An Introduction to Riemann-Finsler Geometry
|
|
- Reynold Hancock
- 5 years ago
- Views:
Transcription
1 D. Bao S.-S. Chern Z. Shen An Introduction to Riemann-Finsler Geometry With 20 Illustrations Springer
2 Contents Preface Acknowledgments vn xiii PART ONE Finsler Manifolds and Their Curvature CHAPTER 1 Finsler Manifolds and the Fundamentals of Minkowski Norms Physical Motivations Finsler Structures: Definitions and Conventions Two Basic Properties of Minkowski Norms A. Euler's Theorem B. A Fundamental Inequality C. Interpretations of the Fundamental Inequality Explicit Examples of Finsler Manifolds A. Minkowski and Locally Minkowski Spaces B. Riemannian Manifolds C. Randers Spaces D. Berwald Spaces E. Finsler Spaces of Constant Flag Curvature The Fundamental Tensor and the Cartan Tensor 22 * References for Chapter 1 25 CHAPTER 2 The Chern Connection Prologue The Vector Bundle TT*TM and Related Objects Coordinate Bases Versus Special Orthonormal Bases The Nonlinear Connection on the Manifold TM \ The Chern Connection on -K*TM 37
3 xvi Contents 2.5 Index Gymnastics A. The Slash (...) s and the Semicolon (...) ;s B. Covariant Derivatives of the Fundamental Tensor g C. Covariant Derivatives of the Distinguished 46 * References for Chapter 2 48 CHAPTER 3 Curvature and Schur's Lemma Conventions and the hh-, hv-, uu-curvatures First Bianchi Identities from Torsion Freeness Formulas for R and P in Natural Coordinates First Bianchi Identities from "Almost" ^-compatibility A. Consequences from the dx k A dx l Terms B. Consequences from the dx k A j?6y l Terms C. Consequences from the j^sy k A jsy 1 Terms Second Bianchi Identities Interchange Formulas or Ricci Identities Lie Brackets among the ~ and the F- ^ Derivatives of the Geodesic Spray Coefficients G x The Flag Curvature A. Its Definition and Its Predecessor B. An Interesting Family of Examples of Numata Type Schur's Lemma 75 * References for Chapter 3 80 CHAPTER 4 Finsler Surfaces and a Generalized Gauss Bonnet Theorem Prologue Minkowski Planes and a Useful Basis A. Rund's Differential Equation and Its Consequence B. A Criterion for Checking Strong Convexity The Equivalence Problem for Minkowski Planes The Berwald Frame and Our Geometrical Setup on SM The Chern Connection and the Invariants I, J, K The Riemannian Arc Length of the Indicatrix A Gauss-Bonnet Theorem for Landsberg Surfaces 105 * References for Chapter 4 110
4 Contents PART TWO Calculus of Variations and Comparison Theorems 111 CHAPTER 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature 5.1 The First Variation of Arc Length Ill 5.2 The Second Variation of Arc Length Geodesies and the Exponential Map Jacobi Fields How the Flag Curvature's Sign Influences Geodesic Rays 135 * References for Chapter Ill CHAPTER 6 The Gauss Lemma and the Hopf-Rinow Theorem The Gauss Lemma A. The Gauss Lemma Proper B. An Alternative Form of the Lemma C. Is the Exponential Map Ever a Local Isometry? Finsler Manifolds and Metric Spaces A. A Useful Technical Lemma B. Forward Metric Balls and Metric Spheres C. The Manifold Topology Versus the Metric Topology D. Forward Cauchy Sequences, Forward Completeness Short Geodesies Are Minimizing The Smoothness of Distance Functions A. On Minkowski Spaces B. On Finsler Manifolds Long Minimizing Geodesies The Hopf-Rinow Theorem 168 * References for Chapter CHAPTER 7 The Index Form and the Bonnet-Myers Theorem Conjugate Points The Index Form What Happens in the Absence of Conjugate Points? A. Geodesies Are Shortest Among "Nearby" Curves B. A Basic Index Lemma What Happens If Conjugate Points Are Present? The Cut Point Versus the First Conjugate Point 186
5 xviii Contents 7.6 Ricci Curvatures A. The Ricci Scalar Ric and the Ricci Tensor Ricij B. The Interplay between Ric and Ric tj The Bonnet-Myers Theorem 194 * References for Chapter CHAPTER 8 The Cut and Conjugate Loci, and Synge's Theorem Definitions The Cut Point and the First Conjugate Point Some Consequences of the Inverse Function Theorem The Manner in Which Cy and i y Depend on y Generic Properties of the Cut Locus Cut x Additional Properties of Cut x When M Is Compact Shortest Geodesies within Homotopy Classes Synge's Theorem 221 * References for Chapter CHAPTER 9 The Cartan Hadamard Theorem and Rauch's First Theorem Estimating the Growth of Jacobi Fields When Do Local Diffeomorphisms Become Covering Maps? Some Consequences of the Covering Homotopy Theorem The Cartan-Hadamard Theorem Prelude to Rauch's Theorem A. Transplanting Vector Fields B. A Second Basic Property of the Index Form C. Flag Curvature Versus Conjugate Points Rauch's First Comparison Theorem Jacobi Fields on Space Forms Applications of Rauch's Theorem 253 * References for Chapter 9 256
6 Contents xix PART THREE Special Finsler Spaces over the Reals 257 CHAPTER 10 Berwald Spaces and Szabo's Theorem for Berwald Surfaces Prologue Berwald Spaces Various Characterizations of Berwald Spaces Examples of Berwald Spaces A Fact about Flat Linear Connections Characterizing Locally Minkowski Spaces by Curvature Szabo's Rigidity Theorem for Berwald Surfaces A. The Theorem and Its Proof B. Distinguishing between y-local and y-global 279 * References for Chapter CHAPTER 11 Randers Spaces and an Elegant Theorem The Importance of Randers Spaces Randers Spaces, Positivity, and Strong Convexity A Matrix Result and Its Consequences The Geodesic Spray Coefficients of a Randers Metric The Nonlinear Connection for Randers Spaces A Useful and Elegant Theorem The Construction of y-global Berwald Spaces A. The Algorithm B. An Explicit Example in Three Dimensions 306 * References for Chapter CHAPTER 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem Prologue Characterizations of Constant Flag Curvature Useful Interpretations of E and E Growth Rates of Solutions of E + A E = Akbar-Zadeh's Rigidity Theorem Formulas for Machine Computations of K A. The Geodesic Spray Coefficients B. The Predecessor of the Flag Curvature 330
7 xx Contents 12.5 C. Maple Codes for the Gaussian Curvature A Poincare Disc That Is Only Forward Complete A. The Example and Its Yasuda-Shimada Pedigree B. The Finsler Function and Its Gaussian Curvature C. Geodesies; Forward and Backward Metric Discs D. Consistency with Akbar-Zadeh's Rigidity Theorem Non-Riemannian Projectively Flat S 2 with K \ A. Bryant's 2-parameter Family of Finsler Structures B. A Specific Finsler Metric from That Family 345 * References for Chapter CHAPTER 13 Riemannian Manifolds and Two of Hopf's Theorems The Levi-Civita (Christoffel) Connection Curvature A. Symmetries, Bianchi Identities, the Ricci Identity B. Sectional Curvature C. Ricci Curvature and Einstein Metrics Warped Products and Riemannian Space Forms A. One Special Class of Warped Products B. Spheres and Spaces of Constant Curvature C. Standard Models of Riemannian Space Forms Hopf's Classification of Riemannian Space Forms The Divergence Lemma and Hopf's Theorem The Weitzenbock Formula and the Bochner Technique 378 * References for Chapter CHAPTER 14 Minkowski Spaces, the Theorems of Deicke and Brickell Generalities and Examples The Riemannian Curvature of Each Minkowski Space The Riemannian Laplacian in Spherical Coordinates Deicke's Theorem The Extrinsic Curvature of the Level Spheres of F The Gauss Equations The Blaschke-Santalo Inequality The Legendre Transformation A Mixed-Volume Inequality, and Brickell's Theorem 412 * References for Chapter Bibliography 419 Index 427
Flag Curvature. Miguel Angel Javaloyes. Cedeira 31 octubre a 1 noviembre. Universidad de Granada. M. A. Javaloyes (*) Flag Curvature 1 / 23
Flag Curvature Miguel Angel Javaloyes Universidad de Granada Cedeira 31 octubre a 1 noviembre M. A. Javaloyes (*) Flag Curvature 1 / 23 Chern Connection S.S. Chern (1911-2004) M. A. Javaloyes (*) Flag
More informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
More informationGeometry for Physicists
Hung Nguyen-Schafer Jan-Philip Schmidt Tensor Analysis and Elementary Differential Geometry for Physicists and Engineers 4 i Springer Contents 1 General Basis and Bra-Ket Notation 1 1.1 Introduction to
More informationRANDERS METRICS WITH SPECIAL CURVATURE PROPERTIES
Chen, X. and Shen, Z. Osaka J. Math. 40 (003), 87 101 RANDERS METRICS WITH SPECIAL CURVATURE PROPERTIES XINYUE CHEN* and ZHONGMIN SHEN (Received July 19, 001) 1. Introduction A Finsler metric on a manifold
More informationEinstein Finsler Metrics and Ricci flow
Einstein Finsler Metrics and Ricci flow Nasrin Sadeghzadeh University of Qom, Iran Jun 2012 Outline - A Survey of Einstein metrics, - A brief explanation of Ricci flow and its extension to Finsler Geometry,
More informationRiemann Finsler Geometry MSRI Publications Volume 50, Index
Riemann Finsler Geometry MSRI Publications Volume 50, 2004 Index Note. This is a partial index and the level of coverage varies from article to article. If you don t find your entry here, it doesn t mean
More informationSubmanifolds of. Total Mean Curvature and. Finite Type. Bang-Yen Chen. Series in Pure Mathematics Volume. Second Edition.
le 27 AIPEI CHENNAI TAIPEI - Series in Pure Mathematics Volume 27 Total Mean Curvature and Submanifolds of Finite Type Second Edition Bang-Yen Chen Michigan State University, USA World Scientific NEW JERSEY
More informationOrigin, Development, and Dissemination of Differential Geometry in Mathema
Origin, Development, and Dissemination of Differential Geometry in Mathematical History The Borough of Manhattan Community College -The City University of New York Fall 2016 Meeting of the Americas Section
More informationHow curvature shapes space
How curvature shapes space Richard Schoen University of California, Irvine - Hopf Lecture, ETH, Zürich - October 30, 2017 The lecture will have three parts: Part 1: Heinz Hopf and Riemannian geometry Part
More informationDifferential Geometry II Lecture 1: Introduction and Motivation
Differential Geometry II Lecture 1: Introduction and Motivation Robert Haslhofer 1 Content of this lecture This course is on Riemannian geometry and the geometry of submanifol. The goal of this first lecture
More informationIndex. Bertrand mate, 89 bijection, 48 bitangent, 69 Bolyai, 339 Bonnet s Formula, 283 bounded, 48
Index acceleration, 14, 76, 355 centripetal, 27 tangential, 27 algebraic geometry, vii analytic, 44 angle at a corner, 21 on a regular surface, 170 angle excess, 337 angle of parallelism, 344 angular velocity,
More informationAs always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing).
An Interlude on Curvature and Hermitian Yang Mills As always, the story begins with Riemann surfaces or just (real) surfaces. (As we have already noted, these are nearly the same thing). Suppose we wanted
More informationHadamard s Theorem. Rich Schwartz. September 10, The purpose of these notes is to prove the following theorem.
Hadamard s Theorem Rich Schwartz September 10, 013 1 The Result and Proof Outline The purpose of these notes is to prove the following theorem. Theorem 1.1 (Hadamard) Let M 1 and M be simply connected,
More informationFundamentals of Differential Geometry
- Serge Lang Fundamentals of Differential Geometry With 22 luustrations Contents Foreword Acknowledgments v xi PARTI General Differential Theory 1 CHAPTERI Differential Calculus 3 1. Categories 4 2. Topological
More informationCENTROAFFINE HYPEROVALOIDS WITH EINSTEIN METRIC
CENTROAFFINE HYPEROVALOIDS WITH EINSTEIN METRIC Udo Simon November 2015, Granada We study centroaffine hyperovaloids with centroaffine Einstein metric. We prove: the hyperovaloid must be a hyperellipsoid..
More informationStrictly convex functions on complete Finsler manifolds
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 126, No. 4, November 2016, pp. 623 627. DOI 10.1007/s12044-016-0307-2 Strictly convex functions on complete Finsler manifolds YOE ITOKAWA 1, KATSUHIRO SHIOHAMA
More informationOn isometries of Finsler manifolds
On isometries of Finsler manifolds International Conference on Finsler Geometry February 15-16, 2008, Indianapolis, IN, USA László Kozma (University of Debrecen, Hungary) Finsler metrics, examples isometries
More informationCHAPTER 1 PRELIMINARIES
CHAPTER 1 PRELIMINARIES 1.1 Introduction The aim of this chapter is to give basic concepts, preliminary notions and some results which we shall use in the subsequent chapters of the thesis. 1.2 Differentiable
More informationDifferential Geometry, Lie Groups, and Symmetric Spaces
Differential Geometry, Lie Groups, and Symmetric Spaces Sigurdur Helgason Graduate Studies in Mathematics Volume 34 nsffvjl American Mathematical Society l Providence, Rhode Island PREFACE PREFACE TO THE
More informationRichard A. Mould. Basic Relativity. With 144 Figures. Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest
Richard A. Mould Basic Relativity With 144 Figures Springer-Verlag New York Berlin Heidelberg London Paris Tokyo Hong Kong Barcelona Budapest Contents Preface vii PARTI 1. Principles of Relativity 3 1.1
More informationTHE DOUBLE COVER RELATIVE TO A CONVEX DOMAIN AND THE RELATIVE ISOPERIMETRIC INEQUALITY
J. Aust. Math. Soc. 80 (2006), 375 382 THE DOUBLE COVER RELATIVE TO A CONVEX DOMAIN AND THE RELATIVE ISOPERIMETRIC INEQUALITY JAIGYOUNG CHOE (Received 18 March 2004; revised 16 February 2005) Communicated
More informationUNIVERSITY OF DUBLIN
UNIVERSITY OF DUBLIN TRINITY COLLEGE JS & SS Mathematics SS Theoretical Physics SS TSM Mathematics Faculty of Engineering, Mathematics and Science school of mathematics Trinity Term 2015 Module MA3429
More informationRiemannian geometry - Wikipedia, the free encyclopedia
Page 1 of 5 Riemannian geometry From Wikipedia, the free encyclopedia Elliptic geometry is also sometimes called "Riemannian geometry". Riemannian geometry is the branch of differential geometry that studies
More informationSome Open Problems in Finsler Geometry
Some Open Problems in Finsler Geometry provided by Zhongmin Shen March 8, 2009 We give a partial list of open problems concerning positive definite Finsler metrics. 1 Notations and Definitions A Finsler
More informationOn characterizations of Randers norms in a Minkowski space
On characterizations of Randers norms in a Minkowski space XIAOHUAN MO AND LIBING HUANG Key Laboratory of Pure and Applied Mathematics School of Mathematical Sciences Peking University, Beijing 100871,
More informationOn homogeneous Randers spaces with Douglas or naturally reductive metrics
On homogeneous Randers spaces with Douglas or naturally reductive metrics Mansour Aghasi and Mehri Nasehi Abstract. In [4] Božek has introduced a class of solvable Lie groups with arbitrary odd dimension.
More informationKlaus Janich. Vector Analysis. Translated by Leslie Kay. With 108 Illustrations. Springer
Klaus Janich Vector Analysis Translated by Leslie Kay With 108 Illustrations Springer Preface to the English Edition Preface to the First German Edition Differentiable Manifolds 1 1.1 The Concept of a
More informationSYNGE-WEINSTEIN THEOREMS IN RIEMANNIAN GEOMETRY
SYNGE-WEINSTEIN THEOREMS IN RIEMANNIAN GEOMETRY AKHIL MATHEW Abstract. We give an exposition of the proof of a few results in global Riemannian geometry due to Synge and Weinstein using variations of the
More informationOn Berwald Spaces which Satisfy the Relation Γ k ij = p k g ij for Some Functions p k on TM
International Mathematical Forum, 2, 2007, no. 67, 3331-3338 On Berwald Spaces which Satisfy the Relation Γ k ij = p k g ij for Some Functions p k on TM Dariush Latifi and Asadollah Razavi Faculty of Mathematics
More informationNEW TOOLS IN FINSLER GEOMETRY: STRETCH AND RICCI SOLITONS
NEW TOOLS IN FINSLER GEOMETRY: STRETCH AND RICCI SOLITONS MIRCEA CRASMAREANU Communicated by the former editorial board Firstly, the notion of stretch from Riemannian geometry is extended to Finsler spaces
More informationSpacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds
Spacelike surfaces with positive definite second fundamental form in 3-dimensional Lorentzian manifolds Alfonso Romero Departamento de Geometría y Topología Universidad de Granada 18071-Granada Web: http://www.ugr.es/
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationHomogeneous Geodesics of Left Invariant Randers Metrics on a Three-Dimensional Lie Group
Int. J. Contemp. Math. Sciences, Vol. 4, 009, no. 18, 873-881 Homogeneous Geodesics of Left Invariant Randers Metrics on a Three-Dimensional Lie Group Dariush Latifi Department of Mathematics Universit
More informationMonochromatic metrics are generalized Berwald
Monochromatic metrics are generalized Berwald Nina Bartelmeß Friedrich-Schiller-Universität Jena Closing Workshop GRK 1523 "Quantum and Gravitational Fields" 12/03/2018 ina Bartelmeß (Friedrich-Schiller-Universität
More informationFisica Matematica. Stefano Ansoldi. Dipartimento di Matematica e Informatica. Università degli Studi di Udine. Corso di Laurea in Matematica
Fisica Matematica Stefano Ansoldi Dipartimento di Matematica e Informatica Università degli Studi di Udine Corso di Laurea in Matematica Anno Accademico 2003/2004 c 2004 Copyright by Stefano Ansoldi and
More informationAn Overview of Mathematical General Relativity
An Overview of Mathematical General Relativity José Natário (Instituto Superior Técnico) Geometria em Lisboa, 8 March 2005 Outline Lorentzian manifolds Einstein s equation The Schwarzschild solution Initial
More informationInvariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem
PETER B. GILKEY Department of Mathematics, University of Oregon Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem Second Edition CRC PRESS Boca Raton Ann Arbor London Tokyo Contents
More informationChapter 3. Riemannian Manifolds - I. The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves
Chapter 3 Riemannian Manifolds - I The subject of this thesis is to extend the combinatorial curve reconstruction approach to curves embedded in Riemannian manifolds. A Riemannian manifold is an abstraction
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an n-dimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationOn Einstein Kropina change of m-th root Finsler metrics
On Einstein Kropina change of m-th root insler metrics Bankteshwar Tiwari, Ghanashyam Kr. Prajapati Abstract. In the present paper, we consider Kropina change of m-th root metric and prove that if it is
More informationGeometric inequalities for black holes
Geometric inequalities for black holes Sergio Dain FaMAF-Universidad Nacional de Córdoba, CONICET, Argentina. 3 August, 2012 Einstein equations (vacuum) The spacetime is a four dimensional manifold M with
More informationDifferential Geometry of Warped Product. and Submanifolds. Bang-Yen Chen. Differential Geometry of Warped Product Manifolds. and Submanifolds.
Differential Geometry of Warped Product Manifolds and Submanifolds A warped product manifold is a Riemannian or pseudo- Riemannian manifold whose metric tensor can be decomposes into a Cartesian product
More informationИзвестия НАН Армении. Математика, том 46, н. 1, 2011, стр HOMOGENEOUS GEODESICS AND THE CRITICAL POINTS OF THE RESTRICTED FINSLER FUNCTION
Известия НАН Армении. Математика, том 46, н. 1, 2011, стр. 75-82. HOMOENEOUS EODESICS AND THE CRITICAL POINTS OF THE RESTRICTED FINSLER FUNCTION PARASTOO HABIBI, DARIUSH LATIFI, MEERDICH TOOMANIAN Islamic
More informationHomework for Math , Spring 2012
Homework for Math 6170 1, Spring 2012 Andres Treibergs, Instructor April 24, 2012 Our main text this semester is Isaac Chavel, Riemannian Geometry: A Modern Introduction, 2nd. ed., Cambridge, 2006. Please
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationLECTURE 8: THE SECTIONAL AND RICCI CURVATURES
LECTURE 8: THE SECTIONAL AND RICCI CURVATURES 1. The Sectional Curvature We start with some simple linear algebra. As usual we denote by ( V ) the set of 4-tensors that is anti-symmetric with respect to
More informationDIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17
DIFFERENTIAL GEOMETRY, LECTURE 16-17, JULY 14-17 6. Geodesics A parametrized line γ : [a, b] R n in R n is straight (and the parametrization is uniform) if the vector γ (t) does not depend on t. Thus,
More informationTRANSITIVITY OF FINSLER GEODESIC FLOWS IN COMPACT SURFACES WITHOUT CONJUGATE POINTS AND HIGHER GENUS
TRANSITIVITY OF FINSLER GEODESIC FLOWS IN COMPACT SURFACES WITHOUT CONJUGATE POINTS AND HIGHER GENUS JOSÉ BARBOSA GOMES AND RAFAEL O. RUGGIERO Abstract. We show that the geodesic flow of a compact, C 4,
More informationON RANDERS SPACES OF CONSTANT CURVATURE
Vasile Alecsandri University of Bacău Faculty of Sciences Scientific Studies and Research Series Mathematics and Informatics Vol. 25(2015), No. 1, 181-190 ON RANDERS SPACES OF CONSTANT CURVATURE H. G.
More informationMath 230a Final Exam Harvard University, Fall Instructor: Hiro Lee Tanaka
Math 230a Final Exam Harvard University, Fall 2014 Instructor: Hiro Lee Tanaka 0. Read me carefully. 0.1. Due Date. Per university policy, the official due date of this exam is Sunday, December 14th, 11:59
More informationWITH SOME CURVATURE PROPERTIES
Khayyam J. Math. DOI:10.034/kjm.019.8405 ON GENERAL (, β)-metrics WITH SOME CURVATURE PROPERTIES BANKTESHWAR TIWARI 1, RANADIP GANGOPADHYAY 1 GHANASHYAM KR. PRAJAPATI AND Communicated by B. Mashayekhy
More informationRICCI SOLITONS ON COMPACT KAHLER SURFACES. Thomas Ivey
RICCI SOLITONS ON COMPACT KAHLER SURFACES Thomas Ivey Abstract. We classify the Kähler metrics on compact manifolds of complex dimension two that are solitons for the constant-volume Ricci flow, assuming
More informationOn the interplay between Lorentzian Causality and Finsler metrics of Randers type
On the interplay between Lorentzian Causality and Finsler metrics of Randers type Erasmo Caponio, Miguel Angel Javaloyes and Miguel Sánchez Universidad de Granada Spanish Relativity Meeting ERE2009 Bilbao,
More informationOn Douglas Metrics. Xinyue Chen and Zhongmin Shen. February 2, 2005
On Douglas Metrics Xinyue Chen and Zhongmin Shen February 2, 2005 1 Introduction In Finsler geometry, there are several important classes of Finsler metrics. The Berwald metrics were first investigated
More informationA Tour of Subriemannian Geometries,Their Geodesies and Applications
Mathematical Surveys and Monographs Volume 91 A Tour of Subriemannian Geometries,Their Geodesies and Applications Richard Montgomery American Mathematical Society Contents Introduction Acknowledgments
More informationTwo simple ideas from calculus applied to Riemannian geometry
Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University
More informationThe geometry of a positively curved Zoll surface of revolution
The geometry of a positively curved Zoll surface of revolution By K. Kiyohara, S. V. Sabau, K. Shibuya arxiv:1809.03138v [math.dg] 18 Sep 018 Abstract In this paper we study the geometry of the manifolds
More information1 First and second variational formulas for area
1 First and second variational formulas for area In this chapter, we will derive the first and second variational formulas for the area of a submanifold. This will be useful in our later discussion on
More informationMetric Structures for Riemannian and Non-Riemannian Spaces
Misha Gromov with Appendices by M. Katz, P. Pansu, and S. Semmes Metric Structures for Riemannian and Non-Riemannian Spaces Based on Structures Metriques des Varietes Riemanniennes Edited by J. LaFontaine
More informationWARPED PRODUCT METRICS ON R 2. = f(x) 1. The Levi-Civita Connection We did this calculation in class. To quickly recap, by metric compatibility,
WARPED PRODUCT METRICS ON R 2 KEVIN WHYTE Let M be R 2 equipped with the metric : x = 1 = f(x) y and < x, y >= 0 1. The Levi-Civita Connection We did this calculation in class. To quickly recap, by metric
More informationIsometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields
Chapter 15 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields The goal of this chapter is to understand the behavior of isometries and local isometries, in particular
More informationA brief introduction to Semi-Riemannian geometry and general relativity. Hans Ringström
A brief introduction to Semi-Riemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationBERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS
BERGMAN KERNEL ON COMPACT KÄHLER MANIFOLDS SHOO SETO Abstract. These are the notes to an expository talk I plan to give at MGSC on Kähler Geometry aimed for beginning graduate students in hopes to motivate
More informationarxiv: v1 [math.dg] 12 Feb 2013
On Cartan Spaces with m-th Root Metrics arxiv:1302.3272v1 [math.dg] 12 Feb 2013 A. Tayebi, A. Nankali and E. Peyghan June 19, 2018 Abstract In this paper, we define some non-riemannian curvature properties
More informationUNIQUENESS RESULTS ON SURFACES WITH BOUNDARY
UNIQUENESS RESULTS ON SURFACES WITH BOUNDARY XIAODONG WANG. Introduction The following theorem is proved by Bidaut-Veron and Veron [BVV]. Theorem. Let (M n, g) be a compact Riemannian manifold and u C
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationIntroduction to Differential Geometry
More about Introduction to Differential Geometry Lecture 7 of 10: Dominic Joyce, Oxford University October 2018 EPSRC CDT in Partial Differential Equations foundation module. These slides available at
More information5.2 The Levi-Civita Connection on Surfaces. 1 Parallel transport of vector fields on a surface M
5.2 The Levi-Civita Connection on Surfaces In this section, we define the parallel transport of vector fields on a surface M, and then we introduce the concept of the Levi-Civita connection, which is also
More informationNEW LIFTS OF SASAKI TYPE OF THE RIEMANNIAN METRICS
NEW LIFTS OF SASAKI TYPE OF THE RIEMANNIAN METRICS Radu Miron Abstract One defines new elliptic and hyperbolic lifts to tangent bundle T M of a Riemann metric g given on the base manifold M. They are homogeneous
More informationThe Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 5 2009, 045, 7 pages The Explicit Construction of Einstein Finsler Metrics with Non-Constant Flag Curvature Enli GUO, Xiaohuan MO and
More informationConformal transformation between some Finsler Einstein spaces
2 2013 3 ( ) Journal of East China Normal University (Natural Science) No. 2 Mar. 2013 Article ID: 1000-5641(2013)02-0160-07 Conformal transformation between some Finsler Einstein spaces ZHANG Xiao-ling
More informationSection 2. Basic formulas and identities in Riemannian geometry
Section 2. Basic formulas and identities in Riemannian geometry Weimin Sheng and 1. Bianchi identities The first and second Bianchi identities are R ijkl + R iklj + R iljk = 0 R ijkl,m + R ijlm,k + R ijmk,l
More informationGeometric and Spectral Properties of Hypoelliptic Operators
Geometric and Spectral Properties of Hypoelliptic Operators aster Thesis Stine. Berge ay 5, 017 i ii Acknowledgements First of all I want to thank my three supervisors; Erlend Grong, Alexander Vasil ev
More informationA PSEUDO-GROUP ISOMORPHISM BETWEEN CONTROL SYSTEMS AND CERTAIN GENERALIZED FINSLER STRUCTURES. Robert B. Gardner*, George R.
A PSEUDO-GROUP ISOMORPHISM BETWEEN CONTROL SYSTEMS AND CERTAIN GENERALIZED FINSLER STRUCTURES Robert B. Gardner*, George R. Wilkens Abstract The equivalence problem for control systems under non-linear
More informationDerivatives in General Relativity
Derivatives in General Relativity One of the problems with curved space is in dealing with vectors how do you add a vector at one point in the surface of a sphere to a vector at a different point, and
More informationIvan G. Avramidi. Heat Kernel Method. and its Applications. July 13, Springer
Ivan G. Avramidi Heat Kernel Method and its Applications July 13, 2015 Springer To my wife Valentina, my son Grigori, and my parents Preface I am a mathematical physicist. I have been working in mathematical
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the Chern-Weil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationRiemannian Curvature Functionals: Lecture I
Riemannian Curvature Functionals: Lecture I Jeff Viaclovsky Park City athematics Institute July 16, 2013 Overview of lectures The goal of these lectures is to gain an understanding of critical points of
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationThe Riemann Legacy. Riemannian Ideas in Mathematics and Physics KLUWER ACADEMIC PUBLISHERS. Krzysztof Maurin
The Riemann Legacy Riemannian Ideas in Mathematics and Physics by Krzysztof Maurin Division of Mathematical Methods in Physics, University of Warsaw, Warsaw, Poland KLUWER ACADEMIC PUBLISHERS DORDRECHT
More informationMetric and comparison geometry
Surveys in Differential Geometry XI Metric and comparison geometry Jeff Cheeger and Karsten Grove The present volume surveys some of the important recent developments in metric geometry and comparison
More informationH-projective structures and their applications
1 H-projective structures and their applications David M. J. Calderbank University of Bath Based largely on: Marburg, July 2012 hamiltonian 2-forms papers with Vestislav Apostolov (UQAM), Paul Gauduchon
More information7 The cigar soliton, the Rosenau solution, and moving frame calculations
7 The cigar soliton, the Rosenau solution, and moving frame calculations When making local calculations of the connection and curvature, one has the choice of either using local coordinates or moving frames.
More informationDifferential Geometry Exercises
Differential Geometry Exercises Isaac Chavel Spring 2006 Jordan curve theorem We think of a regular C 2 simply closed path in the plane as a C 2 imbedding of the circle ω : S 1 R 2. Theorem. Given the
More informationRepresentation theory and the X-ray transform
AMSI Summer School on Integral Geometry and Imaging at the University of New England, Lecture 1 p. 1/18 Representation theory and the X-ray transform Differential geometry on real and complex projective
More informationLeft-invariant Einstein metrics
on Lie groups August 28, 2012 Differential Geometry seminar Department of Mathematics The Ohio State University these notes are posted at http://www.math.ohio-state.edu/ derdzinski.1/beamer/linv.pdf LEFT-INVARIANT
More informationRiemannian Curvature Functionals: Lecture III
Riemannian Curvature Functionals: Lecture III Jeff Viaclovsky Park City Mathematics Institute July 18, 2013 Lecture Outline Today we will discuss the following: Complete the local description of the moduli
More informationFoundation Modules MSc Mathematics. Winter Term 2018/19
F4A1-V3A2 Algebra II Prof. Dr. Catharina Stroppel The first part of the course will start from linear group actions and study some invariant theory questions with several applications. We will learn basic
More informationProjectively Flat Fourth Root Finsler Metrics
Projectively Flat Fourth Root Finsler Metrics Benling Li and Zhongmin Shen 1 Introduction One of important problems in Finsler geometry is to study the geometric properties of locally projectively flat
More informationElements of differential geometry
Elements of differential geometry R.Beig (Univ. Vienna) ESI-EMS-IAMP School on Mathematical GR, 28.7. - 1.8. 2014 1. tensor algebra 2. manifolds, vector and covector fields 3. actions under diffeos and
More informationInfinitesimal Einstein Deformations. Kähler Manifolds
on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds
More informationA NOTE ON FISCHER-MARSDEN S CONJECTURE
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 3, March 1997, Pages 901 905 S 0002-9939(97)03635-6 A NOTE ON FISCHER-MARSDEN S CONJECTURE YING SHEN (Communicated by Peter Li) Abstract.
More informationIsometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields
Chapter 16 Isometries, Local Isometries, Riemannian Coverings and Submersions, Killing Vector Fields 16.1 Isometries and Local Isometries Recall that a local isometry between two Riemannian manifolds M
More informationUmbilic cylinders in General Relativity or the very weird path of trapped photons
Umbilic cylinders in General Relativity or the very weird path of trapped photons Carla Cederbaum Universität Tübingen European Women in Mathematics @ Schloss Rauischholzhausen 2015 Carla Cederbaum (Tübingen)
More informationTHREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE
THREE-MANIFOLDS OF CONSTANT VECTOR CURVATURE ONE BENJAMIN SCHMIDT AND JON WOLFSON ABSTRACT. A Riemannian manifold has CVC(ɛ) if its sectional curvatures satisfy sec ε or sec ε pointwise, and if every tangent
More informationGROVE SHIOHAMA TYPE SPHERE THEOREM IN FINSLER GEOMETRY
Kondo, K. Osaka J. Math. 52 (2015), 1143 1162 GROVE SHIOHAMA TYPE SPHERE THEOREM IN FINSLER GEOMETRY KEI KONDO (Received September 30, 2014) Abstract From radial curvature geometry s standpoint, we prove
More informationEquivalence, Invariants, and Symmetry
Equivalence, Invariants, and Symmetry PETER J. OLVER University of Minnesota CAMBRIDGE UNIVERSITY PRESS Contents Preface xi Acknowledgments xv Introduction 1 1. Geometric Foundations 7 Manifolds 7 Functions
More informationConnections for noncommutative tori
Levi-Civita connections for noncommutative tori reference: SIGMA 9 (2013), 071 NCG Festival, TAMU, 2014 In honor of Henri, a long-time friend Connections One of the most basic notions in differential geometry
More informationMathematical Relativity, Spring 2017/18 Instituto Superior Técnico
Mathematical Relativity, Spring 2017/18 Instituto Superior Técnico 1. Starting from R αβµν Z ν = 2 [α β] Z µ, deduce the components of the Riemann curvature tensor in terms of the Christoffel symbols.
More informationOn a Class of Weakly Berwald (α, β)- metrics of Scalar flag curvature
International Journal of Mathematics Research. ISSN 0976-5840 Volume 5, Number 1 (2013), pp. 97 108 International Research Publication House http://www.irphouse.com On a Class of Weakly Berwald (α, β)-
More information