was partially supported by NSF grant DMS-1105549. L. Ni Department of Mathematics, University of California at San Diego, La Jolla, CA 92093, USA e-mail: [email protected] © Springer Science+Business Media New York 2014 491 R. Howe et al. (eds.), Symmetry: Representation Theory and Its Applications, Progress in Mathematics 257, DOI 10.1007/978-1-4939-1590-3__17 492 Lei Ni Egregium of Gauss [11] asserts that it is in fact an invariant depending only on the first fundamental form, namely the metric of the given surface.
    [Show full text]
  • Sphere Theorems in Geometry
    Surveys in Differential Geometry XIII Sphere theorems in geometry Simon Brendle and Richard Schoen 1. The topological sphere theorem The sphere theorem in differential geometry has a long history, dating back to a paper by H.E. Rauch in 1951. In that paper [64], Rauch posed the question of whether a compact, simply connected Riemannian manifold M whose sectional curvatures lie in the interval (1, 4] is necessarily homeo- morphic to the sphere. Around 1960, M. Berger and W. Klingenberg gave an affirmative answer to this question: Theorem 1.1 (M. Berger [3]; W. Klingenberg [48]). Let M beacom- pact, simply connected Riemannian manifold whose sectional curvatures lie in the interval (1, 4]. Then M is homeomorphic to Sn. More generally, Berger [4] proved that a compact, simply connected Riemannian manifold whose sectional curvatures lie in the interval [1, 4] is either homeomorphic to Sn or isometric to a compact symmetric space of rank one. K. Grove and K. Shiohama proved that the upper bound on the sectional curvature can be replaced by a lower bound on the diameter: Theorem 1.2 (K. Grove, K. Shiohama [33]). Let M be a compact Rie- mannian manifold with sectional curvature greater than 1. If the diameter of M is greater than π/2, then M is homeomorphic to Sn. There is an interesting rigidity statement in the diameter sphere the- orem. To describe this result, suppose that M is a compact Riemannian manifold with sectional curvature K ≥ 1 and diameter diam(M) ≥ π/2. A theorem of D. Gromoll and K. Grove [27] asserts that M is either home- omorphic to Sn, or locally symmetric, or has the cohomology ring of the The first author was partially supported by a Sloan Foundation Fellowship and by NSF grant DMS-0605223.
    [Show full text]