Draft Edition 2 Draft Copyright '2022 Gabriel Lugo The 2021 published version of this book is licensed under a Creative Commons CCBY-NC-NDlicense. Toviewacopyofthelicense,visithttp://creativecommons.org/licenses. Suggested citation: Lugo, Gabriel. Differential Geometry in Physics. Wilm- ington. University of North Carolina Wilmington William Madison Randall Library, 2021. doi: https: //doi.org/10.5149/9781469669267 Lugo ISBN 978-1-4696-6924-3 (cloth: alk. paper) ISBN 978-1-4696-6925-0 (paperback: alk. paper) ISBN 978-1-4696-6926-7 (open access ebook) Cover illustration: The Hopf fibration. Published by the UNC Wilmington William Madison Randall Library Distributed by UNC Press www.uncpress.org Publication of this book was supported by a grant from the Thomas W. Ross Fund from the University of North Carolina Press. Escherdetailonfollowingpagefrom“Belvedere”plate230,[1]. Allotherfigures andillustrationswereproducedbytheauthorortakenfromphotographstaken by the author and family members. Exceptions are included with appropriate attribution. iii This book is dedicated to my family, for without their love and support, this work would have not been possible. Most importantly, let us not forget that our greatest gift are our children and our children’s children, who stand as a reminder,thatcuriosityandimaginationiswhatdrivesourintellectualpursuits, detachingusfromthemundaneandconfinementtoEarthlyvalues,andbringing us closer to the divine and the infinite. G. Lugo (2021) iv Contents Preface viii 1 Vectors and Curves 1 1.1 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Differentiable Maps . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Curves in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3.1 Parametric Curves . . . . . . . . . . . . . . . . . . . . . . 10 1.3.2 Velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3.3 Frenet Frames . . . . . . . . . . . . . . . . . . . . . . . . 15 1.4 Fundamental Theorem of Curves . . . . . . . . . . . . . . . . . . 22 1.4.1 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.4.2 Natural Equations . . . . . . . . . . . . . . . . . . . . . . 28 2 Differential Forms 36 2.1 One-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 2.2 Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2.2.1 Tensor Products . . . . . . . . . . . . . . . . . . . . . . . 39 2.2.2 Inner Product. . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2.3 Minkowski Space . . . . . . . . . . . . . . . . . . . . . . . 44 2.2.4 Wedge Products and 2-Forms . . . . . . . . . . . . . . . . 45 2.2.5 Determinants . . . . . . . . . . . . . . . . . . . . . . . . 48 2.2.6 Vector Identities . . . . . . . . . . . . . . . . . . . . . . . 51 2.2.7 n-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.3 Exterior Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.1 Pull-back . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.3.2 Stokes’ Theorem in Rn . . . . . . . . . . . . . . . . . . . 62 2.4 The Hodge (cid:63) Operator . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4.1 Dual Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.4.2 Laplacian . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.3 Maxwell Equations . . . . . . . . . . . . . . . . . . . . . . 73 3 Connections 77 3.1 Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.2 Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . . . 80 3.3 Covariant Derivative . . . . . . . . . . . . . . . . . . . . . . . . . 83 v vi CONTENTS 3.4 Cartan Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4 Theory of Surfaces 95 4.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.2 The First Fundamental Form . . . . . . . . . . . . . . . . . . . . 99 4.3 The Second Fundamental Form . . . . . . . . . . . . . . . . . . . 108 4.4 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 4.4.1 Classical Formulation of Curvature . . . . . . . . . . . . . 114 4.4.2 Covariant Derivative Formulation of Curvature . . . . . . 116 4.5 Fundamental Equations . . . . . . . . . . . . . . . . . . . . . . . 122 4.5.1 Gauss-Weingarten Equations . . . . . . . . . . . . . . . . 122 4.5.2 Curvature Tensor, Gauss’s Theorema Egregium . . . . . . 128 5 Geometry of Surfaces 138 5.1 Surfaces of Constant Curvature . . . . . . . . . . . . . . . . . . . 138 5.1.1 Ruled and Developable Surfaces . . . . . . . . . . . . . . 138 5.1.2 Surfaces of Constant Positive Curvature . . . . . . . . . . 138 5.1.3 Surfaces of Constant Negative Curvature . . . . . . . . . 138 5.1.4 B¨acklund Transforms . . . . . . . . . . . . . . . . . . . . 138 5.2 Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 5.2.1 Minimal Area Property . . . . . . . . . . . . . . . . . . . 138 5.2.2 Conformal Mappings . . . . . . . . . . . . . . . . . . . . . 138 5.2.3 Isothermal Coordinates . . . . . . . . . . . . . . . . . . . 138 5.2.4 Stereographic Projection . . . . . . . . . . . . . . . . . . . 138 5.2.5 Minimal Surfaces by Conformal Maps . . . . . . . . . . . 138 6 Riemannian Geometry 139 6.1 Riemannian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . 139 6.2 Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 6.3 Sectional Curvature . . . . . . . . . . . . . . . . . . . . . . . . . 150 6.4 Big D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 6.4.1 Linear Connections . . . . . . . . . . . . . . . . . . . . . . 154 6.4.2 Affine Connections . . . . . . . . . . . . . . . . . . . . . . 155 6.4.3 Exterior Covariant Derivative . . . . . . . . . . . . . . . . 157 6.4.4 Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . 160 6.5 Lorentzian Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . 162 6.6 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 6.7 Geodesics in GR . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 6.8 Gauss-Bonnet Theorem . . . . . . . . . . . . . . . . . . . . . . . 181 References 187 Index 191 CONTENTS vii Preface These notes were developed as part of a course on differential geometry which the author has taught for many years at UNCW. The first five chapters plus chapter six, constitute the foundation of the three-hour course. The course is cross-listedatthelevelofseniorsandfirstyeargraduatestudents. Inadditionto applied mathematics majors, the class usually attracts a good cohort of double majorsinmathematicsandphysics. Materialfromotherchaptershaveinspired a number of honors and master level theses. This book should be accessible to students who have completed traditional training in advanced calculus, linear algebra, and differential equations. Students who master the entirety of this materialwillhavegainedinsightonverypowerfultoolsinmathematicalphysics at the graduate level. There are many excellent texts in differential geometry but very few have an early introduction to differential forms and their applications to physics. It is the purpose of these notes to: 1. Provideabridgebetweentheverypracticalformulationofclassicaldiffer- ential geometry created by early masters of the late 1800’s, and the more elegant but less intuitive modern formulation in terms of manifolds, bun- dles and differential forms. In particular, the central topic of curvature is presented in three different but equivalent formalisms. 2. Present the subject of differential geometry with an emphasis on making the material readable to physicists who may have encountered some of the concepts in the context of classical or quantum mechanics, but wish to strengthen the rigor of the mathematics. A source of inspiration for this goal is rooted in the shock to this author as a graduate student in the 70’s at Berkeley, at observing the gasping failure of communications between the particle physicists working on gauge theories and differential geometers working on connection on fiber bundles. They seemed to be completely unaware at the time, that they were working on the same subject. 3. Make the material as readable as possible for those who stand at the boundary between theoretical physics and applied mathematics. For this reason, it will be occasionally necessary to sacrifice some mathematical rigor or depth of physics, in favor of ease of comprehension. viii 4. Provide the formal geometrical background for the mathematical theory of general relativity. 5. Introduceexamplesofotherapplicationsofdifferentialgeometrytophysics thatmightnotappearintraditionaltextsusedincoursesformathematics students. For example, several students at UNCW have written masters’ theses in the theory of solitons, but usually they have followed the path of Lie symmetries in the style of Olver. We hope that the elegance of B¨acklundtransformswillattractstudentstoageometricapproachtothe subject. The book is also a stepping stone to other interconnected ar- eas of mathematics such as representation theory, complex variables and algebraic topology. G. Lugo (2021) The main change in this second edition is the inclusion of exercises and projects suitable for a course using this textbook. The edition includes correc- tions of errors and misprints that seem to have a way of infiltrating on the first pass of most mathematics books. A list of known errors and misprints is found at the course web site, http://people.uncw.edu/lugo/courses/DiffGeom/index.htm. Theauthorisgratefultoanyreaderspointingoutotherneededcorrectionsand welcomes suggestions for revisions that would improve the content. G. Lugo (2022) Chapter 1 Vectors and Curves 1.1 Tangent Vectors 1.1.1 Definition Euclidean n-space Rn is defined as the set of ordered n- tuplesp(p1,...,pn),wherepi ∈R,fori=1,...,n. Wemayassociateaposition vector p = (p1,...,pn) with any given point p in n-space. Given any two n- tuples p=(p1,...,pn), q=(q1,...,qn) and any real number c, we define two operations: p+q = (p1+q1,...,pn+qn), (1.1) cp = (cp1,...,cpn). These two operations of vector sum and multiplication by a scalar satisfy all the 8 properties needed to give the set V =Rn a natural structure of a vector space. It is common to use the same notation Rn for the space of n-tuples and for the vector space of position vectors. Technically, we should write p ∈ Rn when we think of Rn as a metric space and p ∈ Rn when we think of it as vector space, but as most authors, we will freely abuse the notation. 1 1.1.2 Definition Let xi be the real valued functions in Rn such that xi(p)=pi for any point p = (p1,...,pn). The functions xi are then called the natural coordinate functions. When convenient, we revert to the usual names for the coordinates, x1 = x, x2 = y and x3 = z in R3. A small awkwardness might 1Inthesenoteswewillusethefollowingindexconventions: (cid:136) InRn,indicessuchasi,j,k,l,m,n,runfrom1ton. (cid:136) Inspace-time,indicessuchasµ,ν,ρ,σ,runfrom0to3. (cid:136) OnsurfacesinR3,indicessuchasα,β,γ,δ,runfrom1to2. (cid:136) SpinorindicessuchasA,B,A˙,B˙ runfrom1to2. 1