University of Texas at Tyler Scholar Works at UT Tyler Math Theses Math Spring 5-5-2015 Differential Geometry: Curvature and Holonomy Austin Christian Follow this and additional works at: https://scholarworks.uttyler.edu/math_grad Part of the Mathematics Commons Recommended Citation Christian, Austin, "Differential Geometry: Curvature and Holonomy" (2015). Math Theses. Paper 5. http://hdl.handle.net/10950/266 This Thesis is brought to you for free and open access by the Math at Scholar Works at UT Tyler. It has been accepted for inclusion in Math Theses by an authorized administrator of Scholar Works at UT Tyler. For more information, please contact [email protected]. DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY by AUSTIN CHRISTIAN A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Mathematics David Milan, Ph.D., Committee Chair College of Arts and Sciences The University of Texas at Tyler May 2015 c Copyright by Austin Christian 2015 All rights reserved Acknowledgments There are a number of people that have contributed to this project, whether or not they were aware of their contribution. For taking me on as a student and learning differential geometry with me, I am deeply indebted to my advisor, David Milan. Without himself being a geometer, he has helped me to develop an invaluable intuition for the field, and the freedom he has afforded me to study things that I find interesting has given me ample room to grow. For introducing me to differential geometry in the first place, I owe a great deal of thanks to my undergraduate advisor, Robert Huff; our many fruitful conversations, mathematical and otherwise, con- tinue to affect my approach to mathematics. For their helpful comments and criticisms, I would also like to thank my committee members, Sheldon Davis and Regan Beckham. Less conspicuously, my classmates have helped me to mature mathematically and to hone my expository skills; they ought to share the credit for what clarity may be found in my exposition, and bear none of the blame for its shortcomings. I would be remiss to omit my father, who introduced me to mathematics, my mother, who has encouraged my efforts, or my siblings, grandmothers, and other family members, all of whom have been supportive throughout my education. Finally, the greatest gratitude is owed to my wonderful wife, Kristin. She encour- aged me to write when I was tired, forgave the late nights I spent on this project, and, most importantly, agreed to the penury of graduate school. I would be foolish to believe that this thesis could have been completed without her support, making it as much her accomplishment as my own. Contents List of Figures ..................................... iii Abstract ......................................... iv 1 Introduction ..................................... 1 2 Manifolds: Topological and Smooth ...................... 4 2.1 Topological Manifolds . 4 2.2 Partitions of Unity . 9 2.3 Topological Imbeddings . 13 2.4 Smooth Manifolds . 14 3 The Tangent Space ................................. 20 3.1 Tangent Vectors . 20 3.2 Smooth Maps and Their Differentials . 28 3.3 Smooth Vector Fields . 33 3.4 Local Flows . 39 4 Lie Groups ...................................... 47 4.1 The Classical Matrix Groups . 47 4.2 Lie Groups and Lie Algebras . 52 4.3 More on Lie Groups . 58 5 Bundles ........................................ 64 5.1 Vector Bundles . 64 5.2 Frames and Distributions . 67 5.3 Fiber Bundles . 71 6 Differential Forms .................................. 75 6.1 Tensors . 75 6.2 Alternating Tensors . 78 6.3 Differential Forms . 84 i 7 Riemannian Geometry ............................... 96 7.1 Affine Connections . 96 7.2 Riemannian Manifolds . 114 7.3 The Volume Form . 122 7.4 Gauss Curvature . 127 8 Curvature and Holonomy ............................. 133 8.1 Connections on Principal Bundles . 133 8.2 The Curvature Form and the Structural Equation . 139 8.3 The Ambrose-Singer Theorem . 145 ii List of Figures 2.1 Stereographic projection of S1........................... 5 2.2 Stereographic projection of S2........................... 8 2.3 A transition map. 15 3.1 The tangent bundle of S1............................. 31 2 3.2 Flow lines in R .................................. 42 3.3 The Lie Derivative. 44 4.1 A special linear transformation. 48 4.2 The Lie group S1.................................. 53 4.3 Integral curves of vector fields that do not commute. 63 5.1 The M¨obiusbundle. 65 7.1 A vector field along a curve. 97 1 2 7.2 Parallel transport about S in R ......................... 99 7.3 A covariant derivative. 101 7.4 Parallel transport on S2..............................107 8.1 A vertical vector. 134 8.2 Horizontal motion in our new connection. 137 iii Abstract DIFFERENTIAL GEOMETRY: CURVATURE AND HOLONOMY Austin Christian Thesis chair: David Milan, Ph.D. The University of Texas at Tyler May 2015 We develop the basic language of differential geometry, including smooth manifolds, bundles, and differential forms. Once this background is established, we explore paral- lelism in smooth manifolds | in particular, in Riemannian manifolds | and conclude by presenting a proof of the Ambrose-Singer theorem, which relates parallelism (holonomy) to curvature in principal bundles. iv Chapter 1 Introduction In the spring semester of 2014, David Milan | an analyst by trade | agreed to mentor the author in an independent study of differential geometry at UT Tyler. Over the course of that semester, the independent study covered much of the material found in [5], and the study spilled into the summer with the reading of [1], which was supported by many of the sources found in the References section of this paper. This thesis is essentially a report on that independent study, with Chapters 2 through 7 covering the background material learned from textbooks and selected readings, and with Chapter 8 reproducing Ambrose and Singer's proof of their theorem on holonomy found in [1]. In its earliest form, geometry could rightly be called the study of flat space. Cer- tainly Euclid's Elements concerned itself primarily with the structure of two-dimensional space and the manner in which its elements relate to one another. Eventually, however, flat two-dimensional (and even three-dimensional) Euclidean space could no longer contain mathematical curiosities. A study was needed of spaces which \live" in 3-space while having the local appearance of 2-space | for example, the surface on which we live. For centuries surfaces were studied by a variety of mathematicians, but, as is the case in many fields of mathematics, it was Gauss who made the first substantial progress in studying the metric properties of surfaces at an infinitesimal level. In 1822, in response to a prize problem of the Royal Society of Sciences of Copenhagen, Gauss presented what is today called a conformal representation of one surface upon another [17, p. 463]. This is a map between surfaces which preserves the \metric" properties, such as length and angle. Then, in 1827, Gauss proved the remarkable fact that the curvature of a surface, a measure of its shape, is an intrinsic property, meaning that it can be determined from infinitesimal measurements on the surface [11]. But it was a student of Gauss | Riemann | that considered both Euclidean space and surfaces to in fact be special cases of manifolds. In 1854, Riemann presented his habilitation thesis at the University of G¨ottingen,which was entitled \On the Hypotheses which Lie at the Foundations of Geometry." In it, he introduced the notion of an \n-fold extended manifold" and argued that Space...constitutes only a particular case of a triply extended magnitude. A nec- essary sequel of this is that the propositions of geometry are not derivable from general concepts of quantity, but that those properties by which space is dis- tinguished from other conceivable triply extended magnitudes can be gathered only from experience [17, pp. 411-412]. From Riemann's viewpoint, we should not be studying geometry in Euclidean n-space, but studying geometry in any space of dimension n, and determing the characteristics of our 1 space which make its geometry different from the Euclidean case. The goal of the present paper is to develop the background in smooth manifolds (mani- folds on which differential calculus makes sense) necessary to study two important concepts: curvature and parallelism. There are various definitions for curvature on a manifold, but all of them attempt to describe the \shape" of the manifold. Parallelism, on the other hand, regulates what motions in a manifold we consider to be \straight". In Chapter 7 we will study each of these notions in the case of a Riemannian manifold, which is a smooth manifold on which the usual geometric properties are defined. In Chapter 8 we present a more general look at each of these concepts, and relate them to one another with the Ambrose-Singer theorem [1]. In Chapters 2 and 3 we introduce the basic terminology of smooth manifold theory. This begins with defining topological manifolds, giving topological results, and developing smooth manifolds from the topological variety. We then proceed to characterize the tan- gent space of a smooth manifold, interpreting this space both as a collection of infinitesimal curves in the manifold and as the collection of directional derivatives of real-valued func- tions on the manifold. All of the material in these chapters is entirely standard. To a great extent these chapters follow corresponding chapters in [5], as this was the text used in the aforementioned independent study. It should be noted, however, that we flesh out many of the details which were left to the reader in that text. In many cases this means provid- ing proofs which were designated exercises by [5]; in such cases, the originality of the proof is denoted by a y symbol, though this practice is abandoned beyond these first two chapters.