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Introduction to Differential Geometry

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Introduction to Differential Geometry Introduction to Differential Geometry Lecture Notes for MAT367 Contents 1 Introduction ................................................... 1 1.1 Some history . .1 1.2 The concept of manifolds: Informal discussion . .3 1.3 Manifolds in Euclidean space . .4 1.4 Intrinsic descriptions of manifolds . .5 1.5 Surfaces . .6 2 Manifolds ..................................................... 11 2.1 Atlases and charts . 11 2.2 Definition of manifold . 17 2.3 Examples of Manifolds . 20 2.3.1 Spheres . 21 2.3.2 Products . 21 2.3.3 Real projective spaces . 22 2.3.4 Complex projective spaces . 24 2.3.5 Grassmannians . 24 2.3.6 Complex Grassmannians . 28 2.4 Oriented manifolds . 28 2.5 Open subsets . 29 2.6 Compact subsets . 31 2.7 Appendix . 33 2.7.1 Countability . 33 2.7.2 Equivalence relations . 33 3 Smooth maps .................................................. 37 3.1 Smooth functions on manifolds . 37 3.2 Smooth maps between manifolds . 41 3.2.1 Diffeomorphisms of manifolds . 43 3.3 Examples of smooth maps . 45 3.3.1 Products, diagonal maps . 45 3.3.2 The diffeomorphism RP1 =∼ S1 ......................... 45 -3 -2 Contents 3.3.3 The diffeomorphism CP1 =∼ S2 ......................... 46 3.3.4 Maps to and from projective space . 47 n+ n 3.3.5 The quotient map S2 1 ! CP ........................ 48 3.4 Submanifolds . 50 3.5 Smooth maps of maximal rank . 55 3.5.1 The rank of a smooth map . 56 3.5.2 Local diffeomorphisms . 57 3.5.3 Level sets, submersions . 58 3.5.4 Example: The Steiner surface . 62 3.5.5 Immersions . 64 3.6 Appendix: Algebras . 69 4 The tangent bundle ............................................. 71 4.1 Tangent spaces . 71 4.2 Tangent map . 76 4.2.1 Definition of the tangent map, basic properties . 76 4.2.2 Coordinate description of the tangent map . 78 4.2.3 Tangent spaces of submanifolds . 80 4.2.4 Example: Steiner’s surface revisited . 84 4.3 The tangent bundle . 85 5 Vector fields ................................................... 87 5.1 Vector fields as derivations . 87 5.2 Vector fields as sections of the tangent bundle . 89 5.3 Lie brackets . 90 5.4 Related vector fields . 94 5.5 Flows of vector fields . 96 5.6 Geometric interpretation of the Lie bracket . 104 5.7 Frobenius theorem . 107 5.8 Appendix: Derivations . 110 6 Differential forms ..............................................113 m 6.1 Review: Differential forms on R ............................. 113 6.2 Dual spaces . 115 6.3 Cotangent spaces . 117 6.4 1-forms . 118 6.5 Pull-backs of function and 1-forms . 120 6.6 Integration of 1-forms . 122 6.7 2-forms . 123 6.8 k-forms . 124 6.8.1 Definition . 124 6.8.2 Wedge product . 125 6.8.3 Exterior differential . 126 6.9 Lie derivatives and contractions . 128 6.9.1 Pull-backs . 130 Contents -1 6.10 Integration of differential forms . 132 6.11 Integration over oriented submanifolds . 133 6.12 Stokes’ theorem . 133 6.13 Volume forms . 139 A Topology of manifolds ..........................................141 A.1 Topological notions . 141 A.2 Manifolds are second countable. 142 A.3 Manifolds are paracompact . 142 A.4 Partitions of unity . 143 B Vector bundles .................................................147 B.1 Tangent bundle . 147 B.1.1 Vector bundles . 148 B.1.2 Tangent bundles . 150 B.1.3 Some constructions with vector bundles . 151 B.2 Dual bundles . 154 Chapter 1 Introduction 1.1 Some history In the words of S.S. Chern, ”the fundamental objects of study in differential geome- try are manifolds.” 1 Roughly, an n-dimensional manifold is a mathematical object n that “locally” looks like R . The theory of manifolds has a long and complicated history. For centuries, manifolds have been studied as subsets of Euclidean space, given for example as level sets of equations. The term ‘manifold’ goes back to the 1851 thesis of Bernhard Riemann, “Grundlagen fur¨ eine allgemeine Theorie der Functionen einer veranderlichen¨ complexen Grosse”¨ (“foundations for a general theory of functions of a complex variable”) and his 1854 habilitation address “Uber¨ die Hypothesen, welche der Geometrie zugrunde liegen” (“on the hypotheses un- derlying geometry”). 2 However, in neither reference Riemann makes an attempt to give a precise defi- nition of the concept. This was done subsequently by many authors, including Rie- 1 Page 332 of Chern, Chen, Lam: Lectures on Differential Geometry, World Scientific 2 http://en.wikipedia.org/wiki/Bernhard_Riemann 1 2 1 Introduction mann himself. 3 Henri Poincare´ in his 1895 work analysis situs, introduces the idea of a manifold atlas. 4 The first rigorous axiomatic definition of manifolds was given by Veblen and White- head only in 1931. We will see below that the concept of a manifold is really not all that compli- cated; and in hindsight it may come as a bit of a surprise that it took so long to evolve. Quite possibly, one reason is that for quite a while, the concept as such was mainly regarded as just a change of perspective (away from level sets in Eu- clidean spaces, towards the ‘intrinsic’ notion of manifolds). Albert Einstein’s theory of General Relativity from 1916 gave a major boost to this new point of view; In his theory, space-time was regarded as a 4-dimensional ‘curved’ manifold with no dis- tinguished coordinates (not even a distinguished separation into ‘space’ and ‘time’); a local observer may want to introduce local xyzt coordinates to perform measure- ments, but all physically meaningful quantities must admit formulations that are coordinate-free. At the same time, it would seem unnatural to try to embed the 4- dimensional curved space-time continuum into some higher-dimensional flat space, in the absence of any physical significance for the additional dimensions. Some years later, gauge theory once again emphasized coordinate-free formulations, and provided physics motivations for more elaborate constructions such as fiber bundles and connections. Since the late 1940s and early 1950s, differential geometry and the theory of manifolds has developed with breathtaking speed. It has become part of the ba- sic education of any mathematician or theoretical physicist, and with applications in other areas of science such as engineering or economics. There are many sub- branches, for example complex geometry, Riemannian geometry, or symplectic ge- ometry, which further subdivide into sub-sub-branches. 3 See e.g. the article by Scholz http://www.maths.ed.ac.uk/ aar/papers/scholz.pdf for the long list of names involved. 4 http://en.wikipedia.org/wiki/Henri_Poincare 1.2 The.
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