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Homework
(1) Show that the induced topology and the quotient topology fulfill the axioms of a topological space. (2) Prove that S^1 = {x^2 +y^2 = 1} with the induced topology (from R^2 ) is homeomorphic to S^1 = [0, 1]/ ∼ with the quotient topology. (3) Prove that S^1 (with either topology in Problem (2)) is a topological manifold. (4) Give an example of a Hausdorff, second countable topological space which is not a topological manifold. (5) Let φ : V → W be a linear map between vector spaces (over a fixed ground field k). Prove the following: (a) Ker(φ) is a vector subspace of V. (b) Im(φ) is a vector subspace of W. (6) Let V ⊂ W be a vector subspace. Prove that the quotient W/V is a vector space. (7) Let V ∗^ (called the dual of V ) be the set of linear maps from V to the base field k. Prove that V ∗^ is a vector space over k. Assuming V is finite-dimensional, exhibit a basis for V ∗^ in terms of a basis for V. (8) Prove that Sn^ = {x^20 + · · · + x^2 n = 1} ⊂ Rn+1^ is a smooth n-dimensional manifold, by taking stereographic projections. (9) Complete the proof that RPn^ is a smooth n-dimensional manifold. (10) Define CPn^ = (Cn+1^ − {(0,... , 0)})/ ∼, where (z 0 ,... , zn) ∼ (tz 0 ,... , tzn), t ∈ C−{ 0 }. Prove that CPn^ is a smooth 2n-dimensional manifold. (Recall that C →∼ R^2 , where z = x + iy 7 → (x, y).) (11) Complete the proof that T 2 = R^2 /Z^2 is a smooth manifold of dimension 2. (12) Prove that Sn^ = {x^20 +· · ·+x^2 n = 1} ⊂ Rn+1^ is an n-dimensional manifold by showing it is a regular value of some map. (13) Prove that if M, N are manifolds, f : M → N is a submersion, and U ⊂ M is open, then f (U) is open in N. (14) Prove that if M is compact and N is connected, every submersion f : M → N is surjective. Prove that there is no submersion from a compact manifold to Rn. (15) Define SL(2, R) = {A ∈ M 2 | det(A) = 1}, where Mn is the set of n by n matrices with real entries. Prove that SL(2, R) is a smooth manifold. What is its dimension? (SL(n, R) is called the special linear group over R of dimension n.) (16) Let N be an n-dimensional submanifold of M (of dimension m) and x ∈ N. Prove that there exists a local coordinate system {x 1 ,... , xm} for M on U ∋ x so that U ∩ N = {xn+1 = 0,... , xm = 0}. (17) Define the orthogonal group O(n) = {A ∈ Mn | AAT^ = I}, where AT^ is the transpose of A. Prove that O(n) is a manifold as follows: Consider the map φ : Mn → Sym(n) which sends A 7 → AAT^ , where Sym(n) is the set of symmetric n × n matrices (i.e., BT^ = B). 1
(a) Show that Sym(n) is a manifold. Compute its dimension. (b) Compute the derivative of φ and show φ is a submersion. (c) What is the dimension of O(n)? (d) Prove that O(n) is compact. (18) Let F (x 1 ,... , xn) be a homogeneous polynomial of degree d in n (real) variables, i.e.,
F (tx 1 ,... , txn) = td^ · F (x 1 ,... , xn). (a) Prove Euler’s identity: ∑^ n
i=
xi ·
∂F
∂xi
= d · F.
(b) Prove that the set F −^1 (a), a 6 = 0, is a submanifold of Rn. (19) Give a detailed proof of the equivalence of the three definitions of TpM given in class. Pay special attention to good exposition. (20) Recall that Fp is the set of germs of functions on a manifold M which vanish at p ∈ M. Let F (^) pk be the ideal of C∞(p) generated by f 1 · · · fk, where fi ∈ Fp. (This means that every element of F (^) pk is a sum
i gifi^1 · · ·^ fik,^ gi^ ∈^ C
∞(p), f ij ∈ Fp.) (a) Prove that, in every coordinate system (x 1 ,... , xn), an element f ∈ F (^) pk has a Taylor expansion which vanishes up to order k. (b) Compute the dimension of F (^) pk /F (^) pk +1. (c) Construct a smooth manifold E π → M whose fiber at p ∈ M is F (^) p^1 /F (^) p^3. (This involves writing down coordinate charts and computing transition functions.) (21) Consider the cotangent bundle π : T ∗M → M. In class we gave an atlas for T ∗M in terms of π−^1 (Uα), where {Uα} was an atlas for M. Compute the Jacobian for the transition functions on the overlaps π−^1 (Uα) ∩ π−^1 (Uβ ). (22) Prove that d(f g) = f dg + gdf. (23) Consider the map i : S^1 = [0, 2 π]/(0 ∼ 2 π) → R^2 , θ 7 → (cos θ, sin θ). Compute i∗((x^2 + y)dx + (3 + xy^2 )dy). (24) In class we defined the derivative map as follows: Let φ : M → N be a smooth map between manifolds. Then the derivative φ∗ : TpM → Tφ(p)N is given by X 7 → X ◦ φ∗, where X : C∞(p) → R is a derivation and φ∗^ is the pullback C∞(φ(p)) → C∞(p). Give an equivalent definition for φ∗ in terms of Definition 1 of a tangent space. (25) Let φ : M → N be a smooth map between manifolds. Prove that the following diagram commutes: Ω^0 (N) φ∗ → Ω^0 (M) d ↓ ↓ d Ω^1 (N) φ∗ → Ω^1 (M)
(26) Let φ : [a, b] → [c, d] be a diffeomorphism with coordinates s for [a, b] and t for [c, d]. A global 1-form ω on [c, d] can be written as f (t)dt, for some smooth function f (t). (a) Write φ∗ω in terms of coordinates s on [a, b].
given ω =
i 1 ,...,ik
fi 1 ,...,ik (y)dyi 1... dyik.
Show that φ∗ω is well-defined. (39) Compute all the de Rham cohomology groups of S^2. Then inductively compute all the de Rham cohomology groups of Sn. (40) Suppose the manifold M is the disjoint union of manifolds M 1 and M 2. Then prove that HdRk (M) = HdRk (M 1 ) ⊕ HdRk (M 2 ). (41) Let φ : M → N be a smooth map between manifolds. If ω ∈ Ωk(N), then prove that d(φ∗ω) = φ∗(dω).
(42) Complete the proof that the short exact sequence of cochain maps 0 → A φ → B ψ → C → 0 gives rise to a long exact sequence. (This problem should be done carefully and completely — it’s worth 30 points.) (43) Let M be an oriented manifold of dimension n and ω ∈ Ωn(M). Let {fα} be a parti- tion of unity subordinate to an open cover {Uα}. We defined
M ω^ =^
α
Uα φ
∗ α(fαω), where φα : Uα → Rn^ are the coordinate functions. Prove that this definition does not depend on the partition of unity or the open cover chosen. (44) Using the version of Stokes’ Theorem given in class, prove the classical Stokes’ The- orem: Let S be a compact, oriented 2-manifold (i.e., a surface) with boundary in R^3 and let F = (F 1 , F 2 , F 3 ) be a smooth vector field defined on a neighborhood of S. Then: (^) ∫
S
〈curl F, n〉dA =
∂S
F 1 dx + F 2 dy + F 3 dz,
where n is a unit normal to S, 〈·〉 is the standard inner product, dA = n 1 dydz + n 2 dzdx + n 3 dxdy, curl F =
∂F 3 ∂y −^
∂F 2 ∂z ,^
∂F 1 ∂z −^
∂F 3 ∂x ,^
∂F 2 ∂x −^
∂F 1 ∂y
The Cauchy Integral Formula. We will now define complex-valued forms on a complex manifold M. Complex-valued differential forms are simply sums ω = ω 1 + iω 2 , where ω 1 and ω 1 are real. The wedge product extends in the obvious way, and dω def = dω 1 + idω 2 ,
M
ω def =
M
ω 1 + i
M
ω 2.
Note that Stokes’ Theorem is valid for complex-valued forms since it is valid sepa- rately for the real part and the complex part.
On C = R^2 , let z = x + iy be the complex coordinate and z = x − iy be its complex conjugate. Then dz = dx + idy and dz = dx − idy.
(45) Consider the function f : C → C, z 7 → f (z). Show that ω = f (z)dz is closed (dω = 0) if and only if f (z) = f (x, y) (f is viewed as R^2 → R^2 ) satisfies the Cauchy-Riemann
equation ∂f ∂y
= i
∂f ∂x
Recall that a smooth function f satisfies the Cauchy-Riemann equation if and only if f is holomorphic. Also recall that rational functions in z are holomorphic where defined. For example, f (z) = (^) z−^1 a is holomorphic on C − {a}.
(46) Suppose f is a holomorphic function on a domain Ω ⊂ C. Prove that if γ 0 and γ 1 are homotopic curves in Ω, then ∫
γ 0
f (z)dz =
γ 1
f (z)dz.
Two curves γ 0 , γ 1 : S^1 → Ω are said to be homotopic if there exists a smooth Γ : S^1 × [0, 1] → Ω with Γ(θ, t) = γt(θ). (47) Let C be a circle of radius R around a ∈ C. Then prove that ∫
C
z − a
dz = 2πi.
(48) Let f (z) be a holomorphic function on Ω, let C ⊂ Ω be a circle of radius R around a ∈ Ω, and let γ be a smooth closed curve which is homotopic to C inside Ω − {a}. Then prove that ∫
γ
f (z) z − a
dz = 2πif (a).
[Hint: By Exercise (46), the integral only depends on the curve up to homotopy. Therefore, we may assume γ = C. Shrink the radius R and take the limit.] (49) The vector field ( (^) x 2 −+yy 2 , (^) x (^2) +xy 2 ) (defined almost everywhere) on R^2 has curl zero, but it cannot be written as the gradient of any function. Explain what this means in terms of de Rham cohomology. (50) Let V be an R-vector space. Prove that the interior product iv :
∧k V ∗^ →
∧k− 1 V ∗, v ∈ V , where f 1 ∧ · · · ∧ fk 7 →
l(−1)
l+1f 1 ∧... fl(v) · · · ∧ fk, is well-defined.
(51) Let M be a manifold and X a global vector field on M. Show that L = d◦iX +iX ◦d : Ωk(M) → Ωk(M) is a derivation, i.e., it satisfies L(α ∧ β) = L(α) ∧ β + α ∧ L(β). Here iX denotes the interior product with X. (52) Compute the de Rham cohomology of a compact 2-dimensional manifold (surface) of genus g without boundary. [For this problem, you may feel free to draw pictures of the surface of genus g and its decomposition into pieces. You will probably need to use the Mayer-Vietoris sequence a couple of times, as well as the homotopy properties.] Hint: remove a disk from the surface and see what you are left with. (53) For each integer n, exhibit a smooth map S^1 → S^1 of degree n. (You must prove that your map has degree n.)
(b) 〈R(X, Y )Z, W 〉 = 〈R(Z, W )X, Y 〉. (c) R(X, Y )Z + R(Y, Z)X + R(Z, X)Y = 0. (69) Let Σ = {x^2 +y^2 +z^2 = R^2 } ⊂ R^3 , and let g be the metric induced from the standard Euclidean metric on R^3 to Σ. Compute: (a) the induced metric g and (b) the Levi-Civita connection of (Σ, g), locally near (0, 0 , R) using the coordinates (x, y), given by projecting onto the first two coordinates of R^3.
