Math 225A: Smooth Manifolds

Spivak, Problem 1-3. Show that every manifold is locally compact and locally path-connected, and that a connected manifold is path-connected.

Spivak, Problems 1-5 and 1-6. Using invariance of domain, show: (a) the neighborhood $U$ in the definition of a manifold is necessarily open; (b) the integer $n$ in the definition is uniquely determined at each point; (c) if $M$ is connected, the dimension of $M$ at $x$ is the same for all $x \in M$; (d) a subset of an $n$-manifold is an $n$-manifold if and only if it is open.

Spivak, Problem 1-11. Show that every connected manifold has a countable base for its topology and a countable dense subset.

Spivak, Problem 1-12. Let $P \colon S^{n-1} \setminus \{(0,\ldots,0,1)\} \to \mathbb{R}^{n-1}$ be stereographic projection from the north pole. Find an explicit formula for $P$ and for $P^{-1}$, and show that $P$ is a homeomorphism.

Spivak, Problem 1-15. (a) Show that $\mathbb{P}^1$ is homeomorphic to $S^1$. (b) Regarding $\mathbb{P}^{n-1} \subset \mathbb{P}^n$ in the obvious way, show that $\mathbb{P}^n \setminus \mathbb{P}^{n-1}$ is homeomorphic to the open unit ball in $\mathbb{R}^n$.

Additional exercise. Let $X$ be the quotient of $\mathbb{R} \times \{0,1\}$ obtained by identifying $(t,0)$ with $(t,1)$ for all $t \ne 0$ (the line with two origins). Show that $X$ is locally homeomorphic to $\mathbb{R}$ and second countable, but not Hausdorff.

Spivak, Problem 2-1(a). Show that being $C^\infty$-related is not an equivalence relation on charts.

Spivak, Problem 2-4. How many distinct $C^\infty$ structures are there on $\mathbb{R}$? (Distinct maximal atlases, not structures up to diffeomorphism.)

Spivak, Problem 2-6. Check that the stereographic projections from $(0,\ldots,0,1)$ and $(0,\ldots,0,-1)$ are $C^\infty$-related to each of the $2n$ charts $f_i, g_i$ that delete the $i$-th coordinate on the hemispheres $\{\pm x^i > 0\}$ of $S^{n-1}$.

Spivak, Problem 2-8. Let $M_1, M_2$ be smooth manifolds. Show that a map $f \colon N \to M_1 \times M_2$ is $C^\infty$ if and only if $\pi_1 \circ f$ and $\pi_2 \circ f$ are $C^\infty$, and that the product structure is the only smooth structure on $M_1 \times M_2$ with this property. Show also that the slice maps $p_1 \mapsto (p_1, \bar p_2)$ are $C^\infty$.

Spivak, Problem 2-9. Let $g \colon S^n \to \mathbb{P}^n$ be the quotient map $p \mapsto [p]$. Show that $f \colon \mathbb{P}^n \to M$ is $C^\infty$ if and only if $f \circ g$ is $C^\infty$.

Spivak, Problem 2-16. Show that the function $f \colon \mathbb{R} \to \mathbb{R}$ with $f(x) = e^{-1/x}$ for $x > 0$ and $f(x) = 0$ for $x \le 0$ is $C^\infty$.

Spivak, Problem 2-18. On $\mathbb{R}^2$ consider the coordinate system $y^1(a,b) = a$, $y^2(a,b) = a + b$. (a) Compute $\partial f/\partial y^1(a,b)$ from the definition. (b) Compute it again using the chain rule (Proposition 2-3). Notice that $\partial f/\partial y^1 \ne \partial f/\partial x^1$ even though $y^1 = x^1$, where $x$ is the identity coordinate system.

Spivak, Problem 2-19. Express the Laplacian $\dfrac{\partial^2}{\partial x^2} + \dfrac{\partial^2}{\partial y^2}$ in polar coordinates. (Answer: $\frac{1}{r}\left[\frac{\partial}{\partial r}\left(r \frac{\partial}{\partial r}\right) + \frac{\partial}{\partial \theta}\left(\frac{1}{r}\frac{\partial}{\partial \theta}\right)\right]$.)

Additional exercise. Derive the implicit function theorem from the inverse function theorem, by applying the latter to $F(x,y) = (x, f(x,y))$.

Additional exercise. Let $y^1, \ldots, y^n$ be smooth functions defined near a point $p$ of an $n$-manifold $M$, and let $x$ be a chart around $p$. Show that $(y^1,\ldots,y^n)$ is a coordinate system on some neighborhood of $p$ if and only if the matrix $\left(\partial y^i/\partial x^j(p)\right)$ is invertible.

Spivak, Problem 2-26. (a) If $U \subset \mathbb{R}^k$ is open and $f \colon U \to \mathbb{R}^{n-k}$ is $C^\infty$, show that the graph of $f$ is a submanifold of $\mathbb{R}^n$. (b) Show that every submanifold of $\mathbb{R}^n$ is locally of this form, after renumbering the coordinates.

Spivak, Problem 2-20. If $f \colon M^n \to N^m$ is $C^1$ and $m > n$, show that $f(M)$ has measure zero (assuming $M$ has at most countably many components).

Additional exercise. Show that for a smooth map $f \colon M \to N$, the set $\{p \in M : \operatorname{rank}_p f \ge k\}$ is open. Give an example where the rank is not locally constant.

Spivak, Problem 2-27. (a) Show that an immersion from one $n$-manifold to another is an open map. (b) If $M$ and $N$ are $n$-manifolds with $M$ compact and $N$ connected, and $f \colon M \to N$ is an immersion, show that $f$ is onto.

Additional exercise. Prove Sard's theorem directly for a $C^1$ function $f \colon \mathbb{R} \to \mathbb{R}$: the set of critical values has measure zero.

Additional exercise. Show that $f(x) = x^2$ on $\mathbb{R}$ does not have constant rank near $0$, and show that there are no charts $x, y$ with $y \circ f \circ x^{-1} = 0$ near $0$. Why does this not contradict Theorem 2-9?

Spivak, Problem 2-24. (a) Let $M$ be an $n$-manifold. Show that $M_1 \subset M$ can be made into a $k$-dimensional submanifold of $M$ if and only if around each point of $M_1$ there is a chart $(x,U)$ of $M$ with $M_1 \cap U = \{p \in U : x^{k+1}(p) = \cdots = x^n(p) = 0\}$. (b) Show that $M_1$ can be made into a closed submanifold if and only if such charts exist around every point of $M$.

Spivak, Problem 2-25. Show that the set $\{(x, |x|) : x \in \mathbb{R}\}$ is not the image of any immersion of $\mathbb{R}$ into $\mathbb{R}^2$.

Spivak, Problem 2-28. Prove Proposition 2-12: if $f \colon M^n \to N$ has constant rank $k$ on a neighborhood of $f^{-1}(y)$, then $f^{-1}(y)$ is a closed submanifold of $M$ of dimension $n-k$ (or is empty).

Spivak, Problem 2-33. (a) Show that $\mathrm{SL}(n,\mathbb{R}) = \{A : \det A = 1\}$ is a closed submanifold of $\mathrm{GL}(n,\mathbb{R})$ of dimension $n^2 - 1$. (b) Let $\psi(A) = A A^t$, a map from $\mathrm{GL}(n,\mathbb{R})$ to the symmetric matrices. Show that $\mathrm{O}(n) = \psi^{-1}(I)$ is a compact submanifold of dimension $n(n-1)/2$. (Hint: $\psi(BA) = \psi(B)$ for $A \in \mathrm{O}(n)$, so it suffices to compute the rank of $\psi$ at $I$.)

Spivak, Problem 2-34. Let $M(m,n;k)$ be the set of $m \times n$ matrices of rank $k$. Show that $M(m,n;k)$ is a submanifold of the space of all $m \times n$ matrices, of dimension $k(m+n-k)$.

Spivak, Problem 2-29. Let $g \colon \mathbb{P}^2 \to \mathbb{R}^3$ be $g([x,y,z]) = (yz, xz, xy)$, whose image is the Steiner surface. Show that $g$ fails to be an immersion at exactly 6 points.

Spivak, Problem 2-17. Show that every closed subset $C$ of a manifold $M$ is the zero set $f^{-1}(0)$ of some $C^\infty$ function $f \colon M \to [0,1]$.

Spivak, Problem 2-30(f). Show that every smooth manifold admits a proper $C^\infty$ function $f \colon M \to \mathbb{R}$ (proper means preimages of compact sets are compact).

Spivak, Problem 2-32. (a) If $M_1 \subset M$ is a closed submanifold, $U \supset M_1$ is open, and $f \colon M_1 \to \mathbb{R}$ is $C^\infty$, show that there is a $C^\infty$ function $\bar f \colon M \to \mathbb{R}$ with $\bar f = f$ on $M_1$ and support in $U$. (b) Show this fails for $M = \mathbb{R}$ and $M_1 = (0,1)$.

Spivak, Problem 2-31. (a) Find a cover of $[0,1]$ which is point-finite (every point is in only finitely many members) but not locally finite. (b) Prove the shrinking lemma for countable point-finite covers.

Additional exercise. Let $C_1, C_2$ be disjoint closed subsets of a manifold $M$. Show there is a $C^\infty$ function $f \colon M \to [0,1]$ with $f = 0$ on $C_1$ and $f = 1$ on $C_2$.

Spivak, Problem 3-21. Show that for $p \in S^2$, the vector $p_p \in \mathbb{R}^3_p$ is not tangent to $S^2$, by showing that $\langle p, c'(0) \rangle = 0$ for every curve $c$ in $S^2$ with $c(0) = p$.

Additional exercise. Let $f \colon \mathbb{R}^N \to \mathbb{R}^k$ be smooth and let $y$ be a regular value, so $M = f^{-1}(y)$ is a submanifold. Show that the tangent space of $M$ at $p$ is the kernel of $f_{*p}$.

Additional exercise. Show that the tangent space of $\mathrm{O}(n)$ at the identity is the space of skew-symmetric matrices, and the tangent space of $\mathrm{SL}(n,\mathbb{R})$ at the identity is the space of matrices of trace zero.

Spivak, Problem 3-15. (a) Show that $f \colon M \to N$ is an immersion if and only if $f_*$ is one-one on each tangent space, and that $\operatorname{rank}_p f$ equals the rank of $f_{*p}$. (b) If $g$ is a diffeomorphism, show that the rank of $f \circ g$ at $a$ equals the rank of $f$ at $g(a)$.

Spivak, Problem 3-17. Show that $T\mathbb{P}^2$ is homeomorphic to the space obtained from the tangent bundle of $S^2 \subset \mathbb{R}^3$ by identifying the vector $v$ at $p$ with the vector $-v$ at $-p$.

Spivak, Problem 3-3. Show that in the definition of an equivalence of bundles it suffices to assume that the map $E_1 \to E_2$ is continuous; its inverse is then automatically continuous.

Spivak, Problem 3-7. (a) Show that for any bundle, the zero section is a section. (b) Show that an $n$-plane bundle is trivial if and only if it has $n$ sections that are linearly independent at every point. (c) Show that locally every $n$-plane bundle has $n$ linearly independent sections.

Spivak, Problem 3-20. Let $T \colon \mathbb{R}^n \to \mathbb{R}^n$ be a linear isomorphism, and form the space obtained from $[0,1] \times \mathbb{R}^n$ by identifying $(0,v)$ with $(1, Tv)$. (a) Show that this is the total space of a vector bundle over $S^1$. (b) Show that this bundle is orientable if and only if $\det T > 0$.

Spivak, Problem 3-23. Given a bundle $\xi = \pi \colon E \to X$ and a continuous $f \colon Y \to X$, let $E' = \{(y,e) \in Y \times E : f(y) = \pi(e)\}$ with $\pi'(y,e) = y$. (a) Show that this is a bundle $f^*\xi$ over $Y$. (b) Show that if a bundle map from $\xi''$ to $\xi$ covering $f$ is an isomorphism on each fibre, then $\xi'' \cong f^*\xi$. (c) Show that $(f \circ g)^*\xi \cong g^* f^*\xi$. (d) If $i \colon A \to X$ is inclusion, show that $i^*\xi \cong \xi|A$.

Spivak, Problem 3-24. Define the Whitney sum $\xi \oplus \eta$ of two bundles over $B$, whose fibre over $p$ is the direct sum of the fibres. Show that $f^*(\xi \oplus \eta) \cong f^*\xi \oplus f^*\eta$, and that $\xi \oplus \eta \cong \Delta^*(\xi \times \eta)$, where $\Delta \colon B \to B \times B$ is the diagonal.

Additional exercise. Show that the tautological line bundle over $\mathbb{P}^1$ (the fibre over a line is that line) is equivalent to the Möbius bundle.

Spivak, Problem 3-8. Check that the relation $(x,v) \sim_p (y,w)$ used to define $TM$ is an equivalence relation, and that the definition of $f_*$ does not depend on the choice of coordinate systems.

Spivak, Problem 3-9. (a) Show that the correspondence between $[x,v]_p$ and equivalence classes of curves makes $f_*$ correspond to $f_\sharp$. (b) Show that under the correspondence between $[x,a]_p$ and derivations, $(f_*\ell)(g) = \ell(g \circ f)$.

Spivak, Problem 3-11. If $g \colon \mathbb{R} \to \mathbb{R}$ is $C^\infty$, show that $g(x) = g(0) + g'(0)x + x^2 h(x)$ for some $C^\infty$ function $h$.

Spivak, Problem 3-12. Let $\mathcal{F}_p$ be the $C^\infty$ functions $f$ on $M$ with $f(p) = 0$, and let $W$ be the span of all products $fg$ with $f, g \in \mathcal{F}_p$. Show that the derivations at $p$ correspond to the dual space $(\mathcal{F}_p / W)^*$, and that if $x$ is a chart with $x(p) = 0$ then $x^1 + W, \ldots, x^n + W$ is a basis of $\mathcal{F}_p / W$.

Spivak, Problem 3-14. If $f \colon M \to N$ and $f_*$ is the zero map on each tangent space, show that $f$ is constant on each component of $M$.

Spivak, Problem 3-27. Show that the Jacobian matrix of $y_* \circ (x_*)^{-1}$ has the block form with $D(y \circ x^{-1})$ in both diagonal blocks and zero in the upper right block. Conclude that the manifold $TM$ is always orientable.

Spivak, Problem 3-16. Let $M$ be an oriented manifold with boundary. (a) Show that for $p \in \partial M$, the subspace $i_*((\partial M)_p)$ of $M_p$ does not depend on the chart. (b) Show that the notion of an outward-pointing vector does not depend on the chart. (c) Show that $\partial M$ has a unique orientation $\partial\mu$ such that $[v_1,\ldots,v_{n-1}] \in \partial\mu$ if and only if $[w, v_1, \ldots, v_{n-1}] \in \mu$ for every outward $w$. (d) Show that for the usual orientation of $\mathbb{H}^n$, the induced orientation on $\mathbb{R}^{n-1} = \partial \mathbb{H}^n$ is $(-1)^n$ times the usual one.

Spivak, Problem 3-18. There is no nowhere-zero vector field on $S^2$. Show that there is one on $S^2$ minus a point (which is diffeomorphic to $\mathbb{R}^2$), and that it can be chosen to look like a magnetic dipole near the missing point.

Spivak, Problem 3-22. Suppose that the restriction of $TM$ to $A$ is trivial for every $A \subset M$ homeomorphic to $S^1$. Show that $M$ is orientable.

Spivak, Problem 3-26. (a) Show that $T(M \times N) \cong \pi_M^* TM \oplus \pi_N^* TN$. (b) Show that if $M$ and $N$ are orientable then so is $M \times N$. (c) Show that if $M \times N$ is orientable then so are $M$ and $N$.

Additional exercise. Show that the Klein bottle is not orientable.

Additional exercise. Let $f \colon \mathbb{R}^n \to \mathbb{R}$ be smooth with regular value $c$. Show that $f^{-1}(c)$ is orientable.

Spivak, Problem 4-1. Let $f \colon M^n \to N^m$ and let $x$, $y$ be charts around $p$ and $f(p)$. Show that $$f_*\left(\frac{\partial}{\partial x^i}\Big|_p\right) = \sum_{j=1}^m \frac{\partial (y^j \circ f)}{\partial x^i}(p)\, \frac{\partial}{\partial y^j}\Big|_{f(p)} \quad\text{and}\quad f^*(dy^j) = \sum_{i=1}^n \frac{\partial (y^j \circ f)}{\partial x^i}\, dx^i.$$

Spivak, Problem 4-2. If $f, g \colon M \to \mathbb{R}$ are $C^\infty$, show that $d(fg) = f\, dg + g\, df$.

Spivak, Problem 4-3. Let $f \colon M \to \mathbb{R}$ be $C^\infty$. For $v \in M_p$ show that $f_*(v) = df(v)_{f(p)} \in \mathbb{R}_{f(p)}$.

Spivak, Problem 4-4. (a) Show that if the ordered bases $v_1,\ldots,v_n$ and $w_1,\ldots,w_n$ of $V$ are equally oriented, then so are their dual bases. (b) Show that a bundle $\xi$ is orientable if and only if $\xi^*$ is orientable.

Spivak, Problem 4-6. (a) Show that the natural isomorphism $V \to V^{**}$ commutes with every linear map $f \colon V \to W$ and its double dual. (b) Show that there is no family of isomorphisms $V \to V^*$, one for each vector space $V$, compatible in the same way with all linear maps $f$. (Hint: there is not even one for $V = \mathbb{R}$.)

Additional exercise. On $\mathbb{R}^2$ minus the non-negative $x$-axis, express $dr$ and $d\theta$ in terms of $dx$ and $dy$, where $(r,\theta)$ are polar coordinates.

Additional exercise. Let $\omega$ be a 1-form. Show that $(X,Y) \mapsto X(\omega(Y)) - Y(\omega(X)) - \omega([X,Y])$ is linear over $C^\infty(M)$ in each variable, where $[X,Y] = XY - YX$. Show that $(X,Y) \mapsto X(\omega(Y))$ alone is not.

Spivak, Problem 4-5 (ix) and (x). The rank of a covariant 2-tensor $a_{ij}$ is the rank of the matrix $(a_{ij})$. Show that the rank does not depend on the coordinate system. Show that the tensor $a_i b_j$ has rank 1, and that the symmetric tensor $a_i b_j + a_j b_i$ has rank 2 when $a$ and $b$ are linearly independent.

Spivak, Problem 4-7. A covariant functor $F$ assigns a vector space $F(V)$ to each vector space $V$ and a linear map $F(f)$ to each linear map $f$, compatibly with identities and compositions. Show that the identity functor, the double dual functor, and $V \mapsto \mathcal{T}^k(V^*)$ are covariant functors, and that $V \mapsto V^*$ and $V \mapsto \mathcal{T}^k(V)$ are contravariant functors.

Additional exercise. Show that $\mathcal{T}^1_1(V)$ is naturally isomorphic to the space of linear maps $V \to V$, that the identity corresponds to the evaluation map $(v,\lambda) \mapsto \lambda(v)$, and that contraction corresponds to the trace.

Additional exercise. Show that a Riemannian metric on a bundle $\xi$ gives an equivalence between $\xi$ and $\xi^*$.

Spivak, Problem 5-1. (a) If $\alpha \colon M \to N$ is $C^\infty$, show that $\alpha_* \colon TM \to TN$ is $C^\infty$. (b) If $\alpha$ is a diffeomorphism and $X$ is a $C^\infty$ vector field on $M$, show that $\alpha_* X$ is a $C^\infty$ vector field on $N$. (c) For $\alpha(t) = t^3$ on $\mathbb{R}$, find a $C^\infty$ vector field $X$ such that $\alpha_* X$ is not $C^\infty$.

Spivak, Problem 5-2. Find a nowhere-zero vector field on $\mathbb{R}$ such that every integral curve is defined only on a bounded interval around $0$.

Spivak, Problem 5-3. Find a complete metric space $(M,\rho)$ and a map $f \colon M \to M$ with $\rho(f(x), f(y)) < \rho(x,y)$ for all $x \ne y$, but with no fixed point.

Spivak, Problem 5-4. Let $f \colon (-c,c) \times U \times V \to \mathbb{R}^n$ be $C^\infty$. Show that near any $(x_0, y_0) \in U \times V$ the second-order equation $\alpha''(t) = f(t, \alpha(t), \alpha'(t))$ with $\alpha(0) = x$, $\alpha'(0) = y$ has a unique solution, depending smoothly on $(t,x,y)$. (Hint: consider the first-order system $\alpha' = \beta$, $\beta' = f(t, \alpha, \beta)$.)

Spivak, Problem 5-6 (a) to (f). For an $n \times n$ matrix $A$ let $\exp A = \sum_{k \ge 0} A^k / k!$. Show that the series converges, that $\exp(TAT^{-1}) = T (\exp A) T^{-1}$, that $\exp(A+B) = \exp A \exp B$ if $AB = BA$, and hence that $\exp A$ is always invertible.

Additional exercise. Find all integral curves of $X = -a^2 \frac{d}{da}$ on $\mathbb{R}$ explicitly. Which of them are defined for all time?

Spivak, Problem 5-6 (g) to (j). Show that the derivative of $\exp$ at $0$ is the identity. If $A(t)$ is a differentiable matrix-valued function with $A(t)A'(t) = A'(t)A(t)$, show that $\frac{d}{dt}\exp(A(t)) = A'(t)\exp(A(t))$. Conclude that if $g(s)g(t) = g(t)g(s)$ for all $s,t$, the equation $\alpha'(t) = g(t)\alpha(t)$ has the solution $\alpha(t) = \exp\left(\int_0^t g(s)\,ds\right)\alpha(0)$.

Spivak, Problem 5-8(b). If $\phi \colon \mathbb{R} \times M \to M$ is a smooth one-parameter group of diffeomorphisms, show that for every $C^\infty$ function $f$ the limit $\lim_{h \to 0} \frac{1}{h}\left[f(\phi_h(p)) - f(p)\right]$ exists and defines a $C^\infty$ function on $M$. Conclude that $\phi$ is generated by a vector field.

Spivak, Problem 5-13. On $\mathbb{R}^3$ let $X = z\frac{\partial}{\partial y} - y\frac{\partial}{\partial z}$, $Y = -z\frac{\partial}{\partial x} + x\frac{\partial}{\partial z}$, $Z = y\frac{\partial}{\partial x} - x\frac{\partial}{\partial y}$. (a) Show that $aX + bY + cZ \mapsto (a,b,c)$ is an isomorphism onto $\mathbb{R}^3$ taking the bracket $[U,V]$ to the cross product. (b) Show that the flow of $aX + bY + cZ$ is a rotation of $\mathbb{R}^3$ about an axis through $0$.

Additional exercise. Compute the flow of $X = x^2 \frac{d}{dx}$ on $\mathbb{R}$, and show that $X$ is not complete.

Additional exercise. Suppose there is an $\varepsilon > 0$ such that for every $p \in M$ the integral curve of $X$ through $p$ is defined on $(-\varepsilon, \varepsilon)$. Show that $X$ is complete.

Additional exercise. Show that the flow of the linear vector field $X_x = Ax$ on $\mathbb{R}^n$ is $\phi_t(x) = e^{tA}x$.

Spivak, Problem 5-10(a). Prove that $L_X(f\omega) = (Xf)\,\omega + f\,L_X\omega$ and $L_X(\omega(Y)) = (L_X\omega)(Y) + \omega(L_X Y)$.

Spivak, Problem 5-11. (a) Show that $\phi^*(df)(Y) = Y(f \circ \phi)$. (b) Show directly from the definition of $L_X$ that $L_X(df) = d(L_X f)$.

Spivak, Problem 5-12. Check the Jacobi identity $[X,[Y,Z]] + [Y,[Z,X]] + [Z,[X,Y]] = 0$.

Spivak, Problem 5-14(b). For tensor fields $A$ and $B$, show that $L_X(A + B) = L_X A + L_X B$ and $L_X(A \otimes B) = L_X A \otimes B + A \otimes L_X B$. In particular $L_X(fA) = X(f)\,A + f\,L_X A$.

Additional exercise. Show that $[fX, gY] = fg[X,Y] + f(Xg)Y - g(Yf)X$. Conclude that the bracket is not a tensor.

Additional exercise. On $\mathbb{R}^3$ compute $\left[x\frac{\partial}{\partial y} - y\frac{\partial}{\partial x},\; y\frac{\partial}{\partial z} - z\frac{\partial}{\partial y}\right]$.

Spivak, Problem 5-7. Let $\chi \colon \mathbb{R}^n \to M$ and $x = \chi^{-1}$. Check that $X = \frac{\partial}{\partial x^1}$ is equivalent to $\chi_*\left(\frac{\partial}{\partial t^1}\right) = X \circ \chi$.

Spivak, Problem 5-16. (a) Let $f \colon \mathbb{R} \to \mathbb{R}$ satisfy $f'(0) = 0$, and let $g(t) = f(\sqrt{t})$ for $t \ge 0$. Show that the right-hand derivative of $g$ at $0$ is $f''(0)/2$. (b) If $c$ is a curve in $M$ with $c'(0) = 0$, show that $c''(0)$, defined by $c''(0)(f) = (f \circ c)''(0)$, is a tangent vector, and that $c''(0) = 2\gamma'(0)$ for $\gamma(t) = c(\sqrt{t})$.

Spivak, Problem 5-17. Let $p$ be a critical point of $f \colon M \to \mathbb{R}$. For $X_p, Y_p \in M_p$ define $f_{**}(X_p, Y_p) = \tilde X_p(\tilde Y f)$, where $\tilde X, \tilde Y$ are vector fields extending $X_p, Y_p$. Show that this is well defined and symmetric, and that in coordinates it is given by the matrix of second partial derivatives $\frac{\partial^2 f}{\partial x^i \partial x^j}(p)$. Conclude that the rank of this matrix is independent of the coordinate system.

Spivak, Problem 5-19. (a) If $M$ is compact and $0$ is a regular value of $f \colon M \to \mathbb{R}$, show that there is a neighborhood $U$ of $0$ and a diffeomorphism $\phi \colon f^{-1}(0) \times U \to f^{-1}(U)$ with $f(\phi(p,t)) = t$. (Hint: use Theorem 5-7 and a partition of unity to build a vector field $X$ with $f_* X = \frac{d}{dt}$.) (b) Conclude that $f^{-1}(t_1)$ and $f^{-1}(t_2)$ are diffeomorphic for $t_1, t_2$ near $0$.

Additional exercise. Find coordinates near $(1,0,0)$ in $\mathbb{R}^3$ in which $x\frac{\partial}{\partial x} + y\frac{\partial}{\partial y} + z\frac{\partial}{\partial z}$ becomes $\frac{\partial}{\partial u^1}$.

Additional exercise. Show that there are no coordinates in which $\frac{\partial}{\partial x}$ and $x\frac{\partial}{\partial y}$ are both coordinate vector fields.

Spivak, Problem 7-2. Let $\overline{\operatorname{Alt}}$ be $\operatorname{Alt}$ without the factor $1/k!$, and define $\omega \,\bar\wedge\, \eta = \overline{\operatorname{Alt}}(\omega \otimes \eta)$. Show that $\bar\wedge$ is not associative. (Try $\omega, \eta \in \Omega^1(V)$ and $\theta \in \Omega^2(V)$.)

Spivak, Problem 7-3(c). A permutation $\sigma$ of $\{1,\ldots,k+l\}$ is a shuffle if $\sigma(1) < \cdots < \sigma(k)$ and $\sigma(k+1) < \cdots < \sigma(k+l)$. Show that $$\omega \wedge \eta(v_1,\ldots,v_{k+l}) = \sum_{\sigma \text{ a shuffle}} \operatorname{sgn}\sigma \cdot \omega(v_{\sigma(1)},\ldots,v_{\sigma(k)})\, \eta(v_{\sigma(k+1)},\ldots,v_{\sigma(k+l)}).$$

Spivak, Problem 7-4. For $v \in V$ and $\omega \in \Omega^k(V)$ define $\iota_v\omega \in \Omega^{k-1}(V)$ by $\iota_v\omega(v_1,\ldots,v_{k-1}) = \omega(v, v_1, \ldots, v_{k-1})$. Show that $\iota_v \iota_w = -\iota_w \iota_v$, and that $\iota_v(\omega_1 \wedge \omega_2) = (\iota_v\omega_1) \wedge \omega_2 + (-1)^k \omega_1 \wedge (\iota_v\omega_2)$ for $\omega_1 \in \Omega^k(V)$.

Spivak, Problem 7-6. An element of $\Omega^k(V)$ is decomposable if it equals $\phi_1 \wedge \cdots \wedge \phi_k$ for some $\phi_i \in V^*$. (a) If $\dim V \le 3$, show that every element of $\Omega^2(V)$ is decomposable. (b) If $\phi_1, \ldots, \phi_4$ are linearly independent, show that $\phi_1 \wedge \phi_2 + \phi_3 \wedge \phi_4$ is not decomposable. (Hint: look at $\omega \wedge \omega$.)

Spivak, Problem 7-8. (a) For $\omega \in \Omega^2(V)$, show that there is a basis $\phi_1,\ldots,\phi_n$ of $V^*$ with $\omega = \phi_1 \wedge \phi_2 + \cdots + \phi_{2r-1} \wedge \phi_{2r}$. (b) Show that the $r$-fold wedge $\omega \wedge \cdots \wedge \omega$ is nonzero and the $(r+1)$-fold wedge is $0$, so $r$ is well defined. (c) Show that $2r$ is the rank of the matrix of $\omega$.

Spivak, Problem 7-13. If $A \colon V \to V$ is linear and $\dim V = n$, show that $A^* \colon \Omega^n(V) \to \Omega^n(V)$ is multiplication by $\det A$. Conclude that $\det(AB) = \det A \det B$.

Spivak, Problem 7-16(b). Generalize Theorem 7-7 and Corollary 7-8 to $k$-forms: if $\omega = \sum_I \omega_I\, dx^I = \sum_J \omega'_J\, dy^J$, express $\omega'_J$ in terms of the $\omega_I$ and the $k \times k$ minors of the matrix $\left(\partial x^i / \partial y^j\right)$.

Spivak, Problem 7-17. Show that $d\left(\sum_{i<j} a_{ij}\, dx^i \wedge dx^j\right) = 0$ if and only if $\frac{\partial a_{ij}}{\partial x^k} - \frac{\partial a_{ik}}{\partial x^j} + \frac{\partial a_{jk}}{\partial x^i} = 0$ for all $i < j < k$.

Spivak, Problem 7-20. Compute that on $\mathbb{R}^2$ minus a ray, $d\theta = \frac{x\,dy - y\,dx}{x^2 + y^2}$.

Additional exercise. Verify Theorem 7-13 directly in coordinates for 1-forms: $d\omega(X,Y) = X(\omega(Y)) - Y(\omega(X)) - \omega([X,Y])$.

Additional exercise. Using Theorem 7-7, show that $dx \wedge dy = r\, dr \wedge d\theta$.

Spivak, Problem 7-27 (part). On $\mathbb{R}^3$, identify 1-forms and 2-forms with vector fields in the usual way. Show that $d$ on 0-forms, 1-forms and 2-forms corresponds to gradient, curl and divergence, and deduce $\operatorname{curl}\operatorname{grad} f = 0$ and $\operatorname{div}\operatorname{curl} F = 0$.

Spivak, Problem 7-18. (a) Show that if $\omega$ is a $k$-form then so is $L_X\omega$. (b) Show that $L_X(\omega_1 \wedge \omega_2) = L_X\omega_1 \wedge \omega_2 + \omega_1 \wedge L_X\omega_2$. (c) Show that $X(\omega(X_1,\ldots,X_k)) = (L_X\omega)(X_1,\ldots,X_k) + \sum_{i=1}^k \omega(X_1,\ldots,[X,X_i],\ldots,X_k)$. (e) Deduce Cartan's formula $\iota_X d\omega = L_X\omega - d(\iota_X \omega)$. (f) Using (e), show that $d(L_X\omega) = L_X(d\omega)$.

Additional exercise. Show that $L_X \iota_Y - \iota_Y L_X = \iota_{[X,Y]}$.

Additional exercise. Show that $L_{fX}\omega = f\,L_X\omega + df \wedge \iota_X\omega$.

Additional exercise. Let $\mathrm{vol} = dx^1 \wedge \cdots \wedge dx^n$ on $\mathbb{R}^n$ and $X = \sum_i a^i \frac{\partial}{\partial x^i}$. Show that $L_X \mathrm{vol} = \left(\sum_i \frac{\partial a^i}{\partial x^i}\right)\mathrm{vol}$. Conclude that the flow of $X$ preserves volume if and only if $X$ has divergence zero.

Additional exercise. For a smooth function $H$ on $\mathbb{R}^2$, let $X_H = \frac{\partial H}{\partial y}\frac{\partial}{\partial x} - \frac{\partial H}{\partial x}\frac{\partial}{\partial y}$. Show that $L_{X_H}(dx \wedge dy) = 0$ and $X_H(H) = 0$. Interpret both statements in terms of the flow.

Spivak, Problem 6-1. (a) Show that a $k$-dimensional distribution on $M$ is the same thing as a $k$-plane subbundle of $TM$. (b) Show that a distribution is $C^\infty$ if and only if it is a $C^\infty$ subbundle.

Spivak, Problem 6-4. In the proof of Proposition 6-4, show that the functions $C^\alpha_{ij}$ with $[X_i, X_j] = \sum_\alpha C^\alpha_{ij} X_\alpha$ are $C^\infty$.

Spivak, Problem 6-5. Let $\Delta_1, \ldots, \Delta_h$ be integrable distributions of dimensions $d_1, \ldots, d_h$ with $M_p = (\Delta_1)_p \oplus \cdots \oplus (\Delta_h)_p$ for all $p$. Show that around each point there is a chart $x$ such that $\Delta_1$ is spanned by $\frac{\partial}{\partial x^1}, \ldots, \frac{\partial}{\partial x^{d_1}}$, $\Delta_2$ by the next $d_2$ coordinate vector fields, and so on.

Spivak, Problem 6-6. Prove Theorem 6-1 from Theorem 6-5, by considering a suitable distribution on $\mathbb{R}^m \times \mathbb{R}^n$.

Additional exercise. Let $\Delta$ on $\mathbb{R}^3$ be spanned by $\frac{\partial}{\partial x} + y\frac{\partial}{\partial z}$ and $\frac{\partial}{\partial y}$. Show, using brackets, that $\Delta$ is not integrable.

Additional exercise. Show that the distribution on $\mathbb{R}^3 \setminus \{0\}$ whose plane at $p$ is orthogonal to $p$ is integrable, and find its maximal integral manifolds.

Additional exercise. Decide which of the following hyperplane fields on $\mathbb{R}^3$ are integrable, and find the leaves of those that are: $\ker(dz - y\,dx)$, $\ker(x\,dy - y\,dx + dz)$, $\ker(yz\,dx + xz\,dy + xy\,dz)$.

Additional exercise. Prove the easy direction of Corollary 7-15: if $\omega^\alpha = \sum_\beta f^\alpha_\beta\, dg^\beta$, then $d\omega^\alpha = \sum_\beta \theta^\alpha_\beta \wedge \omega^\beta$ for suitable 1-forms $\theta^\alpha_\beta$.

Additional exercise. Show that $(f\omega) \wedge d(f\omega) = f^2\, \omega \wedge d\omega$. Why does this mean that the integrability condition depends only on $\ker\omega$?

Additional exercise. Show that every 1-dimensional distribution is integrable, and that every 2-dimensional distribution on a 2-manifold is integrable.

Additional exercise. Consider the 1-dimensional distribution on the torus $T^2 = \mathbb{R}^2/\mathbb{Z}^2$ spanned by $\frac{\partial}{\partial x} + \alpha\frac{\partial}{\partial y}$, with $\alpha$ irrational. Show that every leaf is dense in $T^2$, and that no leaf is an embedded submanifold.

Spivak, Problem 7-19. Let $a_{ij}$ be $n^2$ functions on $\mathbb{R}^n$ with $a_{ij} = a_{ji}$. Show that there are functions $u_1, \ldots, u_n$ near any point with $a_{ij} = \frac{1}{2}\left(\frac{\partial u_i}{\partial x^j} + \frac{\partial u_j}{\partial x^i}\right)$ if and only if $$\frac{\partial^2 a_{ij}}{\partial x^k \partial x^l} - \frac{\partial^2 a_{ik}}{\partial x^j \partial x^l} = \frac{\partial^2 a_{lj}}{\partial x^k \partial x^i} - \frac{\partial^2 a_{lk}}{\partial x^j \partial x^i}$$ for all $i,j,k,l$.

Spivak, Problem 7-21. (a) If $\omega = f\,dx$ is a 1-form on $[0,1]$ with $f(0) = f(1)$, show that there is a unique number $\lambda$ such that $\omega - \lambda\,dx = dg$ for some $g$ with $g(0) = g(1)$. (b) Let $i \colon S^1 \to \mathbb{R}^2 \setminus \{0\}$ be the inclusion and $\sigma' = i^*(d\theta)$. If $c(x) = (\cos 2\pi x, \sin 2\pi x)$, show that $c^*(\sigma') = 2\pi\,dx$. (c) If $\omega$ is a closed 1-form on $S^1$, show that there is a unique $\lambda$ such that $\omega - \lambda\sigma'$ is exact.

Spivak, Problem 7-23. Let $U \subset \mathbb{R}^n$ be open and star-shaped with respect to $0$, and $H(p,t) = tp$. For $\omega = \sum_I \omega_I\, dx^{i_1} \wedge \cdots \wedge dx^{i_k}$ show that $$I(H^*\omega) = \sum_I \sum_{\alpha=1}^k (-1)^{\alpha-1} \left(\int_0^1 t^{k-1}\omega_I(tx)\,dt\right) x^{i_\alpha}\, dx^{i_1} \wedge \cdots \wedge \widehat{dx^{i_\alpha}} \wedge \cdots \wedge dx^{i_k},$$ where the hat means that factor is omitted.

Spivak, Problem 7-24(b). Find a bounded open set $U \subset \mathbb{R}^3$ such that $\mathbb{R}^3 \setminus U$ is connected but $U$ is not contractible to a point.

Spivak, Problem 7-25. Let $U \subset \mathbb{R}^n$ be an open set that is star-shaped with respect to $0$. Is $U$ homeomorphic to $\mathbb{R}^n$? (The obvious proof does not work, since the length of the rays from $0$ to the boundary of $U$ could vary discontinuously.)

Spivak, Problem 8-2. Compute $\int_c d\theta$, where $c(t) = (\cos 2\pi t, \sin 2\pi t)$ on $[0,1]$.

Spivak, Problem 8-3. For an integer $n$ and $R > 0$, let $c_{R,n}(t) = (R\cos 2\pi n t, R\sin 2\pi n t)$ on $[0,1]$. (a) Show that there is a singular 2-cube $c$ in $\mathbb{R}^2 \setminus \{0\}$ with $c_{R_1,n} - c_{R_2,n} = \partial c$. (b) If $c$ is any closed curve in $\mathbb{R}^2 \setminus \{0\}$ (so $c(0) = c(1)$), show that $c - c_{1,n}$ is a boundary for some $n$. (c) Show that this $n$ is unique; it is the winding number of $c$ around $0$.

Spivak, Problem 8-5. The standard $n$-simplex is $\Delta_n = \{x \in \mathbb{R}^n : x^i \ge 0, \sum_i x^i \le 1\}$. Define singular simplices, their faces and a boundary operator $\partial c = \sum_{i=0}^n (-1)^i \partial_i c$, as in Spivak. (a) Describe the images of the faces geometrically. (b) Show that $\partial^2 = 0$.

Additional exercise. Write out $\partial I^3$ explicitly as a sum of six singular 2-cubes with signs, and check $\partial(\partial I^3) = 0$ directly.

Additional exercise. Show that $\frac{1}{2}\int_{\partial I^2}(x\,dy - y\,dx) = 1$.

Spivak, Problem 8-7. Let $\omega$ be a 1-form on a manifold $M$ with $\int_c \omega = 0$ for every closed curve $c$ in $M$. Show that $\omega$ is exact.

Spivak, Problem 8-17(a). Let $M^n$ and $N^m$ be oriented manifolds with the product orientation on $M \times N$. If $\omega$ and $\eta$ are compactly supported top-degree forms on $M$ and $N$, show that $\int_{M \times N} \pi_1^*\omega \wedge \pi_2^*\eta = \int_M \omega \cdot \int_N \eta$.

Spivak, Problem 8-12. (a) Let $M$ be the open unit disk in $\mathbb{R}^2$ together with a proper nonempty portion of its boundary circle, and let $\omega = x\,dy$. Show that $\int_M d\omega \ne \int_{\partial M}\omega$. (b) Find a similar counterexample to Stokes' theorem on $M = (0,1)$ with a 0-form whose support is not compact.

Spivak, Problem 8-13. Let $M$ be a compact orientable $n$-manifold without boundary and $\theta$ an $(n-1)$-form on $M$. Show that $d\theta$ is zero at some point.

Spivak, Problem 8-14. Let $M_1, M_2 \subset \mathbb{R}^n$ be compact $n$-dimensional manifolds with boundary, with $M_2$ contained in the interior of $M_1$. Show that $\int_{\partial M_1}\omega = \int_{\partial M_2}\omega$ for every closed $(n-1)$-form $\omega$ on $M_1$.

Additional exercise. Let $\sigma' = i^*(x\,dy \wedge dz - y\,dx \wedge dz + z\,dx \wedge dy)$ on $S^2$. Compute $\int_{S^2}\sigma' = 4\pi$ using spherical coordinates.

Additional exercise. Show that there is no smooth map $r \colon D^n \to S^{n-1}$ with $r(p) = p$ for all $p \in S^{n-1}$. Deduce that every smooth map $D^n \to D^n$ has a fixed point.

Spivak, Problem 8-20. Prove that a connected non-compact manifold is a union $U_1 \cup U_2 \cup \cdots$ of coordinate neighborhoods with $U_i \cap U_{i+1} \ne \emptyset$, such that the sequence is eventually outside any given compact set.

Spivak, Problem 8-21. Let $f \colon M^n \to N^n$ be a proper map between oriented $n$-manifolds such that $f_{*p}$ is orientation-preserving at every regular point $p$. Show that if $N$ is connected, then either $f$ is onto or every point of $M$ is a critical point of $f$.

Spivak, Problem 8-22. (a) Show that a polynomial map $f(z) = z^n + a_1 z^{n-1} + \cdots + a_n$ from $\mathbb{C}$ to $\mathbb{C}$ is proper ($n \ge 1$). (b) Using the Cauchy-Riemann equations, show that $\det Df(x,y) = |f'(x+iy)|^2 \ge 0$. (c) Conclude, using Problem 8-21, that $f$ is onto (the fundamental theorem of algebra).

Spivak, Problem 8-23. Let $M^{n-1} \subset \mathbb{R}^n$ be a compact oriented manifold. For $p \notin M$ define the winding number $w(p)$ of $M$ around $p$ as the degree of the map $M \to S^{n-1}$, $q \mapsto (q-p)/|q-p|$. Show that $w$ is constant on each component of $\mathbb{R}^n \setminus M$.

Additional exercise. If $f \colon M \to N$ and $g \colon N \to P$ are maps of compact connected oriented $n$-manifolds, show that $\deg(g \circ f) = \deg g \cdot \deg f$.

Additional exercise. Show that $H^0(M)$ has dimension equal to the number of components of $M$, and that $H^n_c(\mathbb{R}^n) \ne 0$.

Spivak, Problem 8-24(a). Let $f, g \colon M \to N$ be smoothly homotopic maps between compact $n$-manifolds, and let $q$ be a regular value of the homotopy $H \colon M \times [0,1] \to N$ (and of $f$ and $g$). Show that the number of points in $f^{-1}(q)$ and in $g^{-1}(q)$ are equal mod 2.

Additional exercise. Complete the proof of Theorem 11-2: check that the connecting map $\delta$ is well defined, and that the long sequence is exact at each of its three kinds of terms.

Additional exercise. For $M = U \cup V$ with partition of unity $\{\phi_U, \phi_V\}$, show that the connecting map of the Mayer-Vietoris sequence sends the class of a closed form $\omega$ on $U \cap V$ to the class of the form equal to $d\phi_V \wedge \omega$ on $U \cap V$ and $0$ elsewhere.

Additional exercise. Show that a smooth map $f \colon S^n \to S^n$ of degree different from $(-1)^{n+1}$ has a fixed point.

Additional exercise. Write down a nowhere-zero vector field on $S^n$ for $n$ odd.

Spivak, Problem 11-1. Find $H^k(S^1 \times \cdots \times S^1)$ ($n$ factors) by induction on $n$. (Answer: $\dim H^k = \binom{n}{k}$.)

Spivak, Problem 11-2. (a) Use the Mayer-Vietoris sequence to determine $H^k(M \setminus \{p\})$ in terms of $H^k(M)$, for $M$ a connected manifold. (b) For connected $n$-manifolds $M$ and $N$, find the cohomology of the connected sum $M \# N$ in terms of that of $M$ and $N$. (c) Find the Euler characteristic of the $n$-holed torus. (Answer: $2 - 2n$.)

Spivak, Problem 11-3. Find $H^k$ of: (a) the Möbius strip; (b) $\mathbb{P}^2$; (c) $\mathbb{P}^n$; (d) the Klein bottle. (Warning: the answer to (c) printed in Spivak is not correct for de Rham cohomology. The correct answer is $H^0(\mathbb{P}^n) = \mathbb{R}$, $H^n(\mathbb{P}^n) = \mathbb{R}$ when $n$ is odd, and all other groups are $0$.)

Spivak, Problem 11-5. For any triangulation of a compact 2-manifold $M$ with $\alpha_k$ simplices of dimension $k$, show that $3\alpha_2 = 2\alpha_1$, $\alpha_1 = 3(\alpha_0 - \chi(M))$, $\alpha_0(\alpha_0 - 1)/2 \ge \alpha_1$, and hence $\alpha_0 \ge \frac{1}{2}\left(7 + \sqrt{49 - 24\chi(M)}\right)$.

Spivak, Problem 11-6(a). Find $H^k_c(S^n \times \mathbb{R}^m)$ by induction on $n$, using the Mayer-Vietoris sequence for compact supports.

Additional exercise. Compute the cohomology of $\mathbb{R}^2$ with $k$ points removed, of $S^2$ with $k$ points removed, and of $S^1 \times S^2$.