Differential Geometry in Physics Gabriel Lugo University of North Carolina Wilmington i 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, that curiosity and imagination is what drives our intellectual pursuits, detaching us from the mundane and confinement to Earthly values, and bringing us closer to the divine and the infinite. The 1992 edition of this document was reproduced and assigned an ISBN code, by the Printing services of the University of North Carolina Wilmington, from a camera-ready copy supplied by the author. Thereafter, the document has been made available for free download as the main text for the Differential Geometry course offered in this department. The text was generated on a desktop computer using LATEX. ©1992,1998, 2006, 2020, 2021 Escher detail above from \Belvedere" plate 230, [?]. All other figures and illustrations were produced by the author or taken from photographs taken by the author and family members. Exceptions are included with appropriate attribution. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or trans- mitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the written permission of the author. Printed in the United States of America. ii Preface These notes were developed as part a course on differential geometry at the advanced undergraduate, first year graduate level, which the author has taught for several years. 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 purpose of these notes to: 1. Provide a bridge between the very practical formulation of classical differential geometry created by early masters of the late 1800's, and the more elegant but less intuitive modern formulation in terms of manifolds, bundles 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. 4. Provide the formal geometrical background which constitute the foundation of the mathematical theory of general relativity. 5. Introduce examples of other applications of differential geometry to physics that might not appear in traditional texts used in courses for mathematics students. 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 material will have gained insight in a very powerful tool in mathematical physics at the graduate level. Gabriel Lugo, Ph. D. Mathematical Sciences and Statistics UNCW Wilmington, NC 28403 [email protected] iii iv Contents Preface iii 1 Vectors and Curves 1 1.1 Tangent Vectors . 1 1.2 Differentiable Maps . 6 1.3 Curves in R3 ............................................. 8 1.3.1 Parametric Curves . 8 1.3.2 Velocity . 10 1.3.3 Frenet Frames . 12 1.4 Fundamental Theorem of Curves . 17 1.4.1 Isometries . 19 1.4.2 Natural Equations . 22 2 Differential Forms 27 2.1 One-Forms . 27 2.2 Tensors . 29 2.2.1 Tensor Products . 29 2.2.2 Inner Product . 30 2.2.3 Minkowski Space . 33 2.2.4 Wedge Products and 2-Forms . 34 2.2.5 Determinants . 37 2.2.6 Vector Identities . 39 2.2.7 n-Forms . 41 2.3 Exterior Derivatives . 42 2.3.1 Pull-back . 44 2.3.2 Stokes' Theorem in Rn ................................... 48 2.4 The Hodge ? Operator . 51 2.4.1 Dual Forms . 51 2.4.2 Laplacian . 55 2.4.3 Maxwell Equations . 56 3 Connections 61 3.1 Frames . 61 3.2 Curvilinear Coordinates . 63 3.3 Covariant Derivative . 66 3.4 Cartan Equations . 70 4 Theory of Surfaces 75 4.1 Manifolds . 75 4.2 The First Fundamental Form . 78 4.3 The Second Fundamental Form . 85 4.4 Curvature . 89 4.4.1 Classical Formulation of Curvature . 89 v 0 CONTENTS 4.4.2 Covariant Derivative Formulation of Curvature . 91 4.5 Fundamental Equations . 95 4.5.1 Gauss-Weingarten Equations . 95 4.5.2 Curvature Tensor, Gauss's Theorema Egregium . 101 5 Geometry of Surfaces 109 5.1 Surfaces of Constant Curvature . 109 5.1.1 Ruled and Developable Surfaces . 109 5.1.2 Surfaces of Constant Positive Curvature . 109 5.1.3 Surfaces of Constant Negative Curvature . 109 5.1.4 B¨acklund Transforms . 109 5.2 Minimal Surfaces . 109 5.2.1 Minimal Area Property . 109 5.2.2 Conformal Mappings . 109 5.2.3 Isothermal Coordinates . 109 5.2.4 Stereographic Projection . 109 5.2.5 Minimal Surfaces by Conformal Maps . 109 6 Riemannian Geometry 111 6.1 Riemannian Manifolds . 111 6.2 Submanifolds . 113 6.3 Sectional Curvature . 119 6.4 Big D . 122 6.4.1 Linear Connections . 122 6.4.2 Affine Connections . 123 6.4.3 Exterior Covariant Derivative . 125 6.4.4 Parallelism . 127 6.5 Lorentzian Manifolds . 129 6.6 Geodesics . 133 6.7 Geodesics in GR . 136 6.8 Gauss-Bonnet Theorem . 144 References 147 Index 148 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-tuples p(p1; : : : ; pn), where pi 2 R, for each i = 1; : : : ; n. We may associate a position vector p = (p1; : : : ; pn) with any given point a 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) c p = (c p1; : : : ; c pn): 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 2 Rn when think of Rn as a metric space and p 2 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 occur in the transition to modern notation. In classical vector calculus, a point in Rn is often denoted by x, in which case, we pick up the coordinates with the slot projection functions ui : Rn ! R defined by ui(x) = xi: 1.1.3 Definition A real valued function in Rn is of class Cr if all the partial derivatives of the function up to order r exist and are continuous. The space of infinitely differentiable (smooth) functions will be denoted by C1(Rn) or F (Rn). 1In these notes we will use the following index conventions: In Rn, indices such as i; j; k; l; m; n, run from 1 to n. In space-time, indices such as µ, ν; ρ, σ, run from 0 to 3. On surfaces in R3, indices such as α, β; γ; δ, run from 1 to 2. Spinor indices.