The Lumenary · First-year graduate

Smooth manifolds

Name: ______________________

Date: ______________________

You will learn to

Define smooth manifolds by atlases, and prove that spheres, projective spaces and the matrix groups $O(n)$ and $SL(n, \R)$ are smooth manifolds, using charts and the regular value theorem
Work with tangent vectors as derivations, differentials of smooth maps, vector fields and flows, and explain why vectors and covectors transform by different laws
Compute with differential forms, the wedge product, the exterior derivative and pullbacks
Prove Stokes's theorem for compactly supported forms on oriented manifolds with boundary
Compute de Rham cohomology of simple manifolds, prove $H^1_{\mathrm{dR}}(S^1) \cong \R$ , and state de Rham's theorem

1.
What is the dimension of the orthogonal group $O(5)$ as a smooth manifold?
2.
Which of these subsets are smooth embedded submanifolds of the ambient Euclidean space? (Choose all that apply.)
- A.The hyperboloid $x^2 + y^2 - z^2 = 1$ in $\R^3$
- B.The double cone $x^2 + y^2 - z^2 = 0$ in $\R^3$
- C.The graph of $y = |x|$ in $\R^2$
- D. $SL(3, \R)$ in $M_3(\R) \cong \R^9$
- E.The union of the coordinate axes in $\R^2$
3.
To show that $O(n)$ is a manifold we used $F(A) = A^{\mathsf T}A$ with codomain $\mathrm{Sym}_n$ , the symmetric matrices. What goes wrong if we take the codomain to be all of $M_n(\R)$ ?
- A. $dF_A$ is never surjective onto $M_n(\R)$ , so $I$ is not a regular value
- B. $F$ is no longer smooth
- C. $O(n)$ would get the wrong dimension, $n^2 - n^2 = 0$
- D.Nothing goes wrong; any codomain containing the image works
4.
Coordinates change from $x$ to $\tilde x$ near $p$ , with Jacobian matrix $J = \partial \tilde x/\partial x$ at $p$ . A tangent vector has components $v$ in the old chart and $Jv$ in the new one. How do the components of a covector $\omega$ (for example $df$ ) change?
- A. $\tilde\omega = J\omega$
- B. $\tilde\omega = (J^{\mathsf T})^{-1}\omega$
- C. $\tilde\omega = J^{-1}\omega$
- D.The components do not change; a covector is a number
5.
In the proof that $\dim T_pM = n$ we wrote $f(x) = f(a) + \sum_i (x^i - a^i) g_i(x)$ and applied a derivation $v$ at $a$ . Why does the term $(x^i - a^i)(a) \cdot v(g_i)$ vanish?
- A.Because the Leibniz rule evaluates the other factor at $a$ , and $x^i - a^i$ is $0$ at $a$
- B.Because $v(g_i) = 0$ for every smooth function $g_i$
- C.Because derivations kill constants
- D.By the chain rule
6.
Let $R = [0, 2] \times [0, 3]$ with the anticlockwise boundary. Use Stokes's theorem to compute $\displaystyle\oint_{\partial R} x^2 y\, dx + \Big(\frac{x^3}{3} + 2x\Big) dy$ .
7.
What is the dimension of $H^1_{\mathrm{dR}}(\R^2 \setminus \{p_1, p_2, p_3\})$ , the first de Rham cohomology of the plane with three points removed?
8.
Which Lean statement correctly formalises " $d\theta$ is not exact on the circle $\R/2\pi\Z$ "?
- A.¬ ∃ g : ℝ → ℝ, Function.Periodic g (2 * Real.pi) ∧ ∀ x, HasDerivAt g 1 x
- B.¬ ∃ g : ℝ → ℝ, ∀ x, HasDerivAt g 1 x
- C.∫ x in (0 : ℝ)..2 * Real.pi, (1 : ℝ) = 2 * Real.pi
- D.∃ g : ℝ → ℝ, Function.Periodic g (2 * Real.pi) ∧ ¬ ∀ x, HasDerivAt g 1 x
9.
Write period(P, Q, n) that approximates the integral of the $1$ -form $P(x, y)\,dx + Q(x, y)\,dy$ round the unit circle $(\cos t, \sin t)$ , $0 \le t \le 2\pi$ , anticlockwise, by the rectangle rule with n equally spaced values of $t$ . Then check: the period of $d\theta = (x\,dy - y\,dx)/(x^2 + y^2)$ is $2\pi$ , and the period of the exact form $d(x^2 y)$ is $0$ .
Computational — Practice

Answer key — Smooth manifolds

1.
$I$ is a regular value of $F : M_5(\R) \to \mathrm{Sym}_5$ , $F(A) = A^{\mathsf T}A$ (Corollary 4), so $\dim O(5) = 25 - 15 = 10 = 5 \cdot 4/2$ . Equivalently, the tangent space at $I$ is the space of skew-symmetric $5 \times 5$ matrices, which have $10$ free entries above the diagonal.
2.
The hyperboloid and $SL(3, \R)$ are level sets of regular values, so they are embedded submanifolds (of dimensions $2$ and $8$ ). The double cone and the cross are not even topological manifolds (removing the special point leaves too many components). The graph of $|x|$ is a topological manifold but has a corner, so it is not a smooth embedded submanifold of $\R^2$ .
3.
Every value of $dF_A(H) = A^{\mathsf T}H + H^{\mathsf T}A$ is symmetric, so $dF_A : M_n(\R) \to M_n(\R)$ has image inside $\mathrm{Sym}_n \subsetneq M_n(\R)$ (for $n \ge 2$ ) and is not surjective. The hypothesis of the regular value theorem fails. Restricting the codomain to $\mathrm{Sym}_n$ , where $F$ really takes its values, restores surjectivity: $H = \tfrac12 AS$ maps to $S$ .
4.
From $d\tilde x^j = \sum_i (\partial\tilde x^j/\partial x^i)\, dx^i$ we get $\omega_i = \sum_j \tilde\omega_j\, \partial\tilde x^j/\partial x^i$ , i.e. $\omega = J^{\mathsf T}\tilde\omega$ , so $\tilde\omega = (J^{\mathsf T})^{-1}\omega$ . Check: $\tilde\omega^{\mathsf T}\tilde v = \omega^{\mathsf T}J^{-1}Jv = \omega^{\mathsf T}v$ .
5.
By the Leibniz rule at $a$ , $v\big((x^i - a^i)g_i\big) = (a^i - a^i)\, v(g_i) + g_i(a)\, v(x^i)$ (using $v(a^i) = 0$ ). The first term is zero because $x^i - a^i$ vanishes at $a$ . What remains is $\partial_i f(a)\, v(x^i)$ , which is why $v = \sum_i v(x^i)\,\partial_i|_a$ .
6.
$d\omega = \big(\partial_x(x^3/3 + 2x) - \partial_y(x^2 y)\big)\, dx \wedge dy = (x^2 + 2 - x^2)\, dx\wedge dy = 2\, dx \wedge dy$ . By Stokes, $\oint_{\partial R}\omega = \int_R 2\, dA = 2 \cdot 6 = 12$ .
7.
The angle forms $d\theta_1, d\theta_2, d\theta_3$ are closed, and $d\theta_k$ has period $2\pi$ round $p_k$ and $0$ round the others, so they are linearly independent in cohomology. Every closed $1$ -form $\omega$ is cohomologous to $\sum_k \frac{c_k}{2\pi} d\theta_k$ , where $c_k$ is its period round $p_k$ : the difference has zero periods, hence is exact. So $H^1_{\mathrm{dR}} \cong \R^3$ ; the lesson's second experiment shows the two-puncture case.
8.
Functions on $\R/2\pi\Z$ correspond to $2\pi$ -periodic functions on $\R$ , and $d\theta$ pulls back to the constant form $1\,d\theta$ . So " $d\theta$ is not exact" says: no $2\pi$ -periodic $g$ has $g' = 1$ everywhere — option (a). Option (b) drops periodicity and becomes false; (c) is a true lemma used in the proof; (d) is a trivial existential.
9.
```
import math

def period(P, Q, n):
    h = 2 * math.pi / n
    total = 0.0
    for k in range(n):
        t = k * h
        x, y = math.cos(t), math.sin(t)
        total += P(x, y) * (-math.sin(t)) + Q(x, y) * math.cos(t)
    return total * h
```
For $d\theta$ the integrand is identically $1$ , so the period is $2\pi$ ; for $d(x^2y)$ it is $0$ (exact forms have zero periods by Stokes), and for $(x\,dy - y\,dx)/2$ it is the area $\pi$ . The rectangle rule is extremely accurate here because the integrands are smooth and periodic. A non-zero period proves that $d\theta$ is not exact on the punctured plane — but the computation only estimates the period; the exact value $2\pi$ comes from the pullback $F^*\omega = d\theta$ in polar coordinates.