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differential geometry lectures by p m h wilson notesbydavidmehrle dfm33 cam ac uk cambridgeuniversity mathematicaltripospartiii michaelmas2015 contents lecture 1 2 lecture13 35 lecture 2 4 lecture14 38 lecture 3 6 ...

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                                                                                      Differential Geometry
                                                                                            Lectures by P.M.H. Wilson
                                                                                               NotesbyDavidMehrle
                                                                                                    dfm33@cam.ac.uk
                                                                                                CambridgeUniversity
                                                                                          MathematicalTriposPartIII
                                                                                                     Michaelmas2015
                                                   Contents
                                                   Lecture 1 . . . . . . . . . . . . . .                            2 Lecture13 . . . . . . . . . . . . .                                35
                                                   Lecture 2 . . . . . . . . . . . . . .                            4 Lecture14 . . . . . . . . . . . . .                                38
                                                   Lecture 3 . . . . . . . . . . . . . .                            6 Lecture15 . . . . . . . . . . . . .                                41
                                                   Lecture 4 . . . . . . . . . . . . . .                            9 Lecture16 . . . . . . . . . . . . .                                44
                                                   Lecture 5 . . . . . . . . . . . . . .                          12 Lecture17 . . . . . . . . . . . . .                                 48
                                                   Lecture 6 . . . . . . . . . . . . . .                          15 Lecture18 . . . . . . . . . . . . .                                 51
                                                   Lecture 7 . . . . . . . . . . . . . .                          18 Lecture19 . . . . . . . . . . . . .                                 54
                                                   Lecture 8 . . . . . . . . . . . . . .                          20 Lecture20 . . . . . . . . . . . . .                                 57
                                                   Lecture 9 . . . . . . . . . . . . . .                          23 Lecture21 . . . . . . . . . . . . .                                 60
                                                   Lecture 10            .  .  .  .  .  .  .  .  .  .  .  .  .    26 Lecture22 . . . . . . . . . . . . .                                 64
                                                   Lecture 11            .  .  .  .  .  .  .  .  .  .  .  .  .    29 Lecture23 . . . . . . . . . . . . .                                 68
                                                   Lecture 12            .  .  .  .  .  .  .  .  .  .  .  .  .    32 Lecture24 . . . . . . . . . . . . .                                 70
                                                                                                Last updated April 1, 2016.
                                                                                                                       1
                           Lecture 1                                               8October2015
                           Administrative Stuff
                           There are some Lecture Notes online. They have some stuff that we won’t
                           cover. The best book is Spivak.
                           ManifoldsandVectorSpaces
                           SmoothManifolds
                           Definition 1. If U Ă Rm and δ: U Ñ R, we say that δ is smooth or C8 if has
                           continuous partial derivatives of all orders.
                           Definition 2. A topological space X is called second countable if there exists
                           a countable collection B of open subsets of X such that any open subset of X
                           maybewrittenastheunionofsetsofB.
                           Definition 3. A Hausdorff, second countable topological space X is called
                           a topological manifold of dimension d if each point has an open neighbor-
                           hood(nbhd)homeomorphictoanopensubsetU ofRd byahomeomorphism
                           φ: U   „ φpUqĂRd.
                              ThepairpU,φqofahomeomorphismandopensubsetofMiscalledachart:
                           given open subsets U and V of X with U XV ‰ H, and charts pU,φ q and
                                                                                             U
                           pV,φ q, with φ : U Ñ φpUq Ă Rd and φ : V Ñ φpVq Ă Rd, we have a
                                V        U               ´1         V
                           homeomorphism φ      “φ ˝φ :φ pUXVqÑφ pUXVqofopensubsets
                                             VU     V    U    U             V
                           of Rd.
                              Given a chart pU,φ q and a point p P U, we call U a coordinate neighbor-
                                                U
                           hood of p and we call the functions x : U Ñ R given by π ˝φ  (where π is
                                                              i                    i  U          i
                           the projection onto the i-th coordinate) coordinates of U.
                           Definition 4. A smooth structure on a topological manifold is a collection A
                           of charts pU ,φ q for α P A, such that
                                      α  α
                             (i) tU | α P Au is an open cover of M;
                                   α
                             (ii) for any α, β P A such that U XU ‰ H, the transition function φ  “
                                                           α    β                              βα
                                φ ˝φ´1issmooth. Thechartsφ andφ aresaidtobecompatible;
                                  β   α                       α      β
                            (iii) the collection of charts φ is maximal with respect to (ii). In particular,
                                                        α
                                this means that if a chart φ is compatible with all the φ , then it’s already
                                                                                   α
                                in the collection.
                           Remark 5. Since φ   “ φ´1: φ pU XU q Ñ φ pU XU q, both φ         and φ
                                             αβ    βα   β   α    β      α  α     β        βα      αβ
                           are in fact diffeomorphisms (since by assumption, they are smooth).
                              Thisremarkshowsthatitem(ii)inDefinition4impliesthattransitionfunc-
                           tions are diffeomorphisms.
                              For notation, we sometimes write U  “U XU .
                                                                αβ    α    β
                                                               2
                            Definition 6. A collection of charts tpU ,φ q | α P Au satisfying items (i) and
                                                                    α  α
                            (ii) in Definition 4 is called an atlas.
                            Claim7. Anyatlas A is contained in a unique maximal atlas and so defines a
                            uniquesmoothstructureonthemanifold.
                            Proof. If A “ tpU ,φ q | α P Au is an atlas, we define a new atlas A˚ of all
                                              α   α
                            charts on M compatible with every chart in A. To be compatible with every
                            chart in A means that if pU,φq P A˚, φ    “φ˝φ´1issmoothforallαP A.
                                                                  UU         α
                                                        ˚            α
                                Weshouldjustify that A is an atlas. This involves checking conditions (i)
                            and(ii) in Definition 4.
                                Clearly (i) is satisfied, because A˚ contains A and A covers M.
                                For (ii), we suppose pU,φ q and pV,φ q are elements of A˚. We show that
                                                         U           V
                            the homeomorphism φ        is smooth. It suffices to show that φ    is smooth
                                                   VU                                       VU
                            in a neighborhood of each point φ ppq for φ P A. To that end, consider the
                                                               α        α
                            neighborhood φ pU XUXVqofφ ppqinφ pUXVq. Itsufficestoshowthat
                                            U   α               α       U
                            φ    is smooth when restricted to this neighborhood; that is, we want to show
                              VU
                            that
                                        φ    |           : φ pUXVXU qÑφ pUXVXU q
                                         VU φ pUXVXU q      U           α      V            α
                                              U        α
                            is smooth. LetW “ UXVXU . φ          |      canberealizedasthecomposition
                                                          α   VU φ pWq
                                                                   U
                            of two smooth transition functions as follows:
                             φ    |     “φ ˝φ´1˝φ ˝φ´1|            “pφ ˝φ´1q|          ˝pφ ˝φ´1q |
                              VU φ pWq     V    α     α   U φ pWq       V    α   φ pWq    α    U   φ pWq
                                   U                           U                  α                 U
                                                               φ  |
                                                                VU φ pWq
                                               φ pWq               U            φ pWq
                                                U                                V
                                                 φ   |                      φ   |
                                                  U U φ pWq                  VU φ pWq
                                                   α  U         φ pWq          α α
                                                                 α
                            Sinceeachofφ       andφ      is smoothbyassumption,thensoistheircompos-
                                           U U       VU
                                            α          α
                            ite and so φ   is smooth at φ ppq. Therefore, it is smooth.
                                        VU               α
                                Nowfinally, we need to justify that A˚ is maximal. Clearly any atlas con-
                            taining A must consist of elements of A˚. So A˚ is maximal and unique.
                            Definition 8. A topological manifold M with a smooth structure is called a
                            smoothmanifoldofdimensiond. Sometimesweuse Md todenotedimension
                            d.
                            Remark 9. We can also talk about Ck manifolds for k ą 0. But this course is
                            aboutsmoothmanifolds.
                            Example10.
                               (i) Rd with the chart consisting of one element, the identity, is a smooth
                                  manifold.
                              (ii) Sd Ď Rd`1 is clearly a Hausdorff, second-countable topological space.
                                        `     ~    d                    ´     ~     d
                                  Let U   “ tx P S | xi ą 0u and let U     “ tx P S | xi ă 0u. We have
                                        i                               i
                                                                  3
                                     charts φ : U` Ñ Rd and ψ : U´ Ñ Rd given by just forgetting the i-th
                                              i   i                i   i
                                     coordinate. Note that φ ˝φ´1 (and ψ ˝φ´1) are both maps defined by
                                                              2    1         2    1
                                                                  b
                                               py2,...,yd`1q Ñ ´     1´y2´...´y2         , y3,. . . , yd`1¯.
                                                                          2          d`1
                                     This is smooth on an appropriate subset of
                                                        `     !                  2          2       )
                                                   φ pU q “ py2,...,y        q | y `...`y        ă1
                                                    1   1                 d`1    2          d`1
                                     given by y2 ą 0 (resp. y2 ă 0q. The reason that y2 ą 0 is the appropriate
                                                          `      `     ~     d
                                     subset is because U    XU “txPS |x1ą0andx2ą0u,andwewant
                                       ´1                 1      2 `
                                     φ py ,...,y       q to be in U  so that it’s in the domain of φ .
                                       1    2      d`1             2                                 2
                                     From this it follows that Sd is a smooth manifold. We should be careful
                                                        ~     d                                                `
                                     to note that each x P S has some xi ‰ 0, so lies in one of the sets U       or
                                       ´                                                                       i
                                     U .
                                       i
                                (iii) Similarly the real projective space RPd “ Sd{t˘1u identifying antipodal
                                     points is a smooth manifold.
                               Lecture 2                                                       10 October 2015
                               Example11. Furtherexamples. Continuedfromlasttime.
                                                                                 2           ~     ~
                                (iv) Consider the equivalence relation on R given by x „ y if and only if
                                     x1 ´y1 P Z, x2 ´y2 P Z. Let T denote the quotient topological space the
                                     2-dimensional torus. Any unit square Q in R2 with vertices at pa,bq, pa`
                                     1,bq,pa,b`1q,andpa`1,b`1qdeterminesahomeomorphismπ: intQ „ UpQqĂ
                                     T, withUpQq “ πpintQqopeninT. Theinverseisachart. Giventwodif-
                                     ferent unit squares Q ,Q2, we get the coordinate transform φ         which is
                                                            1                                           21
                                     locally (but not globally) just given by translation. This gives a smooth
                                     structure on T. Similarly define the n-torus Tn “ Rn{Zn as a smooth
                                     manifold.
                               Definition12. Let Mm, Nn besmoothmanifoldswithgivensmoothstructures.
                               Acontinuous map f : M Ñ N is smooth if for each p P M, there are charts
                               pU,φ q,pV,ψ qwith p P U, fppq P V, such that f “ ψ ˝ f ˝φ´1 is smooth.
                                     U        V                                          V        U
                                                           p                              f ppq
                                                           P                                P
                                                      UXf´1pVq                f            V
                                                         φ                                   ψ
                                                          U                                   V
                                                   φ pUXf´1pVqq                          ψ pVq
                                                    U                         f           V
                                   Note that since the coordinate transforms for different charts are diffeo-
                               morphisms,thisimpliesthattheconditionthat f is smoothholdsforallcharts
                               pU1,φ 1q, pV1,ψ 1q with p P U1, fppq P V1.
                                     U          V
                                                                         4
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...Differential geometry lectures by p m h wilson notesbydavidmehrle dfm cam ac uk cambridgeuniversity mathematicaltripospartiii michaelmas contents lecture last updated april october administrative stuff there are some notes online they have that we won t cover the best book is spivak manifoldsandvectorspaces smoothmanifolds denition if u rm and n r say smooth or c has continuous partial derivatives of all orders a topological space x called second countable exists collection b open subsets such any subset maybewrittenastheunionofsetsofb hausdorff manifold dimension d each point an neighbor hood nbhd homeomorphictoanopensubsetu ofrd byahomeomorphism puqrd thepairpu qofahomeomorphismandopensubsetofmiscalledachart given v with xv charts pu q pv puq rd pvq homeomorphism puxvqn puxvqofopensubsets vu chart call coordinate functions where i projection onto th coordinates structure on for tu au ii xu transition function issmooth thecharts aresaidtobecompatible iii maximal respect to in particul...

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