Introduction to differential geometry.pdf

INTRODUCTION TODIFFERENTIAL GEOMETRYJoel W.RobbinUW MadisonDietmar A.SalamonETH Z urich19 December 2021iiPrefaceThese are notes for the lecture course“Differential Geometry I”given by thesecond author at ETH Z urich in the fall semester 2017.They are based ona lecture course1given by the first author at the University of WisconsinMadison in the fall semester 1983.One can distinguish extrinsic differential geometry and intrinsic differ-ential geometry.The former restricts attention to submanifolds of Euclideanspace while the latter studies manifolds equipped with a Riemannian metric.The extrinsic theory is more accessible because we can visualize curves andsurfaces in R3,but some topics can best be handled with the intrinsic theory.The definitions in Chapter 2 have been worded in such a way that it is easyto read them either extrinsically or intrinsically and the subsequent chaptersare mostly(but not entirely)extrinsic.One can teach a self contained onesemester course in extrinsic differential geometry by starting with Chapter 2and skipping the sections marked with an asterisk such as 2.8.Here is a description of the content of the book,chapter by chapter.Chapter 1 gives a brief historical introduction to differential geometry andexplains the extrinsic versus the intrinsic viewpoint of the subject.2Thischapter was not included in the lecture course at ETH.The mathematical treatment of the field begins in earnest in Chapter 2,which introduces the foundational concepts used in differential geometryand topology.It begins by defining manifolds in the extrinsic setting assmooth submanifolds of Euclidean space,and then moves on to tangentspaces,submanifolds and embeddings,and vector fields and flows.3Thechapter includes an introduction to Lie groups in the extrinsic setting and aproof of the Closed Subgroup Theorem.It then discusses vector bundles andsubmersions and proves the Theorem of Frobenius.The last two sectionsdeal with the intrinsic setting and can be skipped at first reading.1Extrinsic Differential Geometry2It is shown in 1.3 how any topological atlas on a set induces a topology.3Our sign convention for the Lie bracket of vector fields is explained in 2.5.7.iiiivChapter 3 introduces the Levi-Civita connection as covariant derivativesof vector fields along curves.4It continues with parallel transport,introducesmotions without sliding,twisting,and wobbling,and proves the Develop-ment Theorem.It also characterizes the Levi-Civita connection in terms ofthe Christoffel symbols.The last section introduces Riemannian metrics inthe intrinsic setting,establishes their existence,and characterizes the Levi-Civita connection as the unique torsion-free Riemannian connection on thetangent bundle.Chapter 4 defines geodesics as critical points of the energy functional andintroduces the distance function defined in terms of the lengths of curves.Itthen examines the exponential map,establishes the local existence of min-imal geodesics,and proves the existence of geodesically convex neighbor-hoods.A highlight of this chapter is the proof of the HopfRinow Theoremand of the equivalence of geodesic and metric completeness.The last sectionshows how these concepts and results carry over to the intrinsic setting.Chapter 5 introduces isometries and the Riemann curvature tensor andproves the Generalized Theorema Egregium,which asserts that isometriespreserve geodesics,the covariant derivative,and the curvature.Chapter 6 contains some answers to what can be viewed as the funda-mental problem of differential geometry:When are two manifolds isometric?The central tool for answering this question is the CartanAmbroseHicksTheorem,which etablishes necessary and sufficient conditions for the exis-tence of a(local)isometry between two Riemannian manifolds.The chapterthen moves on to examine flat spaces,symmetric spaces,and constant sec-tional curvature manifolds.It also includes a discussion of manifolds withnonpositive sectional curvature,proofs of the CartanHadamard Theoremand of Cartans Fixed Point Theorem,and as the main example a discussionof the space of positive definite symmetric matrices equipped with a naturalRiemannian metric of nonpositive sectional curvature.This is the point at which the ETH lecture course ended.However,Chapter 6 contains some additional material such as a proof of the BonnetMyers Theorem about manifolds with positive Ricci curvature,and it endswith brief discussions of the scalar curvature and the Weyl tensor.The logical progression of the book up to this point is linear in thatevery chapter builds on the material of the previous one,and so no chaptercan be skipped except for the first.What can be skipped at first readingare only the sections labelled with an asterisk that carry over the variousnotions introduced in the extrinsic setting to the intrinsic setting.4The covariant derivative of a global vector field is deferred to 5.2.2.vChapter 7 deals with various specific topics that are at the heart of thesubject but go beyond the scope of a one semester lecture course.It beginswith a section on conjugate points and the Morse Index Theorem,whichfollows on naturally from Chapter 4 about geodesics.These results in turnare used in the proof of continuity of the injectivity radius in the secondsection.The third section builds on Chapter 5 on isometries and the Rie-mann curvature tensor.It contains a proof of the MyersSteenrod Theorem,which asserts that the group of isometries is always a finite-dimensional Liegroup.The fourth section examines the special case of the isometry group ofa compact Lie group equipped with a bi-invariant Riemannian metric.Thelast two sections are devoted to Donaldsons differential geometric approachto Lie algebra theory as explained in 17.They build on all the previouschapters and especially on the material in Chapter 6 about manifolds withnonpositive sectional curvature.The fifth section establishes conditions un-der which a convex function on a Hadamard manifold has a critical point.The last section uses these results to show that the Killing form on a simpleLie algebra is nondegenerate,to establish uniqueness up to conjugation ofmaximal compact subgroups of the automorphism group of a semisimple Liealgebra,and to prove Cartans theorem about the compact real form of asemisimple complex Lie algebra.The appendix contains brief discussions of some fundamental notions ofanalysis such as maps and functions,normal forms,and Euclidean spaces,that play a central role throughout this book.We thank everyone who pointed out errors or typos in earlier versions ofthis book.In particular,we thank Charel Antony and Samuel Trautwein formany helpful comments.We also thank Daniel Grieser for his constructivesuggestions concerning the exposition.28 August 2021Joel W.Robbin and Dietmar A.SalamonviContents1What is Differential Geometry?11.1Cartography and Differential Geometry.11.2Coordinates.41.3Topological Manifolds*.81.4Smooth Manifolds Defined*.101.5The Master Plan.132Foundations152.1Submanifolds of Euclidean Space.152.2Tangent Spaces and Derivatives.242.2.1Tangent Space.242.2.2Derivative.282.2.3The Inverse Function Theorem.312.3Submanifolds and Embeddings.342.4Vector Fields and Flows.382.4.1Vector Fields.382.4.2The Flow of a Vector Field.412.4.3The Lie Bracket.462.5Lie Groups.522.5.1Definition and Examples.522.5.2The Lie Algebra of a Lie Group.552.5.3Lie Group Homomorphisms.582.5.4Closed Subgroups.622.5.5Lie Groups and Diffeomorphisms.672.5.6Smooth Maps and Algebra Homomorphisms.692.5.7Vector Fields and Derivations.712.6Vector Bundles and Submersions.722.6.1Submersions.722.6.2Vector Bundles.74viiviiiCONTENTS2.7The Theorem of Frobenius.802.8The Intrinsic Definition of a Manifold*.872.8.1Definition and Examples.872.8.2Smooth Maps and Diffeomorphisms.922.8.3Tangent Spaces and Derivatives.932.8.4Submanifolds and Embeddings.952.8.5Tangent Bundle and Vector Fields.972.8.6Coordinate Notation.992.9Consequences of Paracompactness*.1012.9.1Paracompactness.1012.9.2Partitions of Unity.1032.9.3Embedding in Euclidean Space.1072.9.4Leaves of a Foliation1132.9.5Principal Bundles.1143The Levi-Civita Connection1213.1Second Fundamental Form.1213.2Covariant Derivative.1273.3Parallel Transport.1293.4The Frame Bundle.1373.4.1Frames of a Vector Space.1373.4.2The Frame Bundle.1383.4.3Horizontal Lifts.1413.5Motions and Developments.1463.5.1Motion.1463.5.2Sliding.1493.5.3Twisting and Wobbling.1513.5.4Development.1543.6Christoffel Symbols.1603.7Riemannian Metrics*.1663.7.1Existence of Riemannian Metrics.1663.7.2Two Examples.1693.7.3The Levi-Civita Connection.1703.7.4Basic Vector Fields in the Intrinsic Setting.1734Geodesics1754.1Length and Energy.1754.1.1The Length and Energy Functionals.1754.1.2The Space of Paths.1784.1.3Characterization of Geodesics.180CONTENTSix4.2Distance.1834.3The Exponential Map.1914.3.1Geodesic Spray.1914.3.2The Exponential Map.1924.3.3Examples and Exercises.1954.4Minimal Geodesics.1974.4.1Characterization of Minimal Geodesics.1974.4.2Local Existence of Minimal Geodesics.1994.4.3Examples and Exercises.2034.5Convex Neighborhoods.2054.6Completeness and HopfRinow.2094.7Geodesics in the Intrinsic Setting*.2174.7.1Intrinsic Distance.2174.7.2Geodesics and the Levi-Civita Connection.2194.7.3Examples and Exercises.2205Curvature2235.1Isometries.2235.2The Riemann Curvature Tensor.2325.2.1Definition and GauCodazzi.2325.2.2Covariant Derivative of a Global Vector Field.2345.2.3A Global Formula.2375.2.4Symmetries.2395.2.5Riemannian Metrics on Lie Groups.2415.3Generalized Theorema Egregium.2445.3.1Pushforward.2445.3.2Theorema Egregium.2455.3.3Gauian Curvature.2495.4Curvature in Local Coordinates*.2546Geometry and Topology2576.1The CartanAmbroseHicks Theorem.2576.1.1Homotopy.2576.1.2The Global C-A-H Theorem.2596.1.3The Local C-A-H Theorem.2656.2Flat Spaces.2686.3Symmetric Spaces.2726.3.1Symmetric Spaces.2736.3.2Covariant Derivative of the Curvature.2756.3.3Examples and Exercises.278xCONTENTS6.4Constant Curvature.2806.4.1Sectional Curvature.2806.4.2Constant Sectional Curvature.2816.4.3Hyperbolic Space.2866.5Nonpositive Sectional Curvature.2936.5.1The CartanHadamard Theorem.2936.5.2Cartans Fixed Point Theorem.2996.5.3Positive Definite Symmetric Matrices.3026.6Positive Ricci Curvature*.3096.7Scalar Curvature*.3136.8The Weyl Tensor*.3197Topics in Geometry3277.1Conjugate Points and the Morse Index*.3287.2The Injectivity Radius*.3387.3The Group of Isometries*.3427.3.1The MyersSteenrod Theorem.3427.3.2The Topology on the Space of Isometries.3447.3.3Killing Vector Fields.3477.3.4Proof of the MyersSteenrod Theorem.3517.3.5Examples and Exercises.3607.4Isometries of Compact Lie Groups*.3627.5Convex Functions on Hadamard Manifolds*.3687.5.1Convex Functions and The Sphere at Infinity.3687.5.2Inner Products and Weighted Flags.3777.5.3Lengths of Vectors.3817.6Semisimple Lie Algebras*.3917.6.1Symmetric Inner Products.3917.6.2Simple Lie Algebras.3947.6.3Semisimple Lie Algebras.3997.6.4Complex Lie Algebras.404A Notes411A.1 Maps and Functions.411A.2 Normal Forms.412A.3 Euclidean Spaces.414References417Index423Chapter 1What is DifferentialGeometry?This preparatory chapter contains a brief historical introduction to the sub-ject of differential geometry(1.1),explains the concept of a coordinatechart(1.2),discusses topological manifolds and shows how an atlas on aset determines a topology(1.3),introduces the notion of a smooth structure(1.4),and outlines the master plan for this book(1.5).1.1Cartography and Differential GeometryCarl Friedrich Gau(1777-1855)is the father of differential geometry.Hewas(among many other things)a cartographer and many terms in moderndifferential geometry(chart,atlas,map,coordinate system,geodesic,etc.)reflect these origins.He was led to his Theorema Egregium(see 5.3.1)bythe question of whether it is possible to draw an accurate map of a portionof our planet.Let us begin by discussing a mathematical formulation of thisproblem.Consider the two-dimensional sphere S2sitting in the three-dimensionalEuclidean space R3.It is cut out by the equationx2+y2+z2=1.A map of a small region U S2is represented mathematically by a one-to-one correspondence with a small region in the plane z=0.In this book wewill represent this with the notation :U (U)R2and call such anobject a chart or a system of local coordinates(see Figure 1.1).12CHAPTER 1.WHAT IS DIFFERENTIAL GEOMETRY?UFigure 1.1:A chart.What does it mean that is an“accurate”map?Ideally the user wouldwant to use the map to compute the length of a curve in S2.The length ofa curve connecting two points p,q S2is given by the formulaL()=Z10|(t)|dt,(0)=p,(1)=q,so the user will want the chart to satisfy L()=L()for all curves.It is a consequence of the Theorema Egregium that there is no such chart.Perhaps the user of such a map will be content to use the map to plotthe shortest path between two points p and q in U.This path is called ageodesic and is denoted by pq.It satisfies L(pq)=dU(p,q),wheredU(p,q)=infL()|(t)U,(0)=p,(1)=qso our less demanding user will be content if the chart satisfiesdU(p,q)=dE(p),(q),where dE(p),(q)is the length of the shortest path in the plane.It isalso a consequence of the Theorema Egregium that there is no such chart.Now suppose our user is content to have a map which makes it easy tonavigate close to the shortest path connecting two points.Ideally the userwould use a straight edge,magnetic compass,and protractor to do this.S/he would draw a straight line on the map connecting p and q and steer acourse which maintains a constant angle(on the map)between the courseand meridians.This can be done by the method of stereographic projection.This chart is conformal(which means that it preserves angles).Accordingto Wikipedia stereographic projection was known to the ancient Greeksand a map using stereographic projection was constructed in the early 16thcentury.Exercises 3.7.5,3.7.12,and 6.4.22 use stereographic projection;thelatter exercise deals with the Poincar e model of the hyperbolic plane.Thehyperbolic plane provides a counterexample to Euclids Parallel Postulate.1.1.CARTOGRAPHY AND DIFFERENTIAL GEOMETRY3np(p)Figure 1.2:Stereographic Projection.Exercise 1.1.1.It is more or less obvious that for any surface M R3there is a unique shortest path in M connecting two points if they aresufficiently close.(This will be proved in Theorem 4.5.3.)This shortestpath is called the minimal geodesic connecting p and q.Use this fact toprove that the minimal geodesic joining two points p and q in S2is an arcof the great circle through p and q.(This is the intersection of the spherewith the plane through p,q,and the center of the sphere.)Also prove thatthe minimal geodesic connecting two points in a plane is the straight linesegment connecting them.Hint:Both a great circle in a sphere and a linein a plane are preserved by a reflection.(See also Exercise 4.2.5 below.)Exercise 1.1.2.Stereographic projection is defined by the condition thatfor p S2 n the point(p)lies in the xy-pl