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Analytic and Differential Geometry Mikhail G. Katz∗

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Analytic and Differential geometry Mikhail G. Katz∗ Department of Mathematics, Bar Ilan University, Ra- mat Gan 52900 Israel Email address: [email protected] ∗Supported by the Israel Science Foundation (grants no. 620/00-10.0 and 84/03). Abstract. We start with analytic geometry and the theory of conic sections. Then we treat the classical topics in differential geometry such as the geodesic equation and Gaussian curvature. Then we prove Gauss’s theorema egregium and introduce the ab- stract viewpoint of modern differential geometry. Contents Chapter1. Analyticgeometry 9 1.1. Circle,sphere,greatcircledistance 9 1.2. Linearalgebra,indexnotation 11 1.3. Einsteinsummationconvention 12 1.4. Symmetric matrices, quadratic forms, polarisation 13 1.5.Matrixasalinearmap 13 1.6. Symmetrisationandantisymmetrisation 14 1.7. Matrixmultiplicationinindexnotation 15 1.8. Two types of indices: summation index and free index 16 1.9. Kroneckerdeltaandtheinversematrix 16 1.10. Vector product 17 1.11. Eigenvalues,symmetry 18 1.12. Euclideaninnerproduct 19 Chapter 2. Eigenvalues of symmetric matrices, conic sections 21 2.1. Findinganeigenvectorofasymmetricmatrix 21 2.2. Traceofproductofmatricesinindexnotation 23 2.3. Inner product spaces and self-adjoint endomorphisms 24 2.4. Orthogonal diagonalisation of symmetric matrices 24 2.5. Classification of conic sections: diagonalisation 26 2.6. Classification of conics: trichotomy, nondegeneracy 28 2.7. Characterisationofparabolas 30 Chapter 3. Quadric surfaces, Hessian, representation of curves 33 3.1. Summary: classificationofquadraticcurves 33 3.2. Quadric surfaces 33 3.3. Case of eigenvalues (+++) or ( ),ellipsoid 34 3.4. Determination of type of quadric−−− surface: explicit example 35 3.5. Case of eigenvalues (++ ) or (+ ),hyperboloid 36 3.6. Case rank(S) = 2; paraboloid,− hyperbolic−− paraboloid 37 3.7. Cylinder 38 3.8. Exerciseonindexnotation 40 3.9.GradientandHessian 40 3.10. Minima,maxima,saddlepoints 41 3 4 CONTENTS 3.11. Parametricrepresentationofacurve 42 3.12. Implicitrepresentationofacurve 42 3.13. Implicitfunctiontheorem 43 Chapter4. Curvature,Laplacian,Bateman–Reiss 47 4.1. Unitspeedparametrisation 47 4.2. Geodesiccurvatureofacurve 47 4.3. Tangentandnormalvectors 48 4.4. Osculatingcircleofacurve 50 4.5. Centerofcurvature,radiusofcurvature 51 4.6. Flat Laplacian 52 4.7. Bateman–Reissoperator 52 4.8. Geodesiccurvatureforanimplicitcurve 53 4.9. Curvature of circle via DB 53 4.10. Curvature of parabola via DB 54 4.11. Curvature of hyperbola via DB 54 4.12. Maximalcurvatureofatranscendentalcurve 55 4.13. Exploiting the defining equation in studying curvature 56 4.14. Curvatureofgraphoffunction 57 Chapter5. Surfacesandtheircurvature 59 5.1. Existenceofarclength 59 5.2. Arnold’sobservationonfoldingapage 60 5.3. Regularsurface;Jacobian 61 5.4. Coefficients of first fundamental form of a surface 62 5.5. Metriccoefficientsinsphericalcoordinates 62 5.6. Tangentplane;Grammatrix 63 5.7. First fundamental form Ip of M 64 5.8. The form Ip ofplaneandcylinder 64 5.9. Surfacesofrevolution 65 5.10. Fromtractrixtopseudosphere 66 5.11. Chainruleintwovariables 67 5.12. Measuringlengthofcurvesonsurfaces 67 Chapter6. Gammasymbolsofasurface 69 6.1. Normalvectortoasurface 69 6.2.Γsymbolsofasurface 70 6.3. BasicpropertiesoftheΓsymbols 71 6.4. Derivativesofthemetriccoefficients 71 6.5. IntrinsicnatureoftheΓsymbols 72 6.6. Gammasymbolsforasurfaceofrevolution 73 6.7. Spherical coordinates 74 6.8. Great circles 75 CONTENTS 5 6.9. Position vector orthogonal to tangent vector of S2 77 Chapter7. Clairaut’srelation,geodesicequation 79 7.1. Clairaut’s relation 79 7.2. Spherical sine law 79 7.3. Longitudes;length-minimizingproperty 80 7.4. PreliminariestoproofofClairaut’srelation 82 7.5. ProofofClairaut’srelation 83 7.6. Differentialequationofgreatcircle 83 k 7.7. Metrics conformal to the flat metric and their Γij 85 7.8. Gravitation,ants,andgeodesicsonasurface 86 7.9. Geodesicequationonasurface 87 7.10. Equivalenceofdefinitions 88 Chapter 8. Geodesics on surface of revolution, Weingarten map 91 8.1. Geodesicsonasurfaceofrevolution 91 8.2. Integratinggeodesicequationonsurfaceofrev 93 8.3. Areaintegrationinpolarcoordinates 95 8.4. Integration in cylindrical coordinates in R3 95 8.5. Integrationinsphericalcoordinates 96 8.6. Measuringareaonsurfaces 97 8.7. Directional derivative as derivative along path 98 8.8. Extending v.f. along surface to an open set in R3 99 8.9. Differentiating normal vectors n and N 100 8.10. Hessianofafunctionatacriticalpoint 100 8.11. DefinitionoftheWeingartenmap 101 Chapter 9. Gaussian curvature; second fundamental form 103 9.1. PropertiesofWeingartenmap 103 9.2. Weingartenmapisself-adjoint 104 9.3. RelationtotheHessianofafunction 105 9.4. Weingartenmapofsphere 106 9.5. Weingartenmapofcylinder 106 i 9.6. Coefficients L j ofWeingartenmap 107 9.7. Gaussian curvature 107 9.8. Secondfundamentalform 108 9.9. Geodesicsandsecondfundamentalform 109 9.10. ThreeformulasforGaussiancurvature 110 9.11. Principalcurvaturesofsurfaces 111 Chapter10. Minimalsurfaces;Theoremaegregium 117 10.1. Meancurvature,minimalsurfaces 117 10.2. Scherk surface 117 6 CONTENTS 10.3. Minimalsurfacesinisothermalcoordinates 119 10.4. Minimalityandharmonicfunctions 121 10.5. Intro to theorema egregium; Riemann’s formula 122 10.6. An identity involving the Γs and the Ls 125 10.7. The theorema egregium ofGauss 126 10.8. Signed curvature as θ′(s) 127 Chapter 11. Signed curvature of curves; total curvature 131 11.1. Signed curvature with respect to arbitrary parameter 131 11.2. Jordancurves;globalgeometry;Gaussmap 132 11.3. ConvexJordancurves;orientation 133 11.4. PropertiesofGaussmap 134 11.5. TotalcurvatureofaconvexJordancurve 135 11.6. Rotationindexofaclosedcurveintheplane 137 11.7. TotalsignedcurvatureofJordancurve 138 11.8. Gauss–Bonnettheoremforaplanedomain 139 11.9. Gaussian curvature as product of signed curvatures 139 11.10. ConnectiontoHessian 140 11.11. MeanversusGaussiancurvature 141 Chapter12. Laplace–Beltrami,Gauss–Bonnet 143 12.1. TheLaplace–Beltramioperator 143 12.2. Laplace-Beltrami formula for Gaussian curvature 143 12.3. Hyperbolicmetricintheupperhalfplane 145 12.4. Areaelementsofthesurface,orientability 146 12.5. Gauss map for surfaces in R3 147 12.6. Sphere S2 parametrized by Gauss map of M 148 12.7. Comparisonoftwoparametrisations 148 12.8. Gauss–Bonnet theorem for surfaces with K > 0 149 12.9. ProofofGauss–BonnetTheorem 151 12.10. Gauss–Bonnetwithboundary 151 12.11. EulercharacteristicandGauss–Bonnet 152 Chapter 13. Rotation index, isothermisation, pseudosphere 155 13.1. Rotation index of closed curves and total curvature 155 13.2. Surfaces of revolution and isothermalisation 155 13.3. GaussianCurvatureofpseudosphere 157 13.4. Transitionfromclassicaltomoderndiffgeom 158 Chapter 14. Duality 159 14.1. Dualityinlinearalgebra;1-forms 159 14.2. Dual vector space 160 14.3. Derivations 161 CONTENTS 7 14.4. Characterisationofderivations 162 14.5. Tangentspaceandcotangentspace 163 14.6. Constructingbilinearformsoutof1-forms 164 14.7. Firstfundamentalform 164 14.8. Dualbasesindifferentialgeometry 165 14.9.Moreondualbases 166 Chapter15. Latticesandtori 169 15.1. Circleviatheexponentialmap 169 15.2. Lattice,fundamentaldomain 170 15.3. Fundamentaldomain 170 15.4. Latticesintheplane 171 15.5. Successiveminimaofalattice 172 15.6. Gram matrix 173 15.7. Sphereandtorusastopologicalsurfaces 173 15.8. Standardfundamentaldomain 174 15.9. Conformal parameter τ ofalattice 175 15.10. Conformal parameter τ oftoriofrevolution 177 15.11. θ-loops and φ-loopsontoriofrevolution 178 15.12. Torigeneratedbyroundcircles 178 15.13. Conformal parameter of tori of revolution, residues 179 Chapter16. Ahyperrealview 181 16.1. Successive extensions N, Z, Q, R, ∗R 181 16.2. Motivatingdiscussionforinfinitesimals 182 16.3. Introductiontothetransferprinciple 183 16.4. Transfer principle 186 16.5. Ordersofmagnitude 187 16.6. Standardpartprinciple 189 16.7. Differentiation 189 Chapter17. Globalandsystolicgeometry 193 17.1. Definitionofsystole 193 17.2. Loewner’storusinequality 197 17.3. Loewner’sinequalitywithremainderterm 198 17.4. Globalgeometryofsurfaces 198 17.5. Definitionofmanifold 199 17.6. Sphereasamanifold 200 17.7. Dual bases 200 17.8. Jacobian matrix 201 17.9. Areaofasurface,independenceofpartition 202 17.10. Conformalequivalence 203 17.11. Geodesicequation 204 8 CONTENTS 17.12. Closed geodesic 205 17.13. Existenceofclosedgeodesic 206 17.14. Surfacesofconstantcurvature 206 17.15. Realprojectiveplane 206 17.16. Simple loops for surfaces of positive curvature 207 17.17. Successiveminima 208 17.18. Flat surfaces 209 17.19. Hyperbolicsurfaces 209 17.20. Hyperbolic plane 210 Chapter18. Elementsofthetopologyofsurfaces 213 18.1. Loops,simplyconnectedspaces 213 18.2. Orientationonloopsandsurfaces 213 18.3. Cyclesandboundaries 214 18.4. Firstsingularhomologygroup 215 18.5. Stablenormin1-dimensionalhomology 216 18.6.Thedegreeofamap 217 18.7. Degreeofnormalmapofanembeddedsurface 218 18.8. Eulercharacteristicofanorientablesurface 218 18.9. Gauss–Bonnettheorem 219 18.10. Change of metric exploiting Gaussian curvature 219 18.11. Gauss map 220 18.12. An identity 221 18.13. Stable systole 222 18.14. Free loops, based loops, and fundamental group 222 18.15. Fundamentalgroupsofsurfaces 223 Chapter19. Pu’sinequality 225 Chapter 20. Approach To Loewner via energy-area identity 227 20.1. Anintegral-geometricidentity 227 20.2. TwoproofsoftheLoewnerinequality 228 20.3. Remarkoncapacities 230 20.4. Atableofoptimalsystolicratiosofsurfaces 230 Chapter21. Aprimeronsurfaces 233 21.1. Hyperellipticinvolution
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