MTH 674 Differential Geometry of Manifolds Midterm Sample Problems

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 differentiable 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 ∈ 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 ∈ 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 ψ(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 ∈ 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 3 4 2 2 1 2 S = {(x, y, z, u) ∈ R | x + y + z + u = 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 let b ∈ 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 ∈ 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, ··· , 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, ψx), (Uy, ψy), (Uz, ψz) be the standard charts on RP . Note that 2 F (R ) ⊂ Uz and that ψz ◦ F (x, y) = (x, y). Hence we can define a vector field Y on Uz −1 that is F -related to X by Y = (ψz )∗(X). We need show that Y can be extended to a 2 smooth vector field on RP . The coordinate expression for Y in ψz 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 ψx and ψy, respectively. 1 The u -component of Y in ψx coordinates is given by y y Y (ψ1) = X(ψ1 ◦ ψ−1) = X( ) = −2 = −2u1. x x z x x Also, 1 1 Y (ψ2) = X( ) = − = −u2. x x x Thus the coordinate expression for Y in ψx 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 ψy 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, 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, 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, ψ).

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 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 → S2 be the map F (x, y, z) = (1 + 4x2 + 4y2)−1/2(−2x, −2y, 1). Is F smooth? 1 c) Let v = ∂/∂ϕ |(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 ∂ϕ |(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 {(0, 0, 0)} → 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 dimensional complex subspaces of C2 = C × C, or alternatively, the orbit space of the action ∗ 2∗ 2 2∗ 1 of C = C r {0} on C = C r {(0, 0)} 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 ∈ Gr(n, 2). Then any two bases {v1, v2}, {w1, w2} for P are related by

X j wi = ai vj, i = 1, 2, (1) j=1,2

j where (ai ) ∈ 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 j (ai ) ∈ GL(2). b) 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). 5

c) 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. d) 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. e) Conclude that every plane P ∈ T (i, j) can be identiﬁed with a unique (n − 2) × 2 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).)

f) Show that each Gij = πGr(T (i, j)) ⊂ Gr(n, 2) is open, and use part e to deﬁne 2n−4 coordinate maps ϕij in each Gij. In particular, show that each ϕij : Gij → R is a homeomorphism. g) Finally compute a representative sample of transition functions to conclude that Gr(n, 2) forms a diﬀerentiable manifold.

n Problem XIII. Let CP and Gr(n, 2) denote the complex projective space and Grass- mann manifold constructed in problems XI and XII. 3 2 4 a) Let S be the unit sphere in C identiﬁed with R , and let the Hopf map πH : 3 2 3 S → S be the restriction of the projection πP to S . Describe the inverse image −1 2 πH (p) of a point p ∈ S . b) Show that the projection πGr : T → Gr(n, 2) is diﬀerentiable. c) Show that each Gr(n, 2) is compact.

Problem XIV. 2 a) Let ϕ3 denote the standard coordinates on U3 = {[x, y, z] ∈ RP | z 6= 0} given by 1 2 1 (u , u ) = ϕ3([x, y, z]) = (x/z, y/z). The coordinate diﬀerential v = ∂/∂u corre- 2 sponds to the directional derivative under some curve α in RP . Geometrically, a curve represents a 1-parameter family of lines in R3. Describe this family for a curve α of your choice. 1 2 u1 u1 b) Let ϕ34, 1 2 = ϕ34(P), denote the coordinates for Gr(4, 2) as constructed in u2 u2 2 problem XII. The coordinate diﬀerential v = ∂/∂u2 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 XV.

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). T b) Let Ψ: Gl(m) → Gl(m) be the mapping Ψ(X) = X X. Describe Ψ∗ : TI Gl(m) → TI Gl(m). 6

−1 c) Let ι: Gl(m) → Gl(m) denote the inverse ι(X) = X . Describe ι∗ : TI Gl(m) → TI Gl(m). d) Let m: Gl(m) × Gl(m) → Gl(m), m(A, B) = AB denote the multiplication

map. Show that the diﬀerential m∗ : TI Gl(m) ⊕ TI Gl(m) → TI Gl(m) is given by m∗(V,W ) = V + W .

Problem XVI. a) Show that [fX, gY ] = fg[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 that vector fields X, Y ∈ X (M) on M are F -related to vector fields Xb, Yb ∈ X (N). Show that the bracket [X,Y ] is F -related to [X,b Yb]. 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. d) Let F and G be as in part c. Show that F (M) ⊂ N is an embedded submanifold.

Problem XVII. a) Let F : M → N be smooth, where M, N are diﬀerentiable manifolds, and suppose that F∗ : TpM → TF (p)N is an isomorphism for all p ∈ M. Prove that F is an 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 XVIII. m a) Let F : Rm → RP be defined by F (x1, x2, . . . , xm) = [x1, x2, . . . , xm, 1]. Show m that F is a smooth map onto a dense open subset of RP . m b) Define similarly G: Cm → CP by G(z1, z2, . . . , zm) = [z1, z2, . . . , zm, 1]. Show m that G is a diffeomorphism onto a dense, open set of CP . 1 1 c) Let pb(z) be a complex polynomial in one variable and let G: C → U ⊂ CP be as in part b, where U denotes the image of G. Define p: U → U by the condition that p ◦ G = G ◦ p. Show that p can be extended to a smooth map on the entire 1 b CP .

Problem XIX. Let U ⊂ R2 be an open set. A proper coordinate patch is a one-to-one immersion x: U → R3 such that the inverse function x−1 : x(U) → U is continuous in the subspace topology of x(U). A surface is a subset S ⊂ R3 equipped with the subspace topology such that for each point p ∈ S, there is a proper coordinate patch whose image contains a neighborhood of p in S. Show that a surface is a diﬀerentiable manifold.

Problem XX. 2 a) Show that the set of all lines in R2 can be identified with an open subset of RP . 7 b) Show that the set of all planes in R3 can be identified with an open subset of 3 RP . c) Show that the space of all oriented lines in R3 can be identified with TS2.