Introduction to differential geometry.pdf
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1、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 autho
2、r 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 extrins
3、ic 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 mo
4、stly(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 his
5、torical 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 d
6、ifferential 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 an
7、d 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
8、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,
9、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 Riemannia
10、n 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 o
11、f 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 cu
12、rvature 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 centra
13、l 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 manifold
14、s.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 sec
15、tional 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 logi
16、cal 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 introduce
17、d 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
18、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 proo
19、f 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 dif
20、ferential 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 h
21、as 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
22、 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 oft
23、his 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 Differentia
24、l 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.
25、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 M
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