Geometry of Manifolds
Geometry of Manifolds MATH 7350
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This 8 page Class Notes was uploaded by Alvena McDermott on Saturday September 19, 2015. The Class Notes belongs to MATH 7350 at University of Houston taught by Andrei Torok in Fall. Since its upload, it has received 36 views. For similar materials see /class/208387/math-7350-university-of-houston in Mathmatics at University of Houston.
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Date Created: 09/19/15
Geometry of Manifolds a previewquot These notes are an informal presentation of a few Differential Geometry topics Why manifolds The simplest77 higher dimensional objects are the vector spaces manifolds versus vector spaces ltgt curves versus straight liries The vector spaces are the simplest manifolds Many objects that one encounters in modern mathematics are manifolds Here are two examples The unit sphere in R3 is a manifold however locally it looks very much like a plane no wonder people thought for a while that the Earth is at This is roughly speaking the general idea of a manifold it can be modeled locally by a vector space According to Einstein7s General Theory of Relativity our physical world ie the space time is a four dimensional manifold which is determined shaped by matter Objects eg planets and stars move on the shortest77 trajectories called geodesics As soon as matter is present the universe differs from the canonical R4 and the geodesics are not always straight lines In our solar system the planets move on geodesics of space time whose space component can be well approximated by the ellipses etc predicted by Newton7s theory of gravity Newton developed his theory to work in the at R3 In some cases it is natural to endow manifolds with additional structures Eg the natural set up for classical mechanics are symplectic manifolds Surfaces in R3 inherit a Riemarmiari structure which describes the distance On the other hand relativity is described using the Minkowski metric 1 Manifolds Unless speci ed otherwise the manifolds are without boundary The following is a correct statement although not really a de nition Last updated Jan 20 2009 De nition 11 An n dlmcnslonal manifold is a Hausdorff topological space that is lo cally di eomorphic to an open subset of R For convenience we will consider manifolds that are second countable and connected Remark 12 We describerecall brie y some of the notions mentioned above 0 A map p U a V is a CTdi eomorphism if it is C it is invertible and its inverse is also 07 We leave it vague what U and V are the familiar case is when they are open sets in the vector spaces R5 The order of smoothness r can be from 1 to 00 For r 0 the corresponding maps are called homeomorphisms and we speak of topological manifolds If U C R and V C R then a map F U a V is given by m functions f U a R F f1 f2 fm each function having n variables A mapping F f1 f2 fm is 07 if each of its components is C If U C R is an open set then a function f U a R is 07 if all of its partial derivatives up to order r exist and these derivatives are continuous De nition 13 A property is local if it is determined by information on a neighborhood of each point Eg the derivative of a function or the radius of curvature of a curve are local quantities A property is pointwise if it is determined by the value only at the point eg the value of a function is a pointwise quantity A property is global if it is determined by the values on the whole set eg the value of the integral fo dd Now we can expand what De nition77 11 says a manifold is a topological space M so that each point p E M has a neighborhood U which is diffeomorphic to an open subset of R Attention we still did not de ne all the terms In other words as far as di erentiability is concerned locally an n dimensional manifold is the same as an open subset of R 11 Examples of manifolds curves and surfaces a One dimensional manifolds E curves Up to diffeomorphism there are only two compact connected 1 dimensional manifolds 1 the circle a manifold without boundary S1L E R2 l 2 y2 1 the standard choice is the unit circle 2 the closed interval a manifold with boundary I 01 we take this interval only for convenience 2 b Two dimensional manifolds E surfaces 1 The unit sphere 82 7 mm 6 R3 1 mg 22 71 2 All a entable compact surfaces are spheres with handles attached Figure 1 Sphere with zero7 one7 and two handles 3 The Mo39bius strip is a nonorientable compact surface with boundary7 which can be described either as 7 glue the two ends of a strip of paper7 after turning one end by 1800 or 7 the unit square I gtlt I subject to the identi cations 07y E 1717y glue points on left and right edges across As described above7 one can easily realize this surface in R3 A 4 V The Klein bottle is a nonorientable compact surface without boundary7 which can be described either as 7 two Mo39bius strips with the boundaries glued together or 7 the unit square I gtlt I subject to the identi cations x7 0 E 71 glue top and bottom edges7 to get a cylinder 07y E 1717y glue points on left and right edges across This surface cannot be embedded ie7 realized in 1R3 12 General manifolds One can de ne a manifold structure on an abstract set For example 0 The set of all lines through the origin in Rn is a manifold 1 that has dimension n It is called the real projective space and denoted RE 0 The set of all planes through the origin in R72 the 2 Grassmanian GAR 7 is a manifold2 of dimension 2n 7 2 1Here is the reason why given a line L the lines nearby can be labeled by their intersection with a hyperplane perpendicular to E that does not contain the origin 2Here is a hint why given a twoplane E07 pick a subspace F0 such that E0 EB F0 R then each twoplane E that intersects trivially F0 can be identi ed with the graph of a linear map from E0 into F0 the plane E0 corresponds to the zero map Thus7 a neighborhood of E0 in G2 Rn can be labeled by the linear maps from a 2dimensional space to a n 7 2dimensional space7 hence n 7 2 X 2 matrices 3 o On A E MatnmlR l AA I the set of all orthogonal linear transformations of R is a manifold and actually a Lie group However one can also realize the abstract manifolds77 as submanifolds of the vector space RN Below are some results not in their strongest form The statement M can be embedded into N means that one can realize the manifold M as a sub manifold of N Theorem 14 Whitney easiest version Any compact manifold can be embedded into RN for some large enough N Theorem 15 Whitney 1936 medium version Any compactn dimensional manifold can be embedded into Rh Theorem 16 Whitney 1944 dif cult version Any compact n dimensional manifold can be embedded into R2 Thus the Klein bottle can be embedded into R4 2 Vector bundles Each manifold M has a naturally de ned tangent bundle TM The vectors tangent to M at a point p E M form a vector space TpM For example if we consider the unit sphere 52 C R3 then one can identify the set of tangent vectors at a point p TPSZ with the plane tangent to the sphere at that point To be precise in order to make this plane a vector space it should be translated to the origin Figure 2 The tangent bundle to the 2 sphere Similarly one can consider other bundles over a manifold For example one can consider the line bundle L over 52 C R3 given at p E 2 by the line normal to the sphere at p Figure 3 A line bundle over the 2 sphere 3 Integration on manifolds Without any additional data7 one can integrate naturally on an orientable manifold of dimension n only n forms ln local coordinatesil7 an n form looks like fx17p277xndp1 Adp2Adxn Here naturally means that the result of the integration does not depend on the choice of local coordinates One central result7 which generalizes Green7s Theorem discussed in Calculus Ill and is related to many conservation laws in Physics7 is Theorem 31 Stokes7 Theorem Let M be an orientable manifold possibly with bound ary of dimension k andw a compactly supported k 7 1 form Then dw w M 8M Here dw denotes the eacterior derivative of the form w 4 Riemannian geometry One way to introduce distance on a manifold M is to endow it with a Riemannian metric7 which is an inner product on each tangent space TPM Using this one can associate length to each curve7 much in the same way as we do it for curves in R3 It also determines a volume form7 and thus allows to integrate functions on M Moreover7 the Riemannian structure is inherited naturally by the submanifolds of M Given a Riemannian structure7 one can speak of curves that minimize distance locally7 called geodesics One can also introduce curvature 5 Global results The quantities discussed so far are of local character all are de ned considering neighbor hoods Therefore7 they do not see the whole manifold But certain quantities are con strained by global data We give three examples the last two results exhibit the connection between local and global quantities 3Explain what these are Given a manifold M one can use forms to de ne its de Rham oohomology HM7R7 which is a topological invariant4 The recently proved Poincare conjecture asked whether certain topological information identi es completely the three sphere 53 within the family ofthree manifolds 5 The positive answer was obtained by Grigori Perelman following the program of Richard Hamilton7 about 100 years after Poincare 7s question Theorem 51 GaussBonnet Given a compact two dimensional Riemannian manifold M without boundary KdA 27rxM M where K is the Gaussian curvature dA is the area element and xM is the Euler charac teristic dij XM Z 71kdim HkMR k0 The remarkable fact about this formula is that the curvature K on the left hand side determined by the Riemannian structure can be changed by bending77 the surface7 however its total integral remains constant7 and is a priori determined by the topology of the surface For example7 the sphere of radius R in R3 has constant curvature K E 1R27 area element given by the usual surface area7 and Euler characteristic xSZ 2 indeed 1 2 x47rR27rgtlt2 No matter how we deform the sphere without tearing it 7 the right hand side remains 47f however7 the curvature will change once the sphere is deformed Theorem 52 Poincar Hopf index theorem Let M be a compact manifold M with out boundary Ifi is a smooth vector eld that has only isolated zeroes points where the vector eld vanishes then 2 indexdp XM PEMl7JPOl The left hand side is of local nature we skip the de nition of the index for the moment7 whereas the right hand side is a global quantity For example7 since the Euler characteristic of the two sphere is non zero7 the above theorem shows that any smooth vector eld on 52 must vanish somewhere which is another way to state the so called Hairy Ball Theorem one cannot comb hair on a two sphere without creating a vortex or other discontinuity 4That is7 if two manifolds are homeomorphic7 then their cohomologies coincide Say a few words about algebraic topology 51s every simply connected compact 3manifold without boundary homeomorphic to a 3sphere Manifolds Topological versus Differentiablequot Here are some facts about the differential smooth structures that a topological manifold of dimension n can support7 and related results Unless stated otherwise7 all manifolds are without boundary Counting eg7 uniqueness is meant in the appropriate sense up to smooth diffeomorphisms for smooth manifolds See also the remarks on pages 14 and 37 of the textbook a 7133 0 Each n dimensional topological manifold has a unique smooth structure J Munkres7 E Moise b n 2 4 o For each 71 there is a connected compact topological manifold that does not admit a smooth structure 0 Each n dimensional compact connected topological manifold admits at most countably many different smooth structures 0 S7 admits exactly 28 different smooth structures JW Milnor and MA Ker vaire 0 Classi cation of smooth even compact manifolds of dimension n 2 4 is very hard 0 Does S4 or CClP Z admit more than one smooth structures 0 The Poincare conjecture 1904 a compact simply connectedl smooth manifold is homeomorphic to the sphere of that dimension 0 n 2 5 proved by S Smale 1961 o n 4 proved by MH Freedman 1982 o n 3 proved by G Perelman 2002720037 using the Ricci ow method of RS Hamilton This makes the geometrization conjecture77 of W Thurston close to being provedz there are eight standard Riemannian models in dimension 3 compare to the dimension 2 case below7 where there are three models Last updated Jan 27 2009 only limited claim of accuracy is made 1A path connected topological space X is simplyconnected if its fundamental group7 7r1X7 is trivial any closed path in X can be continuously shrunk to a point The 1sphere is not simplyconnected but all higher dimensional spheres are 2Might be already proven7 using Perelmanls worki d 711 0 Any 1 dimensional connected smooth manifold is diffeomorphic to either R or 1 with the canonical structure e 712 0 Any 2 dimensional connected compact oriented smooth manifold is diffeomorphic to the sphere 2 with zero or more handles attached Any 2 dimensional compact manifold M admits a Riemannian metric with makes it locally isometric to one of the following because they are quotients of the manifolds listed below7 under a free discrete group action 1 2 with the canonical metric ie7 the one inherited from 52 C R3 this has constant 1 curvature R2 with the Euclidean metric this has constant 0 curvature A D V The hyperbolic plane7 H2 with the Poincare metric this has constant 71 curvature A 0 V The model above that applies to M is determined by the sign of its Euler char acteristic7 XM Zeiydim HiMR 13920 Recall that by Gauss Bonnet fM KdA 270M7 so the Euler characteristic determines the Gauss curvature if the latter is constant For example7 the only compact 2 dimensional smooth manifolds that fall in the rst case above are 52 and MW in the second case above are the 2 torus7 Sl gtlt 81 and the Klein bottle The surfaces of genus at least 2 in the orientable case7 these are the sphere with at least two handles admit a metric with constant curvature 71 The only smooth non compact manifolds that admit a complete metric3 of con stant zero curvature are R2 R gtlt 1 and the Mobius strip without boundary there are no such manifolds for curvature 1 The hyperbolic plane is such an example for curvature 71 f R For n 31 4 R admits a unique smooth structure R4 admits uncountably many non diffeomorphic smooth structures SK Donald son and MH Freedman7 1984 3A Riemannian metric is complete if each geodesic can be extended to in nite time For example7 R and any compact manifold are complete7 but R 0 is not complete With the canonical metric because some geodesics straight lines in this case run in the missing origin in nite time
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