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Differential Geometry

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Differential Geometry Part III: Differential Geometry (Version 2: October 14, 2016) 1. (Fundamentals about Smooth Maps) Verify the following: (i) If U ⊂ M is an open set of a manifold then U has inherits the structure of a manifold such that the inclusion map is smooth. (ii) A function f : Rn ! Rm is smooth as a map between manifolds (where Rn and Rm are given the standard smooth structure) if and only if it is smooth in the usual sense. (iii) Let (U; ') be a chart on a manifold M. Then ': U ! '(U) is a diffeomor- phism. (iv) Compositions of smooth maps are smooth. (v) A smooth map between manifolds is continuous (with respect to the topology defined in lectures). [Throughout this course you are expected to use standard results from analysis without proof.] 2. (Products) Let M1 and M2 be smooth manifolds of dimension m1 and m2 respec- tively. Show that M1 × M2 is naturally a manifold of dimension m1 + m2 and the projections pi : M1 × M2 ! Mi are smooth maps. Show also that if N is another manifold then a map f : N ! M1 × M2 is smooth if and only if pi ◦ f is smooth for i = 1; 2 3. (Topology of manifolds) (i) Show any manifold is locally path connected and locally compact. That is if p 2 V where V is open, then p 2 U ⊂ V for some open U that is path connected, and whose closure is compact. (ii) Show that the open set U in (i) can be taken to be a chart given by a bijection n ' : U ! B1 where B1 is the open unit ball in R and '(p) = 0. Prove the same n with B1 replaced by R .[We will need very little material from the theory of topological spaces, but the statement of this exercise can be useful] 4. (Dimension) Suppose a manifold M is connected. Prove that it has the same dimen- sion at every point. 3 5. (Differentiable structure on R) Do the charts '1(x) = x and '2(x) = x (x 2 R) belong to the same atlas on the set R? Let Rj, j = 1; 2, be the manifold defined by using the chart 'j on the topological space R. Are R1 and R2 diffeomorphic? 6. (Projective Space) Let RP n be the set of lines in Rn+1. Show how RP n can be made into a manifold in such a way that the natural map π : Sn ! RP n taking v 2 Sn to the line spanned by v is smooth. [Hint: Any line in Rn+1 is spanned by some non- zero vector v = (v0; : : : ; vn). Start by defining charts on the set Ui = fv : vi = 1g. Now do the same to show CP n, the space of (complex) lines in Cn+1 , is a real manifold of dimension 2n. 7. (i) Prove that the complex projective line CP 1 is diffeomorphic to the sphere S2. (ii) The natural map (C2 n f0g) ! CP 1 induces a smooth map of manifolds π : S3 ! S2, called the Poincar´emap. Show that the derivative of this map induces surjections on the tangent spaces, and π−1(p) is diffeomorphic to S1 for all p 2 S2. 8. (Tangent Space) Let p 2 U where U ⊂ M is open. Prove that for all sufficiently small open V ⊂ U containing p and f 2 C1(V; R) there exists a g 2 C1(U; R) and an open p 2 W ⊂ U \ V with fjW = gjW [Hint: bump functions]. Next show that the restriction map 1 1 C (U; R) ! C (V; R) f 7! fjV induces a map 1 1 Derp(C (V; R)) ! Derp(C (U; R)) which is an isomorphism for sufficiently small open V ⊂ U containing p. Deduce that the definition of the tangent space given in lectures does not depend on any choice of open set. 9. Show that (i) TS1 is diffeomorphic to S1 × R; (ii) TS3 is diffeomorphic to S3 × R3.[More generally if G is a Lie group, then TG is diffeomorphic to G × Rd,where d = dim G.] 10. Let f : M ! N be a diffeomorphism of manifolds. If X, Y denote smooth vector fields on M, define the corresponding vector fields f∗X, f∗Y on N. Show that f∗ respects the relevant Lie brackets, i.e. that f∗([X; Y ]M ) = [f∗X; f∗Y ]N as vector fields on N. 11. Let x1; : : : ; xn be local coordinates on a manifold, and suppose that vector fields Pn @ Pn @ X and Y are vector fields given locally by X = ai and Y = bi . i=1 @xi i=1 @xi By constructing a suitable flow or otherwise, show from the definition of the Lie derivative that n n X X @bj @aj @ L Y = a − b X i @x i @x @x i=1 j=1 i i j [It may be helpful to use linearity]. Use this to verify that LX Y = [X; Y ]. 12. For which values of c 2 R is the zero locus in R3 of the polynomial z2 − (x2 + y2)2 + c an embedded manifold in R3, and for which values is it an immersed manifold? 13. Show a compact manifold M of dimension n can be embedded in RN for some N. [Hint: Start by using bump functions to prove the existence of a cover of M by finitely many charts (Uα;'α) for 1 ≤ α ≤ m and smooth functions α such that (1) α is supported in Uα and (2) α ≡ 1 on some open set Vα and (3) the Vα cover M. Then m(n+1) define f : M ! R by f(p) = ( 1'1(p); : : : ; m'm(p); 1; : : : ; m). A harder theorem due to Whitney states it is possible to have N = 2n + 1.] 14. Prove that the map 1 ρ(x : y : z) = (x2; y2; z2; xy; yz; zx) x2 + y2 + z2 gives a well-defined embedding of RP 2 into R6. Find on R6 a finite system of polyno- mials, of degree ≤ 2, whose common zero locus is precisely the image of ρ. Construct an embedding of RP 2 in R4.[Hint: Compose ρ with a suitable map.] 15. Show that the following groups are Lie groups (in particular, smooth manifolds): (i) special linear group SL(n; R) = fA 2 GL(n; R) : det A = 1g; (ii) The special unitary group SU(n) = fA 2 SL(n; C): AA∗ = Ig, where A∗ denotes the conjugate transpose of A and I is the n × n identity matrix; (iii) Sp(m) = fA 2 U(2m): AJAt = Jg, where At denotes the transpose of A (no 0 I conjugation!) and J = −I 0 . In each case, find the corresponding Lie algebra. 16. (Alternative definition of a manifold) It is said that the physicist's definition of a manifold is a \Lie group without the group structure". Discuss. The last two exercises go slightly beyond the course and connect with some concepts with other courses which you may be taking. They are not necessarily hard, but will not be examined. 17. (Intrinsic/better definition of the tangent space). Given p 2 M define a relation on smooth functions f; g defined on open sets around p by f ' g if there is an open nhood W around p such that fjW = gjW . This gives an equivalence relation. The 1 equivalence class of a function f is called the germ of f at p and the set Cp of germs 1 at p is an algebra. We define TpM to be the space of derivation Cp ! R which is a vector space. Show that this definition is naturally isomorphic to the definition of the tangent space given in lectures for any chart U [Hint: Given a function f defined on an nhood W of p let ' be a bump function supported on W that takes the value 1 on some W 0 ⊂ W . Then 'f and f have the same germ at p, and start by showing that 1 if v is a derivation of Cp then v('f) = v(f).] 18. (Connection with algebraic geometry) Let M be a manifold and p 2 M. Show that 1 the evaluation map evp : Cp (M) ! R given by evp(f) = f(p) is a well-defined ring 1 homomorphism, and that its kernel is the unique maximal ideal m in Cp (M) (this 1 says that Cp (M) is a local ring). Given v 2 Tp(M) and f 2 m let (v; f) := evp(v(f)). 2 ∗ Show this is a well-defined pairing and use it to show Tp(M) = (m=m ) where the star denotes the dual vector space. [email protected].
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