Totally Geodesic Maps Into Manifolds with No Focal Points
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Arxiv:Math/0701389V3 [Math.DG] 11 Apr 2007 Hoesadcnetrsi Hssbet E H Uvyb B by Survey the See Subject
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 fit 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 significantly 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 find 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. -
Of the American Mathematical Society August 2017 Volume 64, Number 7
ISSN 0002-9920 (print) ISSN 1088-9477 (online) of the American Mathematical Society August 2017 Volume 64, Number 7 The Mathematics of Gravitational Waves: A Two-Part Feature page 684 The Travel Ban: Affected Mathematicians Tell Their Stories page 678 The Global Math Project: Uplifting Mathematics for All page 712 2015–2016 Doctoral Degrees Conferred page 727 Gravitational waves are produced by black holes spiraling inward (see page 674). American Mathematical Society LEARNING ® MEDIA MATHSCINET ONLINE RESOURCES MATHEMATICS WASHINGTON, DC CONFERENCES MATHEMATICAL INCLUSION REVIEWS STUDENTS MENTORING PROFESSION GRAD PUBLISHING STUDENTS OUTREACH TOOLS EMPLOYMENT MATH VISUALIZATIONS EXCLUSION TEACHING CAREERS MATH STEM ART REVIEWS MEETINGS FUNDING WORKSHOPS BOOKS EDUCATION MATH ADVOCACY NETWORKING DIVERSITY blogs.ams.org Notices of the American Mathematical Society August 2017 FEATURED 684684 718 26 678 Gravitational Waves The Graduate Student The Travel Ban: Affected Introduction Section Mathematicians Tell Their by Christina Sormani Karen E. Smith Interview Stories How the Green Light was Given for by Laure Flapan Gravitational Wave Research by Alexander Diaz-Lopez, Allyn by C. Denson Hill and Paweł Nurowski WHAT IS...a CR Submanifold? Jackson, and Stephen Kennedy by Phillip S. Harrington and Andrew Gravitational Waves and Their Raich Mathematics by Lydia Bieri, David Garfinkle, and Nicolás Yunes This season of the Perseid meteor shower August 12 and the third sighting in June make our cover feature on the discovery of gravitational waves -
A Gauss-Bonnet Theorem for Manifolds with Asymptotically Conical Ends and Manifolds with Conical Singularities
A Gauss-Bonnet Theorem for Asymptotically Conical Manifolds and Manifolds with Conical Singularities. Thèse N° 9275 Présentée le 25 janvier 2019 à la Faculté des sciences de base Groupe Troyanov Programme doctoral en mathématiques pour l’obtention du grade de Docteur ès Sciences par ADRIEN GIULIANO MARCONE Acceptée sur proposition du jury Prof. K. Hess Bellwald, présidente du jury Prof. M. Troyanov, directeur de thèse Prof. A. Bernig, rapporteur Dr I. Izmestiev, rapporteur Prof. J. Krieger, rapporteur 2019 Remerciements En tout premier lieu je souhaite remercier mon directeur de thèse le professeur Marc Troyanov. Ses conseils avisés tout au long de ces quatre années ainsi que son soutien, tant moral que mathématique, ont été indispensables à la réalisation de ce travail. Cependant, réduire ce qu’il m’a appris au seul domaine scientifique serait inexact et je suis fier de dire qu’il est devenu pour moi aujourd’hui un ami. Secondly I want to express all my gratitude to the members of my committee: 1) the director: Professor K. Hess Bellwald, whom I know since the first day of my studies and who has always been a source of inspiration; 2) the internal expert: Professor J. Krieger, whose courses were among the best I took at EPFL; 3) the external experts: Professor A. Bernig (Goethe-Universität Frankfurt), who accepted to be part of this jury despite a truly busy schedule, and Dr I. Iz- mestiev (Université de Fribourg) who kindly invited me to the Oberseminar in Fribourg; for accepting to review my thesis. Merci au Professeur Dalang qui a eu la générosité de financer la fin de mon doc- torat. -
On Index Expectation Curvature for Manifolds 11
ON INDEX EXPECTATION CURVATURE FOR MANIFOLDS OLIVER KNILL Abstract. Index expectation curvature K(x) = E[if (x)] on a compact Riemannian 2d- manifold M is an expectation of Poincar´e-Hopfindices if (x) and so satisfies the Gauss-Bonnet R relation M K(x) dV (x) = χ(M). Unlike the Gauss-Bonnet-Chern integrand, these curvatures are in general non-local. We show that for small 2d-manifolds M with boundary embedded in a parallelizable 2d-manifold N of definite sectional curvature sign e, an index expectation K(x) d Q with definite sign e exists. The function K(x) is constructed as a product k Kk(x) of sec- tional index expectation curvature averages E[ik(x)] of a probability space of Morse functions Q f for which if (x) = ik(x), where the ik are independent and so uncorrelated. 1. Introduction 1.1. The Hopf sign conjecture [16, 5, 6] states that a compact Riemannian 2d-manifold M with definite sectional curvature sign e has Euler characteristic χ(M) with sign ed. The problem was stated first explicitly in talks of Hopf given in Switzerland and Germany in 1931 [16]. In the case of non-negative or non-positive curvature, it appeared in [6] and a talk of 1960 [9]. They appear as problems 8) and 10) in Yau's list of problems [25]. As mentioned in [5], earlier work of Hopf already was motivated by these line of questions. The answer is settled positively in the case d = 1 (Gauss-Bonnet) and d = 2 (Milnor), if M is a hypersurface in R2d+1 (Hopf) [15] or in R2d+2 (Weinstein) [23] or in the case of positively sufficiently pinched curvature, because of sphere theorems (Rauch,Klingenberg,Berger) [22, 10] spheres and projective spaces in even dimension have positive Euler characteristic. -
Submanifolds of Euclidean Space with Parallel Second Fundamental Form
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 32, Number 1, March 1972 SUBMANIFOLDS OF EUCLIDEAN SPACE WITH PARALLEL SECOND FUNDAMENTAL FORM JAAK VILMS1 In this paper necessary conditions are given for a complete Rie- mannian manifold M" to admit an isometric immersion into R*+r with parallel second fundamental form. Namely, it is shown that M" must be affinely equivalent either to a totally geodesic sub- manifold of the Grassmann manifold G(n,p), or to a fibre bundle over such a submanifold, with Euclidean space as fibre and the structure being close to a product. (An affine equivalence is a dif- feomorphism that preserves Riemannian connections.) The proof depends on the auxiliary result that the second fundamental form is parallel iff the Gauss map is a totally geodesic map. 1. The main result. Let i :Mn-+Rn+p be an immersion of a manifold Mn into Euclidean space, and endow M with the induced Riemannian metric. The second fundamental form ¡x of the immersion ; is a symmetric section of the bundle of bilinear maps L2(TM, TM; A/M), where TM and NM denote the tangent and normal bundles of M, respectively [4]. We iden- tify this bundle of maps with the isomorphic bundle L(TM, L(TM, NM)), and <xwith the corresponding section. At each point of M, the kernel of x (in the second interpretation) is called the relative nullity space, and its dimension k, the relative nullity index [2]. We say the second fundamental form a is parallel if 7>oc=0 (D always denotes the appropriate covariant derivative). -
Fundamental Theorems in Mathematics
SOME FUNDAMENTAL THEOREMS IN MATHEMATICS OLIVER KNILL Abstract. An expository hitchhikers guide to some theorems in mathematics. Criteria for the current list of 243 theorems are whether the result can be formulated elegantly, whether it is beautiful or useful and whether it could serve as a guide [6] without leading to panic. The order is not a ranking but ordered along a time-line when things were writ- ten down. Since [556] stated “a mathematical theorem only becomes beautiful if presented as a crown jewel within a context" we try sometimes to give some context. Of course, any such list of theorems is a matter of personal preferences, taste and limitations. The num- ber of theorems is arbitrary, the initial obvious goal was 42 but that number got eventually surpassed as it is hard to stop, once started. As a compensation, there are 42 “tweetable" theorems with included proofs. More comments on the choice of the theorems is included in an epilogue. For literature on general mathematics, see [193, 189, 29, 235, 254, 619, 412, 138], for history [217, 625, 376, 73, 46, 208, 379, 365, 690, 113, 618, 79, 259, 341], for popular, beautiful or elegant things [12, 529, 201, 182, 17, 672, 673, 44, 204, 190, 245, 446, 616, 303, 201, 2, 127, 146, 128, 502, 261, 172]. For comprehensive overviews in large parts of math- ematics, [74, 165, 166, 51, 593] or predictions on developments [47]. For reflections about mathematics in general [145, 455, 45, 306, 439, 99, 561]. Encyclopedic source examples are [188, 705, 670, 102, 192, 152, 221, 191, 111, 635]. -
Differential Geometry Proceedings of Symposia in Pure Mathematics
http://dx.doi.org/10.1090/pspum/027.1 DIFFERENTIAL GEOMETRY PROCEEDINGS OF SYMPOSIA IN PURE MATHEMATICS VOLUME XXVII, PART 1 DIFFERENTIAL GEOMETRY AMERICAN MATHEMATICAL SOCIETY PROVIDENCE, RHODE ISLAND 1975 PROCEEDINGS OF THE SYMPOSIUM IN PURE MATHEMATICS OF THE AMERICAN MATHEMATICAL SOCIETY HELD AT STANFORD UNIVERSITY STANFORD, CALIFORNIA JULY 30-AUGUST 17, 1973 EDITED BY S. S. CHERN and R. OSSERMAN Prepared by the American Mathematical Society with the partial support of National Science Foundation Grant GP-37243 Library of Congress Cataloging in Publication Data Symposium in Pure Mathematics, Stanford University, UE 1973. Differential geometry. (Proceedings of symposia in pure mathematics; v. 27, pt. 1-2) "Final versions of talks given at the AMS Summer Research Institute on Differential Geometry." Includes bibliographies and indexes. 1. Geometry, Differential-Congresses. I. Chern, Shiing-Shen, 1911- II. Osserman, Robert. III. American Mathematical Society. IV. Series. QA641.S88 1973 516'.36 75-6593 ISBN0-8218-0247-X(v. 1) Copyright © 1975 by the American Mathematical Society Printed in the United States of America All rights reserved except those granted to the United States Government. This book may not be reproduced in any form without the permission of the publishers. CONTENTS Preface ix Riemannian Geometry Deformations localement triviales des varietes Riemanniennes 3 By L. BERARD BERGERY, J. P. BOURGUIGNON AND J. LAFONTAINE Some constructions related to H. Hopf's conjecture on product manifolds ... 33 By JEAN PIERRE BOURGUIGNON Connections, holonomy and path space homology 39 By KUO-TSAI CHEN Spin fibrations over manifolds and generalised twistors 53 By A. CRUMEYROLLE Local convex deformations of Ricci and sectional curvature on compact manifolds 69 By PAUL EWING EHRLICH Transgressions, Chern-Simons invariants and the classical groups 72 By JAMES L. -
Book of Abstracts
Georgian National Georgian Mathematical Ivane Javakhishvili Academy of Sciences Union Tbilisi State University Caucasian Mathematics Conference CMC I BOOK OF ABSTRACTS Tbilisi, September 5 – 6 2014 Organizers: European Mathematical Society Armenian Mathematical Union Azerbaijan Mathematical Union Georgian Mathematical Union Iranian Mathematical Society Moscow Mathematical Society Turkish Mathematical Society Sponsors: Georgian National Academy of Sciences Ivane Javakhishvili Tbilisi State University Steering Committee: Carles Casacuberta (ex officio; Chair of the EMS Committee for European Solidarity), Mohammed Ali Dehghan (President of Iranian Mathematical Society), Roland Duduchava (President of the Georgian Mathematical Union), Tigran Harutyunyan (President of the Armenian Mathematical Union), Misir Jamayil oglu Mardanov (Representative of the Azerbaijan Mathematical Union), Marta Sanz-Sole (ex officio; President of the European Mathematical Society), Armen Sergeev (Representative of the Moscow Mathematical Society and EMS), Betül Tanbay (President of the Turkish Mathematical Society). Local Organizing Committee: Tengiz Buchukuri (IT Responsible), Tinatin Davitashvili (Scientific Secretary), Roland Duduchava (Chairman), David Natroshvili (Vice chairman), Levan Sigua (Treasurer). Editors: Guram Gogishvili, Vazha Tarieladze, Maia Japoshvili Cover Design: David Sulakvelidze Contents Abstracts of Invited Talks 13 Maria J. Esteban, Symmetry and Symmetry Breaking for Optimizers of Functional Inequalities . 15 Mohammad Sal Moslehian, Recent Developments in Operator Inequalities . 15 Garib N. Murshudov, Some Application of Mathematics to Structural Biology 16 Dmitri Orlov, Categories in Geometry and Physics: Mirror Symmetry, D- Branes and Landau–Ginzburg Models . 17 Samson Shatashvili, Supersymmetric Gauge Theories and the Quantisation of Integrable Systems . 18 Leon A. Takhtajan, On Bott–Chern and Chern–Simons Characteristic Forms 19 Cem Yalçin Yildirim, Small Gaps Between Primes: the GPY Method and Recent Advancements Over it . -
Triangle and Beyond
Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark Triangle and Beyond Xingwang Xu Nanjing University October 10, 2018 Xingwang Xu Geometry, Analysis and Topology 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces Xingwang Xu Geometry, Analysis and Topology 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds Xingwang Xu Geometry, Analysis and Topology 4 Many attempts for higher dimensional complete manifolds 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces Xingwang Xu Geometry, Analysis and Topology 5 My observation 6 Final Remarks Outline Surfaces Riemmanian Manifolds Complete manifolds New Development Final Remark 1 A formula for compact surfaces 2 Chern-Gaussian-Bonnet formula for Riemannian manifolds 3 Generalization to complete Riemann surfaces 4 Many attempts for higher dimensional -
From Betti Numbers to 2-Betti Numbers
Snapshots of modern mathematics № 1/2020 from Oberwolfach From Betti numbers to `2-Betti numbers Holger Kammeyer • Roman Sauer We provide a leisurely introduction to `2-Betti num- bers, which are topological invariants, by relating them to their much older cousins, Betti numbers. In the end we present an open research problem about `2-Betti numbers. 1 A geometric problem Large parts of mathematics are concerned with symmetries of various objects of interest. In topology, classical objects of interest are surfaces, like the one of a donut, which is also known as a torus and commonly denoted by T. An apparent symmetry consists in the possibility of rotating T by arbitrary angles without changing its appearance as is indicated in the following figure. In fact, if we follow the position of an arbitrary point on the torus while rotating it, we notice that the point travels along a circle. Any point not contained in 1 this circle moves around yet another circle parallel to the first one. All these circles are called the orbits of the rotation action. To say what this means in mathematical terms, let us first recall that the (standard) circle consists of all pairs of real numbers (x, y) such that x2 + y2 = 1. Any point (x, y) on the circle can equally be characterized by an angle α ∈ [0, 2π) in radian, meaning by the length of the arc between the points (0, 1) and (x, y). Now, the key observation is that the circle forms a group, which means that we can “add” two points by adding corresponding angles, forfeiting full turns: for π π 3π 3π π 1 example 2 + 4 = 4 , 2 + π = 2 , π + π = 0, and so forth. -
Compact Foliations with Finite Transverse Ls Category 11
COMPACT FOLIATIONS WITH FINITE TRANSVERSE LS CATEGORY STEVEN HURDER AND PAWELG. WALCZAK Abstract. We prove that if F is a foliation of a compact manifold M with all leaves compact submanifolds, and the transverse saturated category of F is finite, then the leaf space M=F is compact Hausdorff. The proof is surprisingly delicate, and is based on some new observations about the geometry of compact foliations. The transverse saturated category of a compact Hausdorff foliation is always finite, so we obtain a new characterization of the compact Hausdorff foliations among the compact foliations as those with finite transverse saturated category. 1. Introduction A compact foliation is a foliation of a manifold M with all leaves compact submanifolds. For codimension one or two, a compact foliation F of a compact manifold M defines a fibration of M over its leaf space M=F which is a Hausdorff space, and has the structure of an orbifold [27, 11, 12, 33, 10]. A compact foliation F with Hausdorff leaf space is said to be compact Hausdorff. Millett [22] and Epstein [12] showed that for a compact Hausdorff foliation F of a manifold M, the holonomy group of each leaf is finite, a property which characterizes them among the compact foliations. If every leaf has trivial holonomy group, then a compact Hausdorff foliation is a fibration. Otherwise, a compact Hausdorff foliation is a \generalized Seifert fibration", where the leaf space M=F is a \V-manifold" [29, 17, 22]. In addition, M admits a Riemannian metric so that the foliation is Riemannian. For codimension three and above, the leaf space M=F of a compact foliation of a compact manifold need not be a Hausdorff space. -
Descending Maps Between Slashed Tangent Bundles
DESCENDING MAPS BETWEEN SLASHED TANGENT BUNDLES IOAN BUCATARU AND MATIAS F. DAHL f ABSTRACT. Suppose T M \ {0} and T M \ {0} are slashed tangent bundles of two smooth manifolds M and Mf, respectively. In this paper we characterize f those diffeomorphisms F : T M \ {0}→ T M \ {0} that can be written as F = f (Dφ)|TM\{0} for a diffeomorphism φ: M → M. When F = (Dφ)|TM\{0} one say that F descends. If M is equipped with two sprays, we use the charac- terization to derive sufficient conditions that imply that F descends to a totally geodesic map. Specializing to Riemann geometry we also obtain sufficient con- ditions for F to descent to an isometry. 1. INTRODUCTION In this paper we study the following differential-topological problem: ( ) Suppose M and M are smooth manifolds, and suppose that F is a diffeo- ∗ morphism betweenf slashed tangent bundles (1) F : T M 0 T M 0 . \ { } → f \ { } Characterize those maps F that can be written as F = (Dφ) for a |TM\{0} diffeomorphism φ: M M, where Dφ is the tangent map of φ. → f Problem ( ) is related to anisotropic boundary rigidity problems on Riemannian manifolds∗ [Cro04, Uhl01, PU05]. It is also the setting for studying conjugate geo- desic flows. For an overview of this topic for Riemann metrics, see [Ber07, p. 495]. When F = (Dφ) TM\{0} one say that map F descends to a map φ: M M [Cro04]. | → f Let us first note that if f is a diffeomorphism between (unslashed) cotangent bun- dles ∗ ∗ arXiv:0903.5133v3 [math.DG] 10 Jun 2009 f : T M T M, → f the analogous problem is well understood.