MTH 674 Differential Geometry of manifolds Midterm Sample Problems Problem I. Homework assignments I and II. Problem II. Define in detail a) A topological manifold M. b) A differentiable manifold M. c) Transition functions. d) An atlas of coordinate charts. n n e) Real and complex projective spaces RP and CP . f) Grassmannians Gr(m; k). g) The product manifold M1 × M2 of two differentiable manifolds M1, M2. h) Coordinate representation of a mapping F : M1 ! M2 from M1 into M2, where M1, M2 are topological manifolds. i) A smooth mapping F : M1 ! M2 from a differentiable manifold M1 into a differ- entiable manifold M2. j) A diffeomorphism F : M1 ! M2 between two manifolds M1 and M2. k) A bump function. l) Partition of unity subordinate to a cover. m) A derivation at a point p 2 M, where M is a differentiable manifold. n) The tangent space Tp(M) to a manifold M at point p. o) The pushforward F∗(v) of a tangent vector v 2 TpM under a mapping F : M ! N. p) Coordinate vectors @=@xi, where (U; x = (x1; : : : ; xm)) is a coordinate chart on a differentiable manifold. q) A smooth vector field X on a differentiable manifold M. r) The rank of a mapping F : M1 ! M2. s) The Lie bracket [X; Y ] of two smooth vector fields on M. t) F -related vector fields on M1, M2, where F : M1 ! M2 is a smooth mapping. u) Quotient topology v) Immersion, submersion w) Immersed, embedded submanifold. x) Slice coordinates y) Lie group and the Lie algebra of a Lie group. z) The Einstein summation convention. Problem III. a) Let M1 = R with the coordinate map φ(x) = x and let M2 = R with the 1=3 coordinate map (x) = x . Show that M1 and M2 are not equal as manifolds but that they are diffeomorphic as manifolds. 1 2 b) The figure eight is the image of the mapping π π F :(−π; π) ! 2;F (t) = (2 cos(t − ); sin 2(t − )): R 2 2 Show that F is an injective immersion but not an embedding. Problem IV. a) Let M1, M2 be smooth manifolds. Show that M1 × M2 is diffeomorphic with M2 × M1 in the standard product manifold structure. 2 b) Let F : RP ! Gr(3; 2) map a line in R3 to the plane perpendicular to it. Show that F is a diffeomorphism. Problem V. Let M be a differentiable manifold. a) State the n-submanifold property for a subset N ⊂ M. b) Let F : M ! P be a smooth mapping with constant rank k. Starting from the rank theorem, show that for any p 2 P the level set Np = fq 2 M j f(q) = pg satisfies the n submanifold property (provided it is not empty). c) Show that the 3-dimensional unit sphere 3 4 2 2 1 2 S = f(x; y; z; u) 2 R j x + y + z + u = 1g is an embedded submanifold of R4. d) Show that the subgroup SL(n) ⊂ Gl(n) of matrices with unit determinant is an embedded submanifold of Gl(n). e) Show that the group SO(n) ⊂ Gl(n) of orthogonal matrices with unit determinant is an embedded submanifold of Gl(n). f) Let M, N be differentiable manifolds and let b 2 N. Show that the mapping ib : M ! M × N, ib(p) = (p; b) is an embedding. Problem VI. Let D be a derivation at p 2 M. a) Suppose that f : M ! R is zero in some neighborhood of the point p. Show that Df = 0 (that is, that D is a local operator). b) Let M = Rn with the coordinates x1,. xn, and let D be a derivation at 0 2 Rn. Show that D can be expressed as a linear combination of the coordinate differentials @=@xi, i = 1; 2; : : : ; n. You may assume without proof that given a smooth function f : Rn ! R with f(0; ··· ; 0) = 0 then there are smooth functions n 1 n Pn i 1 n gi : R ! R such that f(x ; : : : ; x ) = i=1 x gi(x ; : : : ; x ). Problem VII. 2 2 2 a) Let πP : S ! RP , πP(x; y; z) = [x; y; z], be the natural projection, where S is 2 2 the unit sphere and RP the projective space. Show that open sets in RP are 2 precisely the images of open sets in S under the mapping πP. 2 2 b) Show that the projection πP : S ! RP is smooth and everywhere of rank 2. 2 2 c) Let F : R ! RP be the smooth map F (x; y) = [x; y; 1], and let X = x@x − y@y 2 be a smooth vector field on R2. Prove that there is a smooth vector field on RP 3 that is F -related to X, and find its coordinate expressions in the standard charts 2 for RP . 2 Solution. (4c) Let (Ux; x), (Uy; y), (Uz; z) be the standard charts on RP . Note that 2 F (R ) ⊂ Uz and that z ◦ F (x; y) = (x; y). Hence we can define a vector field Y on Uz −1 that is F -related to X by Y = ( z )∗(X). We need show that Y can be extended to a 2 smooth vector field on RP . The coordinate expression for Y in z coordinates is simply X = x@x − y@y. We will 1 2 1 2 2 use coordinates (u ; u ) and (v ; v ) on the images (= R ) of x and y, respectively. 1 The u -component of Y in x coordinates is given by y y Y ( 1) = X( 1 ◦ −1) = X( ) = −2 = −2u1: x x z x x Also, 1 1 Y ( 2) = X( ) = − = −u2: x x x Thus the coordinate expression for Y in x coordinates is 1 2 Y = −2u @u1 − u @u2 : Note that the above expression defines a smooth vector field on all of Ux. Similarly, the coordinate expression of Y in y coordinates is given by 1 2 Y = 2v @v1 + v @v2 : 2 It follows that Y extends to a smooth vector field on all of RP . Problem VIII. 2 2 3 a) Let X = x y@y − z@z, Y = xy@x + y@y − z @z be vector fields on R . Compute [X; Y ]. b) Let (U; φ), (U; ) be two coordinate systems on a 2-dimensional manifold M with the same domain U ⊂ M, and suppose that the transition function is given by (u1; u2) = ◦ φ−1(x1; x2) = ((x1)2 − (x2)2; 2x1x2). A vector field X on M has 2 the coordinate expression X = x @x1 in the chart (U; φ). Find its coordinate expression in the chart (U; ). c) Find the expression for the vector field X = y@x in polar coordinates. Problem IX. Let M ⊂ R3 be the paraboloid determined by the equation z = x2 + y2 with the subspace topology. a) Let ': M ! R2 be given by '(x; y; z) = (x; y). Show that ' provides a global coordinate map for M. b) Let F : M ! S2 be the map F (x; y; z) = (1 + 4x2 + 4y2)−1=2(−2x; −2y; 1): Is F smooth? 1 c) Let v = @=@' j(x;y;z)=(0;1;1). Compute a coordinate expression for F∗(v), where F is as in part b. 4 d) Let f : M ! R, f(x; y; z) = xyz2. Find the coordinate derivative @ 1 f @' j(1;−1;2) at the indicated point. Problem X. The equations x = ρ cos θ sin φ, y = ρ sin θ sin φ, z = ρ cos φ define spherical coordinates ρ, θ, φ on R3. a) Let V = @=@φ, W = @=@ρ be the coordinate partial derivative vector fields. Express V, W in Cartesian coordinates x, y, z. 3 2 b) Let πP : R r f(0; 0; 0)g ! RP be the projection. Find an expression for πP∗(V) 2 in suitable coordinates for RP . 1 Problem XI. The complex projective space CP is defined as the set of all 1 dimen- sional complex subspaces of C2 = C × C, or alternatively, the orbit space of the action ∗ 2∗ 2 2∗ 1 of C = C r f0g on C = C r f(0; 0)g by multiplication. Write πP : C ! CP for 1 1 the projection. Endow CP with the usual quotient topology so that a set U ⊂ CP is open if and only if the union of all lines constituting U (with the origin removed!) is ∗ open in C2 . 1 a) Explain why the quotient topology on CP is second countable and Hausdorff, and show that the projection map πP is open. 1 b) Define an atlas of coordinate charts on CP in analogy with the standard coor- 1 dinate charts for RP constructed in class, and show that these are homeomor- phisms. Also identify the image of each coordinate map. c) Find the transition functions (a.k.a. the change of coordinate maps) for the atlas 1 for CP constructed in part b. 2∗ d) Compute the rank of the projection πP at every point in C . 1 e) Use part c to show that CP is diffeomorphic to the unit sphere S2. Problem XII. Let Gr(n; 2) denote the Grassmannian space of 2 { dimensional (vector) subspaces in Rn. a) Let P 2 Gr(n; 2). Then any two bases fv1; v2g, fw1; w2g for P are related by X j wi = ai vj; i = 1; 2; (1) j=1;2 j where (ai ) 2 GL(2) is an invertible 2 × 2 matrix.