MTH 674 Differential Geometry of manifolds Midterm Sample Problems Problem I. Define in detail a) b) c) d) e) f) g) h) i) j) k) l) m) n) o) p) q) r) s) t) u) v) w) x) A topological manifold M . A differentiable manifold M . An atlas for a differentiable manifold M . The product manifold M1 × M2 of two differentiable manifolds M1 , M2 . Coordinate representation of a mapping F : M1 → M2 from M1 into M2 , where M1 , M2 are topological manifolds. A smooth mapping F : M1 → M2 from a differentiable manifold M1 into a differentiable manifold M2 . A diffeomorphism F : M1 → M2 between two manifolds M1 and M2 . A bump function. Partition of unity subordinate to an open cover M = ∪α∈A Uα of a topological space M . A derivation at a point p ∈ M , where M is a differentiable manifold. The tangent space Tp (M ) to a manifold M at point p. Coordinate vectors ∂/∂xi , where (U, x = (x1 , . . . , xm )) is a coordinate chart on a differentiable manifold. The differential of a smooth mapping F : M1 → M2 , where M1 , M2 are differentiable manifolds. The rank of a mapping F : M1 → M2 . Immersion, submersion. Immersed, imbedded submanifolds. k-submanifold property (a.k.a. local k-slice condition). A smooth vector field X on a differentiable manifold M . The Lie bracket [X, Y ] of two smooth vector fields on M . F -related vector fields on M1 , M2 , where F : M1 → M2 is a smooth mapping. Quotient topology The real projective space RPn and the Grassman manifold Gr(k, n). A Lie group. The Lie algebra of a Lie group. Problem II. a) Let M1 = R with the coordinate map φ(x) = x and let M2 = R with the coordinate map ψ(x) = x1/3 . 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 ), sin 2(t − )). 2 2 Show that F is an injective immersion but not an embedding. F : (−π, π) → R2 , F (t) = (2 cos(t − Problem III. a) Let M1 , M2 be smooth manifolds. Show that M1 × M2 is diffeomorphic with M2 × M1 in the standard product manifold structure. b) Let F : RP2 → Gr(3, 2) map a line in R3 to the plane perpendicular to it. Show that F is a diffeomorphism. Problem IV. 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 ∈ P the level set Np = {q ∈ M | f (q) = p} satisfies the n submanifold property (provided it is not empty). c) Show that the 3-dimensional unit sphere S 3 = {(x, y, z, u) ∈ R4 | x2 + y 2 + z 1 + u2 = 1} 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 choose b ∈ N . Show that the mapping ib : M → M × N , ib (p) = (p, b) is an embedding. Problem V. Let D be a derivation at p ∈ 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 ∈ 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, P· · · , 0) = 0 then there are smooth functions gi : Rn → R so that f (x1 , . . . , xn ) = ni=1 xi gi (x1 , . . . , xn ). Problem VI. a) Let πP : S 2 → RP2 , πP (x, y, z) = [x, y, z], be the natural projection, where S 2 is the unit sphere and RP2 the projective space. Show that open sets in RP2 are precisely the images of open sets in S 2 under the mapping πP . b) Show that the projection πP : S 2 → RP2 is smooth and everywhere of rank 2. 3 c) Let F : R2 → RP2 be the smooth map F (x, y) = [x, y, 1], and let X = x∂x − y∂y be a smooth vector field on R2 . Prove that there is a smooth vector field on RP2 that is F -related to X, and find its coordinate expressions in the standard charts for RP2 . Problem VII. a) Let X = x2 y∂y − z∂z , Y = xy∂x + y∂y − z 2 ∂z be vector fields on R3 . Compute [X, Y ]. b) Let (U, φ), (U, ψ) 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 ) = ψ ◦ φ−1 (x1 , x2 ) = ((x1 )2 − (x2 )2 , 2x1 x2 ). A vector field X on M has the coordinate expression X = x2 ∂x1 in the chart (U, φ). Find its coordinate expression in the chart (U, ψ). c) Find the expression for the vector field X = y∂x in polar coordinates. Problem VIII. Let M ⊂ R3 be the paraboloid determined by the equation z = x2 +y 2 with the subspace topology. a) Let ϕ : M → R2 be given by ϕ(x, y, z) = (x, y). Show that ϕ provides a global coordinate map for M . b) Let F : M → S 2 be the map F (x, y, z) = (1 + 4x2 + 4y 2 )−1/2 (−2x, −2y, 1). Is F smooth? c) Let v = ∂/∂ϕ1 |(x,y,z)=(0,1,1) . Compute a coordinate expression for F∗ (v), where F is as in part b. d) Let f : M → R, f (x, y, z) = xyz 2 . Find the coordinate derivative ∂ f ∂ϕ1 |(1,−1,2) at the indicated point. Problem IX. 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. b) Let πP : R3 r {(0, 0, 0)} → RP2 be the projection. Find an expression for πP∗ (V) in suitable coordinates for RP2 . Problem X. The complex projective space CP1 is defined as the set of all 1 dimensional complex subspaces of C2 = C × C, or alternatively, the orbit space of the action of ∗ ∗ ∗ C = C r {0} on C2 = C2 r {(0, 0)} by multiplication. Write πP : C2 → CP1 for the projection. Endow CP1 with the usual quotient topology so that a set U ⊂ CP1 is open 4 if and only if the union of all lines constituting U (with the origin removed!) is open in ∗ C2 . a) Explain why the quotient topology on CP1 is second countable and Hausdorff, and show that the projection map πP is open. b) Define an atlas of coordinate charts on CP1 in analogy with the standard coordinate charts for RP1 constructed in class, and show that these are homeomorphisms. Also identify the image of each coordinate map. c) Find the transition functions (a.k.a. the change of coordinate maps) for the atlas for CP1 constructed in part b. ∗ d) Compute the rank of the projection πP at every point in C2 . e) Use part c to show that CP1 is diffeomorphic to the unit sphere S 2 . Problem XI. Let Gr(n, 2) denote the Grassmannian space of 2 – dimensional (vector) subspaces in Rn . a) Let P ∈ Gr(n, 2). Then any two bases {v1 , v2 }, {w1 , w2 } for P are related by X j wi = ai v j , i = 1, 2, (1) j=1,2 b) c) d) e) f) g) where (aji ) ∈ GL(2) is an invertible 2 × 2 matrix. Conclude that Gr(n, 2) can be identified with the equivalence classes of linearly independent pairs of vectors {v1 , v2 }, where {v1 , v2 } ∼ {w1 , w2 } provided that equation (1) holds for some (aji ) ∈ GL(2). Let GL(2) act on the space T of 2 × n matrices of rank 2 by left multiplication. Use part a to show that the orbit space T /GL(2) of the action can be identified with Gr(n, 2). Equip T with the subspace topology as an open subset of R2n , and equip Gr(n, 2) with the usual quotient topology. Conclude that the projection πGr : T → Gr(n, 2) is an open mapping and that Gr(n, 2) is Hausdorff. Let T (i, j) ⊂ T denote the set of rank 2 matrices whose minor consisting of the ith and jth columns is invertible. Show that the action of GL(2) on T preserves the sets T (i, j) and that every GL(2) orbit in T (i, j) contains a unique matrix whose i, j-minor is the identity matrix. Conclude that every plane P ∈ T (i, j) can be identified with a unique − 2) × 2 (n a b matrix. (For example, with n = 4 and i = 1, j = 3, the 2×2 matrix would c d correspond to the plane admitting the basis v1 = (1, a, 0, b), v2 = (0, c, 1, d).) Show that each Gij = πGr (T (i, j)) ⊂ Gr(n, 2) is open, and use part e to define coordinate maps ϕij in each Gij . In particular, show that each ϕij : Gij → R2n−4 is a homeomorphism. Finally compute a representative sample of transition functions to conclude that Gr(n, 2) forms a differentiable manifold. 5 Problem XII. Let CPn and Gr(n, 2) denote the complex projective space and Grassmann manifold constructed in problems X and XI. a) Let S 3 be the unit sphere in C2 identified with R4 , and let the Hopf map πH : S 3 → S 2 be the restriction of the projection πP to S 3 . Describe the inverse image πH −1 (p) of a point p ∈ S 2 . b) Show that the projection πGr : T → Gr(n, 2) is differentiable. c) Show that each Gr(n, 2) is compact. Problem XIII. a) Let ϕ3 denote the standard coordinates on U3 = {[x, y, z] ∈ RP2 | z 6= 0} given by (u1 , u2 ) = ϕ3 ([x, y, z]) = (x/z, y/z). The coordinate differential v = ∂/∂u1 corresponds to the directional derivative along some curve α in RP2 . Geometrically, a curve represents a 1-parameter family of lines in R3 . Describe this family for a curve α of yourchoice. u11 u21 denote the coordinates for Gr(4, 2) as constructed in b) Let ϕ34 (P) = u12 u22 problem XI. The coordinate differential v = ∂/∂u22 corresponds to the directional derivative along some curve α in Gr(4, 2), that is, 1-parameter family of planes in R4 . Describe this family for a curve α of your choice. Problem XIV. a) Let LA : Gl(m) → Gl(m), A ∈ Gl(m), be the left translation LA (X) = AX. Describe LA∗ : TI Gl(m) → TA Gl(m). b) Let Ψ : Gl(m) → Gl(m) be the mapping Ψ(X) = X T X. Describe Ψ∗ : TI Gl(m) → TI Gl(m). c) Let ι : Gl(m) → Gl(m) denote the inverse ι(X) = X −1 . Describe ι∗ : TI Gl(m) → TI Gl(m). Problem XV. a) Show that [f X, gY ] = f g[X, Y ] + f (Xg)Y − g(Y f )X, where X, Y ∈ X (M ). b) Let F : M → N be smooth, where M , N are differentiable manifolds. Suppose b Yb ∈ X (N ). that vector fields X, Y ∈ X (M ) on M are F -related to vector fields X, b Yb ]. Show that the bracket [X, Y ] is F -related to [X, c) Let M , N be smooth manifolds and let F : M → N , G : N → M be smooth maps satisfying G ◦ F = id. Given Y ∈ X (N ), define Xp by Xp = G∗ (YF (p) ). Show that the assignment p → Xp defines a smooth vector field on M . Problem XVI. a) Let F : M → N be smooth, where M , N are differentiable manifolds, and suppose that F∗ : Tp M → TF (p) N is an isomorphism for all p ∈ M . Prove that F is an 6 open mapping. Show, in addition, that F must be surjective if M is compact and N is connected. b) Suppose that F : M → N is of constant rank and surjective. Prove that F is a smooth submersion. Problem XVII. a) Let F : Rm → RPm be defined by F (x1 , x2 , . . . , xm ) = [x1 , x2 , . . . , xm , 1]. Show that F is a smooth map onto a dense open subset of RPm . b) Define similarly G : Cm → CPm by G(z 1 , z 2 , . . . , z m ) = [z 1 , z 2 , . . . , z m , 1]. Show that G is a diffeomorphism onto a dense, open set of CPm . c) Let pb(z) be a complex polynomial in one variable and let G : C1 → U ⊂ CP1 be as in part b, where U denotes the image of G. Define p : U → U by the condition that p ◦ G = G ◦ pb. Show that p can be extended to a smooth map on the entire CP1 . Problem XVIII. Let U ⊂ R2 be an open set. A proper coordinate patch is a one-toone immersion x : U → R3 A surface is a subset M ⊂ R3 equipped with the subspace topology such that for each point p ∈ M, there is a proper coordinate patch whose image contains a neighborhood of p in M. Show that a surface is a differentiable manifold. Last modified Friday 2nd May, 2014 at 11:02.