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Book: Lectures on Differential Geometry Lectures on Differential geometry John W. Barrett 1 October 5, 2017 1Copyright c John W. Barrett 2006-2014 ii Contents Preface .................... vii 1 Differential forms 1 1.1 Differential forms in Rn ........... 1 1.2 Theexteriorderivative . 3 2 Integration 7 2.1 Integrationandorientation . 7 2.2 Pull-backs................... 9 2.3 Integrationonachain . 11 2.4 Changeofvariablestheorem. 11 3 Manifolds 15 3.1 Surfaces .................... 15 3.2 Topologicalmanifolds . 19 3.3 Smoothmanifolds . 22 iii iv CONTENTS 3.4 Smoothmapsofmanifolds. 23 4 Tangent vectors 27 4.1 Vectorsasderivatives . 27 4.2 Tangentvectorsonmanifolds . 30 4.3 Thetangentspace . 32 4.4 Push-forwards of tangent vectors . 33 5 Topology 37 5.1 Opensubsets ................. 37 5.2 Topologicalspaces . 40 5.3 Thedefinitionofamanifold . 42 6 Vector Fields 45 6.1 Vectorsfieldsasderivatives . 45 6.2 Velocityvectorfields . 47 6.3 Push-forwardsofvectorfields . 50 7 Examples of manifolds 55 7.1 Submanifolds . 55 7.2 Quotients ................... 59 7.2.1 Projectivespace . 62 7.3 Products.................... 65 8 Forms on manifolds 69 8.1 Thedefinition. 69 CONTENTS v 8.2 dθ ....................... 72 8.3 One-formsandtangentvectors . 73 8.4 Pairingwithvectorfields . 76 8.5 Closedandexactforms . 77 9 Lie Groups 81 9.1 Groups..................... 81 9.2 Liegroups................... 83 9.3 Homomorphisms . 86 9.4 Therotationgroup . 87 9.5 Complexmatrixgroups . 88 10 Tensors 93 10.1 Thecotangentspace . 93 10.2 Thetensorproduct. 95 10.3 Tensorfields. 97 10.3.1 Contraction . 98 10.3.2 Einstein summation convention . 100 10.3.3 Differential forms as tensor fields . 100 11 The metric 105 11.1 Thepull-backmetric . 107 11.2 Thesignature . 108 12 The Lie derivative 115 12.1 Commutator of vector fields . 115 vi CONTENTS 12.2 Liederivativeoftensors . 116 12.3 Infinitesimalmappings . 118 13 Manifold topics 121 13.1 Compactmanifolds . 121 13.2 Manifoldswithboundary . 123 13.3 Orientation . 125 14 Stokes’ theorem 129 14.1 Thehalf-space . 129 14.2 Stokes’ theorem for manifolds . 131 15 Actions of Lie groups 137 15.1 Definitions. 137 15.2 Classificationofactions . 138 16 Differential criteria 143 16.1 The inverse function theorem . 143 16.2 Theranktheorem . 146 17 Lie algebras 151 17.1 Definitionandexamples . 151 17.2 Homomorphisms . 153 17.3 Structureconstants. 154 18 The exponential map 157 18.1 SubgroupsofGL(n) . 158 CONTENTS vii 18.2 Liealgebraofamatrixgroup . 161 19 The covariant derivative 165 19.1 Coordinateformula. 166 19.2 Torsion .................... 169 19.3 The Christoffel connection . 171 20 Lie algebra of a Lie group 175 20.1 Actiononamanifold. 181 Preface The book is a series of lectures about differential geome- try. It is aimed at students who would like to learn enough differential geometry to understand modern discussions of general relativity and theoretical physics (particularly high-energy physics). However the subject matter does not require any physics knowledge and students in pure mathematics have also benefitted from these lectures. The book is not intended to be a complete reference book. In particular, many things are not proved. Some of these are easy and it is expected that the reader will fill in the gaps, with the help of the exercises. Some other things are not proved because it would take too much effort and additional theory. So in a number of places some key results are just stated and a reader interested in a proof viii CONTENTS is referred elsewhere. The hope is that the book provides the conceptual framework and motivation to tackle other perhaps more formidable books. The book is more than just a list of Things You Should Know, though it is at least that. I have paid particular attention to providing clear and simple definitions and the logical structure of the subject. This is by no means as straightforward as it sounds. Treatments of differential geometry vary endlessly about which aspects are treated as the definitions and which are deductions. The order of the material is also not that of a con- ventional textbook. Different topics are deliberately in- terleaved. The main reason for this is that presenting all of one topic in one dose generally results in indigestion. In fact several topics are resumed when enough time has passed to do and digest the exercises from the first dose. A second reason is that the different topics are related and it is desirable to develop different strands in parallel so that interconnecting examples can be used. A good knowledge of multi-variable calculus is an es- sential prerequisite. It is also desirable to have studied enough linear algebra to be familiar with vector spaces, linear independence and bases. It is very helpful, but not essential, for the reader to have studied some geometry be- fore. Some of the students taking the lecture course at the University of Nottingham have studied general relativity CONTENTS ix at an introductory level, and have thus already seen some of the calculations of vector and tensor components. Some other students have taken pure geometry courses covering curves and surfaces in Euclidean space. It is however pos- sible to study this book without having seen either of these subjects before. The exercises are design to force the reader back to reading the definitions and engage with the concepts. None of them involve fiendish calculations or obscure arguments. Some of them introduce some additional material that is not in the main text, on the grounds that students will learn much more by engaging actively than by reading pas- sively. Thus attempting the exercises is an integral part of reading the book. This book is copyright c John W. Barrett 2006-2014. Copying for the purposes of private study is allowed under the conditions currently displayed on my website johnwbarrett.wordpress.com If this text is missing or you cannot find it, then copying is not permitted. Thanks are due to all the students who have pointed out errors and suggested improvements to the text. Please let me know of any further errors or suggestions. Thanks are also due to Josie Barrett for providing unlimited sup- plies of cake. x CONTENTS Chapter 1 Differential forms 1.1 Differential forms in Rn A point in n-dimensional Euclidean space is written with the notation x =(x1,x2,...,xn) Rn. ∈ Thus the coordinate function x1 is a function from Rn to R. In calculus it is common to use the differentials dx1, dx2,..., dxn in formulae for integration and differentiation. Here these differentials will be defined as differential forms, and it will 1 2 CHAPTER 1. DIFFERENTIAL FORMS be shown how to use them in a geometric context. Many of the formulae of elementary calculus are recovered as particular cases. The differentials are to be regarded as abstract alge- braic symbols, to which one can apply the following oper- ations: 1. Multiply them, with the ‘wedge’ product, denoted , such that ∧ dxi dxj = dxj dxi ∧ − ∧ and dxi dxj dxk =dxi dxj dxk , ∧ ∧ ∧ ∧ so that brackets are not necessary. 2. Take linear combinations, with coefficients depend- ing on x Rn. The wedge product is linear in both ∈ factors. As examples, 1. for i = j implies dxi dxi = 0. Also, ∧ 2. implies, for example, that a(x)dx1 + b(x)dx2 dx3 = a(x)dx1 dx3 +b(x)dx2 dx3. ∧ ∧ ∧ A differential form is said to be of degree p if it has p differentials wedged together, and the form is called a p- form. The set of all differential forms of degree p is denoted Ωp(Rn). It forms a vector space. 1.2. THE EXTERIOR DERIVATIVE 3 Example. On R3, the differential forms are: Ω0(R3) f(x) Ω1(R3) α(x)dx1 + β(x)dx2 + γ(x)dx3 Ω2(R3) a(x)dx1 dx2 + b(x)dx2 dx3 + c(x)dx3 dx1 ∧ ∧ ∧ Ω3(R3) g(x)dx1 dx2 dx3 ∧ ∧ On R3, all p-forms for p > 3 are equal to zero because at least two of the differentials in any term must be the same. 1.2 The exterior derivative The exterior derivative is a linear operator d: Ωq(Rn) Ωq+1(Rn) → defined by n ∂h d h(x)dxj dxk ... = dxi dxj dxk .... ∧ ∧ ∂xi ∧ ∧ ∧ Xi=1 For this formula to make sense, it is obviously neces- sary that h is a differentiable function. It is common to assume that all functions can be differentiated an arbitrary number of times; these are called ‘smooth’ functions. 4 CHAPTER 1. DIFFERENTIAL FORMS Example. In R3, a 0-form (function) f(x) has exterior derivative ∂f ∂f ∂f df = dx1 + dx2 + dx3, ∂x1 ∂x2 ∂x3 so if a 1-form corresponds to a vector field by α dx1 + β dx2 + γ dx3 (α,β,γ), 7→ then df is the gradient of the function f. The definition of d for an arbitrary form can be ex- pressed very simply in terms of d for 1-forms. For example, if ω = f(x)dx1, then ∂f dω = dxi dx1 ∂xi ∧ X ∂f = dxi dx1 ∂xi ∧ =dfXdx1. ∧ The exterior derivative has the following properties 1. Leibniz rule. If τ is an m-form and ω is any differ- ential form, then d(τ ω)=(dτ) ω +( 1)mτ dω. ∧ ∧ − ∧ 2. For any differential form σ, d2σ = 0. 1.2. THE EXTERIOR DERIVATIVE 5 Exercises 1. If f(x,y) = x2 + y2 and g(x,y) = xy, calculate df, dg and df dg in terms of dx and dy. ∧ 2. Simplify d(u v) d(u + v) and d udv vdu , − ∧ ∧ where u and v are functions. 3. If f = xy + x2, show that d(df) = 0. Construct a general proof for an arbitrary f(x,y). 4. In R3, vector fields correspond to 1-forms and 2- forms by (a, b, c) adx + bdy + cdz = ω 7→ (a, b, c) ady dz + bdz dx + cdx dy = σ.
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