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Differential Geometry Mikhail G. Katz∗

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Differential geometry Mikhail G. Katz∗ M. Katz, Department of Mathematics, Bar Ilan Univer- sity, Ramat Gan 52900 Israel Email address: [email protected] Abstract. Lecture notes for course 88-826 in differential geom- etry on differentiable manifolds via coordinate charts and tran- sition functions, systoles, exterior differential complex, de Rham cohomology, Wirtinger inequality, Gromov’s systolic inequality for complex projective spaces. Contents Chapter1. Differentiablemanifolds 9 1.1. Definitionofdifferentiablemanifold 9 1.2. Metrizability 11 1.3. Hierarchy of smoothness of manifold M 11 1.4. Opensubmanifolds,Cartesianproducts 12 1.5. Circle, tori 13 1.6. Projective spaces 15 1.7. Derivations,Leibnizrule 17 Chapter 2. Derivations, tangent and cotangent bundles 19 2.1. Thespaceofderivations 19 2.2. Tangentbundleofasmoothmanifold 20 2.3. Vectorfieldasasectionofthebundle 22 2.4. Vectorfieldsincoordinates 22 2.5. Representingavectorbyapath 23 2.6. Vectorfieldsinpolarcoordinates 23 2.7. Source,sink,circulation 24 2.8. Dualityinlinearalgebra 25 2.9. Polar,cylindrical,andsphericalcoordinates 26 2.10. Cotangentspaceandcotangentbundle 28 Chapter3. Metricdifferentialgeometry 31 3.1. Isometries, constructing bilinear forms out of 1-forms 31 3.2. Riemannianmetric,firstfundamentalform 32 3.3. Metric as sum of squared 1-forms; element of length ds 33 3.4. Hyperbolic metric 34 3.5. Surface of revolution in R3 35 3.6. Coordinate change 37 3.7. Conformalequivalence 38 3.8. Lattices,uniformisationtheoremfortori 39 3.9. Isothermalcoordinates 41 3.10. Tori of revolution 42 3.11. Standard fundamental domain, conformal parameter τ 42 3.12. Conformalparameterofrectangularlattices 43 3 4 CONTENTS 3.13. Conformal parameter of standard tori of revolution 44 Chapter 4. Differential forms, exterior derivative and algebra 49 4.1. Differential 1-form as section of cotangent bundle 49 4.2. Fromfunctiontodifferential1-form 50 4.3. Space Ω1(M) of differential 1-forms, exterior derivative 51 4.4. Gradient & exterior derivative; musical isomorphisms 51 4.5. Exterior product and algebra, alternating property 53 4.6. Exterioralgebraoveradim1vectorspace 55 4.7. Areasintheplane;signedarea 55 4.8. Algebraiccharacterisationofsignedarea 56 4.9. Vector product, triple product, and wedge product 57 4.10. Anticommutativityofthewedgeproduct 58 4.11. The k-exterior power; simple/decomposable multivs 58 4.12. Basisanddimensionofexterioralgebra 59 Chapter5. Exteriordifferentialcomplex 61 5.1.Rankofamultivector 61 5.2. Rankof2-multivectors;matrixofcoefficients 61 5.3. Constructionofthetensoralgebra 62 5.4. Constructionoftheexterioralgebra 64 5.5. Exteriorbundleandexteriorderivative 64 5.6. Exteriordifferentialcomplex 66 5.7. Antisymmetricmultilinearfunctions 67 5.8. Case of 2-forms 68 Chapter6. Normsonforms,Wirtingerinequality 71 6.1. Norm on 1-forms 71 6.2. Polarcoordinatesanddualnorms 72 6.3. Dual bases and dual lattices in a Euclidean space 73 6.4. Shortestnonzerovectorinalattice 73 6.5. Euclidean norm on k-multivectors and k-forms 74 6.6. Comass norm 74 6.7. Symplecticformfromacomplexviewpoint 75 6.8. Hermitian product 76 6.9. Orthogonaldiagonalisation 78 6.10. Wirtinger inequality 79 6.11. ProofofWirtingerinequality 80 Chapter 7. Complex projective spaces; de Rham cohomology 85 7.1. Celldecompositionofrealprojectivespace 85 7.2. Complex projective line CP1 86 7.3. Complex projective space CPn 87 CONTENTS 5 7.4. Unitary group, projective space as quotient by action 87 7.5. Hopf bundle 88 7.6. Flags and cell decomposition of CPn 88 7.7. Closure of cells produces compact submanifolds 89 7.8. g, α, and J inacomplexvectorspace 90 7.9. RelationtoHermitianproduct 91 7.10. Explicit formula for Fubini–Study 2-form on CP1 92 7.11. Exteriordifferentialcomplexrevisited 93 7.12. DeRhamcocyclesandcoboundaries 94 7.13. De Rham cohomology of M,Bettinumbers 94 Chapter8. DeRhamcohomologyandtopology 97 8.1. De Rham cohomology in dimensions 0 and n 97 8.2. deRhamcohomologyofacircle 98 8.3. Fubini–Study form and the area of CP1 99 8.4. 2-cohomologygroupofthetorus 100 8.5. 2-cohomologygroupofthesphere 102 8.6. Fubini–Study form and volume form on CPn 104 8.7. Cup product, ring structure in de Rham cohomology 105 8.8. Cohomologyofcomplexprojectivespace 106 8.9. Abelianisationingrouptheory 107 8.10. The fundamental group π1(M) 108 Chapter9. Homologygroups;stablenorm 111 9.1. 1-cyclesonmanifolds 111 9.2. Orientationonamanifold 112 9.3. Interior product y ofaformbyavector 113 9.4. Induced orientation on the boundary ∂M 113 9.5. Surfacesandtheirboundaries 114 9.6. Restriction h⇂∂Σg and1-homologygroup 115 9.7. Lengthofcyclesand1-homologyclasses 116 9.8. Length in 1-homology group of orientable surfaces 117 9.9. Stable norm for higher-dimensional manifolds 118 9.10. Systoleandstable1-systole 119 9.11. 2-homologygroupand2-systole 120 9.12. Stablenormin2-homology 121 Chapter 10. Gromov’s stable systolic inequality for CPn 123 10.1. Pu’sinequalityforrealprojectiveplane 123 10.2. Integrationover1-cyclesandover2-cycles 124 10.3. Duality of homology and de Rham cohomology 126 10.4. VolumeformonRiemannianmanifolds 127 k k 10.5. Comass norm in Ω (M) and in HdR(M) 128 6 CONTENTS 10.6. Dualityofcomassandstablenorm 129 10.7. Fubini–Study metric on complex projective line 129 10.8. Fubini–Study metric on complex projective space 131 10.9. Gromov’s inequality for complex projective space 132 10.10. Homology class C and cohomology class ω 133 10.11. Comass and application of Wirtinger inequality 134 10.12. ProofofGromov’sinequality 134 Chapter11. Loewner’sinequality 137 11.1. Eisensteinintegers,Hermiteconstant 137 11.2. Standardfundamentaldomain 138 11.3. Loewner’storusinequality 139 11.4. Expectedvalueandvariance 140 11.5. Application of computational formula for variance 141 11.6. Flattorusaspencilofparallelgeodesics 142 11.7. ProofofLoewner’storusinequality 143 11.8. Boundary case of equality in Loewner’s inequality 144 11.9. Rectangularlattices 145 Chapter12. Pu’sinequalityandgeneralisations 147 12.1. StatementofPu’sinequality 147 12.2. Tangent map 148 12.3. Riemanniansubmersions 148 12.4. A Stiefel manifold 149 12.5. AmetricontheStiefelmanifold 150 12.6. Adoubleintegration 151 12.7. Aprobabilisticinterpretation 152 12.8. Fromthespheretotherealprojectiveplane 152 12.9. AgeneralisationofPu’sinequality 153 Chapter13. AlternativeproofofPu’sinequality 155 13.1. Hopf fibration h 155 13.2. HopffibrationisaRiemanniansubmersion 156 13.3. Hamiltonquaternions 156 13.4. Complex structures on the algebra H 157 13.5. Fromcomplexstructuretofibration 158 13.6. Lie groups 159 13.7. Lie group SO(3) as quotient of S3 160 13.8. Unitspheretangentbundle 161 13.9. 2-sphereasahomogeneousspace 161 13.10. Geodesicflowonthetangentbundle 162 13.11. Dual real projective plane and its double cover 163 13.12. Apairoffibrations 164 CONTENTS 7 13.13. Double fibration of T uS2, integral geometry on S2 165 Chapter 14. Gromov’s inequality for essential manifolds 169 14.1. Essentialmanifolds 169 14.2. Gromov’sinequalityforessentialmanifolds 169 14.3. Fillingradiusofaloopintheplane 169 14.4. The sup-norm 171 14.5. Riemannian manifolds as spaces with a distance function171 14.6. Kuratowskiembedding 171 14.7. Homologytheoryforgroups 171 14.8. Relativefillingradius 172 14.9. Absolutefillingradius 172 14.10. Systolic freedom 172 Bibliography 173 CHAPTER 1 Differentiable manifolds (1) Course site http://u.math.biu.ac.il/~katzmik/88-826.html (2) The final exam is 90% of the grade and the targilim 10%. (3) The first homework assignment is due on 7 april ’21. 1.1. Definition of differentiable manifold An n-dimensional manifold is a set M possessing certain addi- tional properties. A formal definition appears below as Definition 1.1.2. Here M is assumed to be covered by a collection of subsets (called coor- dinate charts or neighborhoods), typically denoted A or B, and having the following properties. For each coordinate neighborhood A M we have an injective map u: A Rn whose image ⊆ → u(A) Rn ⊆ is an open set in Rn. A coordinate chart is the pair (A, u). The maps are required to satisfy the following compatibility condition. Let u: A Rn, u =(ui) , (1.1.1) → i=1,...,n and similarly v : B Rn, v =(vα) (1.1.2) → α=1,...,n be a pair of coordinate charts. Whenever the overlap A B is nonempty, it has a nonempty image v(A B) in Euclidean space.∩ Both u(A) and u(A B), etc., are assumed∩ to be open subsets of Rn. ∩ 1 Definition 1.1.1. Let v− be the inverse map of the map v of (1.1.2). 1 n Thus v− is a map from the image (in R ) of the injective map v back to M. Restricting to the subset v(A B) Rn, we obtain a one-to-one map ∩ ⊆ 1 n φ = u v− : v(A B) R (1.1.3) ◦ ∩ → from an open set v(A B) Rn to Rn. ∩ ⊆1 n n Similarly, the map v u− from the open set u(A B) R to R is one-to-one. We can now◦ state the formal definition.∩ ⊆ 9 10 1.DIFFERENTIABLEMANIFOLDS Definition 1.1.2. A smooth n-dimensional manifold M is a union M = α I Aα, ∪ ∈ n where I is an index set, together with injective maps uα : Aα R , satisfying the following compatibility condition: the map φ of→ (1.1.3) is differentiable for all choices of coordinate neighborhoods A = Aα and B = A (where α, β I) as above. β ∈ 1 1 Definition 1.1.3. The map φ = u v− is called a transition map. The collection of coordinate charts as◦ above is called an atlas for the manifold M. Definition 1.1.4. A 2-dimensional manifold is called a surface. Note that we have not said anything yet about a topology on M. We define a topology on M as follows. Definition 1.1.5. The coordinate charts induce a topology on M by imposing the usual conditions: n 1 (1) If S R is an open set then v− (S) M is defined to be open;⊆2 ⊆ (2) arbitrary unions of open sets in M are open; (3) finite intersections of open sets are open. Remark 1.1.6. We will usually assume that M is connected. Given the manifold structure as above, connectedness of M is equivalent to path-connectedness3 of M.
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  • Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates

    Arxiv:1412.2393V4 [Gr-Qc] 27 Feb 2019 2.6 Geodesics and Normal Coordinates

    Riemannian Geometry: Definitions, Pictures, and Results Adam Marsh February 27, 2019 Abstract A pedagogical but concise overview of Riemannian geometry is provided, in the context of usage in physics. The emphasis is on defining and visualizing concepts and relationships between them, as well as listing common confusions, alternative notations and jargon, and relevant facts and theorems. Special attention is given to detailed figures and geometric viewpoints, some of which would seem to be novel to the literature. Topics are avoided which are well covered in textbooks, such as historical motivations, proofs and derivations, and tools for practical calculations. As much material as possible is developed for manifolds with connection (omitting a metric) to make clear which aspects can be readily generalized to gauge theories. The presentation in most cases does not assume a coordinate frame or zero torsion, and the coordinate-free, tensor, and Cartan formalisms are developed in parallel. Contents 1 Introduction 2 2 Parallel transport 3 2.1 The parallel transporter . 3 2.2 The covariant derivative . 4 2.3 The connection . 5 2.4 The covariant derivative in terms of the connection . 6 2.5 The parallel transporter in terms of the connection . 9 arXiv:1412.2393v4 [gr-qc] 27 Feb 2019 2.6 Geodesics and normal coordinates . 9 2.7 Summary . 10 3 Manifolds with connection 11 3.1 The covariant derivative on the tensor algebra . 12 3.2 The exterior covariant derivative of vector-valued forms . 13 3.3 The exterior covariant derivative of algebra-valued forms . 15 3.4 Torsion . 16 3.5 Curvature .