Introduction to Differential Geometry

1 Introduction to Differential GeometryLecture Notes for MAT367 Contents1 Introduction .. Some history .. The concept of manifolds : Informal discussion .. manifolds in Euclidean space .. Intrinsic descriptions of manifolds .. Surfaces ..62 manifolds .. Atlases and charts .. Definition of manifold .. Examples of manifolds .. projective spaces .. projective spaces .. Grassmannians .. Oriented manifolds .. Open subsets .. Compact subsets .. Appendix .. relations .. 333 Smooth maps.. Smooth functions on manifolds .. Smooth maps between manifolds .. of manifolds .. Examples of smooth maps .. , diagonal maps .. diffeomorphismRP1 =S1.. diffeomorphismCP1 =S2.. to and from projective space .. quotient mapS2n+1 CPn.. Submanifolds .. Smooth maps of maximal rank .. rank of a smooth map .. diffeomorphisms .. sets, submersions.

2 : The Steiner surface .. Appendix: Algebras .. 694 The tangent bundle.. Tangent spaces .. Tangent map .. of the tangent map, basic properties .. description of the tangent map .. spaces of submanifolds .. : Steiner s surface revisited .. The tangent bundle .. 855 Vector fields.. Vector fields as derivations .. Vector fields as sections of the tangent bundle .. Lie brackets .. Related vector fields .. Flows of vector fields .. Geometric interpretation of the Lie bracket .. Frobenius theorem .. Appendix: Derivations .. 1106 Differential forms.. Review: Differential forms onRm.. Dual spaces .. Cotangent spaces .. 1-forms .. Pull-backs of function and 1-forms .. Integration of 1-forms .. 2-forms .. product .. Differential .. Lie derivatives and contractions .. Integration of Differential forms .. Integration over oriented submanifolds.

3 Stokes theorem .. Volume forms .. 139 ATopology of manifolds .. Topological notions .. manifolds are second countable .. manifolds are paracompact .. Partitions of unity .. 143 BVector bundles.. Tangent bundle .. Vector bundles .. Tangent bundles .. Some constructions with vector bundles .. Dual bundles .. 154 Chapter Some historyIn the words of Chern, the fundamental objects of study in Differential geome-try are manifolds . 1 Roughly, ann-dimensional manifold is a mathematical objectthat locally looks likeRn. The theory of manifolds has a long and complicatedhistory. 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 the1851 thesis of Bernhard Riemann, Grundlagen f ur eine allgemeine Theorie derFunctionen einer ver anderlichen complexen Gr osse ( foundations for a generaltheory of functions of a complex variable ) and his 1854 habilitation address Uberdie Hypothesen, welche der Geometrie zugrunde liegen ( on the hypotheses un-derlying Geometry ).

4 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 Scientific2 Introductionmann Poincar e in his 1895 workanalysis situs, introduces the ideaof amanifold first rigorous axiomatic definition of manifolds was given by Veblen and White-head only in 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 toevolve. Quite possibly, one reason is that for quite a while, the concept as suchwas 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 theoryof General Relativity from 1916 gave a major boost to this new point of view; In histheory, 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 localxyztcoordinates to perform measure-ments, but all physically meaningful quantities must admit formulations that arecoordinate-free.

5 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. Someyears later,gauge theoryonce again emphasized coordinate-free formulations, andprovided physics motivations for more elaborate constructions such as fiber bundlesand the late 1940s and early 1950s, Differential Geometry and the theory ofmanifolds has developed with breathtaking speed. It has become part of the ba-sic education of any mathematician or theoretical physicist, and with applicationsin 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 the article by Scholz aar/ for the long listof names The concept of manifolds : Informal The concept of manifolds : Informal discussionTo repeat, ann-dimensional manifold is something that locally looks likeRn.

6 Theprototype of a manifold is the surface of planet earth:It is (roughly) a 2-dimensional sphere, but we use local charts to depict it as subsetsof 2-dimensional Euclidean describe the entire planet, one uses an atlas with a collection of such charts, suchthat every point on the planet is depicted in at least one such idea will be used to give an intrinsic definition of manifolds , as essentiallya collection of charts glued together in a consistent way. One can then try to de-velop analysis on such manifolds for example, develop a theory of integration anddifferentiation, consider ordinary and partial Differential equations on manifolds , byworking in charts; the task is then to understand the change of coordinates as oneleaves the domain of one chart and enters the domain of that such a chart will always give a somewhat distorted picture of the planet; the distanceson the sphere are never quite correct, and either the areas or the angles (or both) are wrong. Forexample, in the standard maps of the world, Canada always appears somewhat bigger than it reallyis.

7 (Even more so Greenland, of course.)41 manifolds in Euclidean spaceIn multivariable calculus, you will have encountered manifolds as solution sets ofequations. For example, the solution set of an equation of the formf(x,y,z) =ainR3defines a smooth hypersurfaceS R3provided the gradient offis non-vanishing at all points ofS. We call such a value offaregular value, and henceS=f 1(a)aregular level set. Similarly, the joint solution setCof two equationsf(x,y,z) =a,g(x,y,z) =bdefines a smooth curve inR3, provided(a,b)is a regular value of(f,g)in the sensethat the gradients offandgare linearly independent at all points ofC. A familiarexample of a manifold is the 2-dimensional sphereS2, conveniently described as alevel surface insideR3:S2={(x,y,z) R3|x2+y2+z2=1}.There are many ways of introducing local coordinates on the 2-sphere: For exam-ple, one can use spherical polar coordinates, cylindrical coordinates, stereographicprojection, or orthogonal projections onto the coordinate planes.

8 We will discusssome of these coordinates below. More generally, one has then-dimensional sphereSninsideRn+1,Sn={(x0,..,xn) Rn+1|(x0)2+..+(xn)2=1}.The 0-sphereS0consists of two points, the 1-sphereS1is theunit circle. Anotherexample is the2-torus,T2. It is often depicted as a surface of revolution: Given realnumbersr,Rwith 0<r<R, take a circle of radiusrin thex zplane, with centerat(R,0), and rotate about resulting surface6is given by an equation,T2={(x,y,z)|( x2+y2 R)2+z2=r2}.( )Not all surfaces can be realized as embedded inR3; for non-orientable surfacesone needs to allow for self-intersections. This type of realization is referred to as an6 Intrinsic descriptions of manifolds5immersion: We don t allow edges or corners, but we do allow that different parts ofthe surface pass through each other. An example is theKlein bottle7 The Klein bottle is an example of anon-orientable surface: It has only one side. (Infact, the Klein bottle contains a M obius band see exercises.)

9 It is not possible torepresent it as a regular level setf 1(0)of a functionf: For any such surface onehas one side wherefis positive, and another side wherefis Intrinsic descriptions of manifoldsIn this course, we will mostly avoid concrete embeddings of manifolds into , the term embedding is used in an intuitive sense, for example as the real-ization as the level set of some equations. (Later, we will give a precise definition.)There are a number of reasons for why we prefer developing an intrinsic theory Embeddings of simple manifolds in Euclidean space can look quite following one-dimensional manifold8is intrinsically, as a manifold , just a closed curve, that is, a circle. The problemof distinguishing embeddings of a circle intoR3is one of the goals ofknot theory,a deep and difficult area of Such complications disappear if one goes to higher dimensions. For example, theabove knot (and indeed any knot inR3) can be disentangled insideR4(withR3viewed as a subspace).

10 Thus, inR4they The intrinsic description is sometimes much simpler to deal with than the extrin-sic one. For instance, the equation describing the torusT2 R3is not especially7 Introductionsimple or beautiful. But once we introduce the following parametrization of thetorusx= (R+rcos )cos ,y= (R+rcos )sin ,z=rsin ,where , are determined up to multiples of 2 , we recognize thatT2is simplya product:T2=S1 S1.( )That is,T2consists of ordered pairs of points on the circle, with the two factorscorresponding to , . In contrast to ( ), there is no distinction between small circle (of radiusr) and large circle (of radiusR). The new description suggestsan embedding ofT2intoR4which is nicer then the one inR3. (But does ithelp?)4. Often, there is no natural choice of an embedding of a given manifold insideRN,at least not in terms of concrete equations. For instance, while the triple torus9is easily pictured in 3-spaceR3, it is hard to describe it concretely as the level setof an While many examples of manifolds arise naturally as level sets of equations insome Euclidean space, there are also many examples for which the initial con-struction is different.