DIFFERENTIAL GEOMETRY Series of Lecture Notes and Workbooks for Teaching Undergraduate Mathematics Algoritmuselm´elet Algoritmusok bonyolults´aga Analitikus m´odszerek a p´enz ugyekben¨ Bevezet´es az anal´ızisbe Differential Geometry Diszkr´et optimaliz´al´as Diszkr´et matematikai feladatok Geometria Igazs´agoseloszt´asok Interakt´ıv anal´ızis feladatgy˝ujtem´eny matematika BSc hallgat´oksz´am´ara Introductory Course in Analysis Matematikai p´enz ugy¨ Mathematical Analysis-Exercises 1-2 M´ert´ekelm´elet ´es dinamikus programoz´as Numerikus funkcion´alanal´ızis Oper´aci´okutat´as Oper´aci´okutat´asi p´eldat´ar Optim´alis ir´any´ıt´asok Parci´alis differenci´alegyenletek P´eldat´araz anal´ızishez Szimmetrikus kombinatorikai strukt´ur´ak T¨obbv´altoz´osadatelemz´es Bal azs´ Csik os´ DIFFERENTIAL GEOMETRY E¨otv ¨os Lor´and University Faculty of Science Typotex 2014 ➞ 2014–2019, Bal´azs Csik´os, E¨otv ¨os Lor´and University, Faculty of Science Referee: Arp´adKurusa´ Creative Commons NonCommercial-NoDerivs 3.0 (CC BY-NC-ND 3.0) This work can be reproduced, circulated, published and performed for non- commercial purposes without restriction by indicating the author’s name, but it cannot be modified. ISBN 978 963 279 221 7 Prepared under the editorship of Typotex Publishing House (http://www.typotex.hu ) Responsible manager: Zsuzsa Votisky Technical editor: J´ozsef Gerner Made within the framework of the project Nr. T AMOP-4.1.2-08/2/A/KMR-´ 2009-0045, entitled “Jegyzetek ´es p´eldat´arak a matematika egyetemi oktat´a- s´ahoz” (Lecture Notes and Workbooks for Teaching Undergraduate Mathe- matics). KEY WORDS: Curve, Frenet frame, curvature, torsion, hypersurface, funda- mental forms, principal curvature, Gaussian curvature, Minkowski curvature, manifold, tensor field, connection, geodesic curve SUMMARY: The aim of this textbook is to give an introduction to differ- ential geometry. It is based on the lectures given by the author at E ¨otv ¨os Lor´and University and at Budapest Semesters in Mathematics. In the first chapter, some preliminary definitions and facts are collected, that will be used later. The classical roots of modern differential geometry are presented in the next two chapters. Chapter 2 is devoted to the theory of curves, while Chapter 3 deals with hypersurfaces in the Euclidean space. In the last chapter, differentiable manifolds are introduced and basic tools of analysis (differentiation and integration) on manifolds are presented. At the end of Chapter 4, these analytical techniques are applied to study the geometry of Riemannian manifolds. Contents 1 Preliminaries 1 1.1 Categories and Functors . 1 1.2 Linear Algebra . 3 1.2.1 Linear Spaces and Linear Maps . 3 1.2.2 Determinant of Matrices and Linear Endomorphisms . 7 1.2.3 Orientation of a Linear Space . 11 1.2.4 TensorProduct...................... 11 1.2.5 ExteriorPowers...................... 15 1.2.6 Euclidean Linear Spaces . 21 1.2.7 Hodge Star Operator . 30 1.3 Geometry ............................. 33 1.3.1 Affine Geometry . 33 1.3.2 Euclidean Spaces . 40 1.4 Topology . 49 1.4.1 Separation and Countability Axioms . 56 1.4.2 Compactness . 58 1.4.3 Fundamental Group and Covering Spaces . 60 1.5 Multivariable Calculus . 62 1.6 Measure and Integration . 67 1.7 Ordinary Differential Equations . 75 2 Curves in En 81 2.1 The Notion of a Curve . 81 2.2 The Length of a Curve . 83 2.3 Crofton’s Formula . 86 2.4 The Osculating k-planes..................... 92 2.5 Frenet Frames and Curvatures . 97 2.6 Limits of Some Geometrical Quantities . 107 2.7 OsculatingSpheres .... ...... ..... ...... ... 112 2.8 PlaneCurves ........................... 117 i 2.8.1 Evolute, Involute, Parallel Curves . 118 2.8.2 The Rotation Number Theorem . 124 2.8.3 Convex Curves . 127 2.8.4 The Four Vertex Theorem . 129 2.9 Curves in R3 ........................... 131 2.9.1 Orthogonal Projections onto Planes Spanned by Frenet Vectors .......................... 133 2.9.2 Fenchel’s Theorem . 135 2.9.3 The F´ary-Milnor Theorem . 139 3 Hypersurfaces in Rn 141 3.1 General Theory . 141 3.1.1 Definition of a Parameterized Hypersurface . 141 3.1.2 Curvature of Curves on a Hypersurface . 143 3.1.3 The Weingarten Map and the Fundamental Forms . 145 3.1.4 Umbilical Points . 151 3.1.5 The Fundamental Equations of Hypersurface Theory 152 3.1.6 Surface Volume . 161 3.2 Surfaces in R3 ........................... 162 3.2.1 Surfaces of Revolution . 162 3.2.2 Lines of Curvature, Triply Orthogonal Systems . 166 3.2.3 Ruled and Developable Surfaces . 173 3.2.4 Asymptotic Curves on Negatively Curved Surfaces . 179 3.2.5 Surfaces of Constant Negative Curvature . 183 3.2.6 Minimal Surfaces . 195 4 Manifolds 201 4.1 Topological and Differentiable Manifolds . 201 4.1.1 Basic Definitions . 201 4.1.2 Configuration Spaces . 205 4.1.3 Submanifolds of Rn ................... 206 4.1.4 Remarks on the Classification of Manifolds . 209 4.2 The Tangent Bundle . 212 4.2.1 The Tangent Bundle . 217 4.2.2 The Derivative of a Smooth Map . 218 4.3 The Lie Algebra of Vector Fields . 219 4.3.1 The Lie Algebra of Lie Groups . 227 4.3.2 Subspace Distributions and the Frobenius Theorem . 229 4.4 Tensor Bundles and Tensor Fields . 233 4.5 The Lie Derivative . 238 4.6 Differential Forms . 243 4.6.1 Interior Product by a Vector Field . 243 ii 4.6.2 Exterior Differentiation . 245 4.6.3 De Rham Cohomology . 251 4.7 Integration of Differential Forms . 254 4.7.1 Integration on Chains . 255 4.7.2 Integration on Regular Domains . 260 4.7.3 Integration on Riemannian Manifolds . 265 4.8 Differentiation of Vector Fields . 266 4.9 Curvature............................. 275 4.10 Decomposition of Algebraic Curvature Tensors . 290 4.11 Conformal Invariance of the Weyl Tensor . 298 4.12Geodesics ............................. 304 4.13 Applications to Hypersurface Theory . 311 4.13.1 Geodesic Curves on Hypersurfaces . 311 4.13.2 Clairaut’s Theorem . 315 4.13.3 Moving Orthonormal Frames Along a Hypersurface . 317 4.13.4 Relation to Earlier Formulae for Parameterized Hyper- surfaces .......................... 318 4.13.5 The Gauss–Bonnet Formula . 320 4.13.6 Steiner’s Formula . 321 4.13.7 Minkowski’s Formula . 325 4.13.8 Rigidity of Convex Surfaces . 330 Bibliography 336 Index 338 iv Chapter 1 Preliminaries In this chapter, we collect some definitions and facts that will be used later in the text. 1.1 Categories and Functors We shall often use the term natural map or natural isomorphism between two sets carrying certain structures. The concept of naturality can be properly defined within the framework of category theory. Category theory yields a unified way to look at different areas of mathematics and their constructions. Definition 1.1.1. A category consists of C ❼ a class Ob C of objects; ❼ an assignment of a set Mor C(X, Y ) to any pair of objects X, Y Ob C, the elements of which are called morphisms, arrows or maps from∈ X to Y ; ❼ a composition operation Mor C(X, Y ) Mor C(Y, Z ) Mor C(X, Z ), (f, g ) g f for any three objects X, Y,× Z . → 7→ ◦ These should satisfy the following axioms. (i) For any four objects X, Y, Z, W Ob C and any morphisms h Mor (X, Y ), g Mor (Y, Z )∈f Mor (Z, W ), we have ∈ C ∈ C ∈ C (f g) h = f (g h). ◦ ◦ ◦ ◦ (ii) There is a (unique) morphism 1 X Mor C(X, X ) for any object X, called the identity morphism of X,∈ such that for any morphisms f ∈ 1 2 1. Preliminaries Mor (X, Y ) and g Mor (Y, X ), the identities C ∈ C f 1 = f and 1 g = g ◦ X X ◦ hold. ➳ Definition 1.1.2. A morphism f Mor C(X, Y ) is called an isomorphism if it has a two-sided inverse, that is a morphism∈ g Mor (Y, X ) with f g = 1 ∈ C ◦ Y and g f = 1 X . Two objects are isomorphic if there is an isomorphism be- tween◦ them. The morphism f is an endomorphism of X if Y = X. The set Mor C(X, X ) of endomorphisms of X is also denoted by End C(X). The morphism f is an automorphism if f is both an endomorphism and an isomor- phism. Automorphisms of an object X form a group Aut C(X) with respect to the composition operation. ➳ A basic example is the category of sets, in which the objects are the sets, Mor( X, Y ) is the set of all maps from X to Y , is the ordinary composition ◦ of maps, 1 X is the identity map of X. Isomorphisms of this category are the bijective maps. Two sets are isomorphic in this category if and only if they have the same cardinality. Aut( X) is the group of permutations of the elements of X. Definition 1.1.3. A diagram is a directed graph, the vertices of which are objects of a category and the edges are labeled by morphisms from the initial point of the edge to the endpoint. Any directed path in a diagram gives rise to morphism from the initial point of the path to its endpoint obtained as the composition of morphisms attached to the consecutive edges of the path. A diagram is said to be commutative when, for each pair of vertices X and Y and for any two directed paths from X to Y , the compositions of the edge labels of the paths are equal morphism from X to Y . ➳ Definition 1.1.4. A covariant functor F from a category to a category associates to each object X of the category an object F (XC) Ob ; and toD C ∈ D each morphism f Mor C(X, Y ) a morphism F (f) Mor D(F (X), F (Y )) in such a way that ∈ ∈ ❼ F (1 ) = 1 for any object X Ob ; X F (X) ∈ C ❼ F (g f) = F (g) F (f) for any f Mor (X, Y ) and g Mor (Y, Z ).