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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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