Chapter 11 Exercises (Contd.)

Lee Smooth Manifolds

Problem 1 (E10-8). Let ${\text{Vec}}_1$ be the category whose objects are finite-dimensional real vector spaces and whose morphisms are linear isomorphisms. If $\mathcal{F}$ is a covariant functor from ${\mathsf{Vec}}_1$ to itself, for each finite-dimensional vector space $V$ we get a map $\mathcal{F}:\mathrm{GL}(V)\to \mathrm{GL}(\mathcal{F}(V))$ sending each isomorphism $A:V\to V$ to the induced isomorphism $\mathcal{F}(A):\mathcal{F}(V)\to \mathcal{F}(V)$. We say $\mathcal{F}$ is a smooth functor if this map is smooth for every $V$. Given a smooth vector bundle $E\to M$ and a smooth functor $\text{F}:{\mathsf{Vec}}_1\to {\text{Vec}}_1$, show that there is a smooth vector bundle $\mathcal{F}(E)\to M$ whose fiber at each point $p\in M$ is $\mathcal{F}(E_p)$.

Proof) It suffices to show the Smooth Bundle Chart Lemma applies to the disjoint union

$$\mathcal{F}(E)=\underset{p\in M}{⨆}\mathcal{F}(E_p)$$

with the natural projection $\pi :\mathcal{F}(E_p)\mapsto p$. First, as a critical bookkeeping measure, let $n=\dim (E_p)$ and $m=\dim \mathcal{F}(E_p)$, and fix a basis in $\mathcal{F}({\mathbb{R}}^n)$ for a fixed linear isomorphism $\iota :\mathcal{F}({\mathbb{R}}^n)\to {\mathbb{R}}^m$. Conjugation by $\iota$ gives a Lie group isomorphism (and thus diffeomorphism) $C_{\iota }:GL(\mathcal{F}({\mathbb{R}}^n))\to GL(\mathbb{R},m)$ given by $C_{\iota }(f)=\iota \circ f\circ {\iota }^{-1}$.

Take a trivializing cover $\{U_{\alpha }\}$ of $M$; each trivialization ${\Phi }_{\alpha }:{\pi }^{-1}(U_{\alpha })\to U_{\alpha }\times {\mathbb{R}}^n$ restricts to a linear isomorphism ${\Phi }_{\alpha }{|}_p:E_p\to {\mathbb{R}}^n$ which $\mathcal{F}$ lifts to $\mathcal{F}({\Phi }_{\alpha }{|}_p):\mathcal{F}(E_p)\to \mathcal{F}({\mathbb{R}}^n)$. Composing with $\iota$, we can define a new trivialization on $U_{\alpha }$ given by

$${\Phi }_{\alpha }^{\prime }(p,v)=(p,\iota \circ \mathcal{F}({\Phi }_{\alpha }{|}_p)(v))$$

These trivializations are the bijections we need. Take any $\alpha ,\beta$ and consider some $p\in U_{\alpha }\cap U_{\beta }$. Then, the transition function between $U_{\alpha }$ and $U_{\beta }$ is given by

$${\tau }_{\beta \alpha }^{\prime }(p)=\iota \circ \mathcal{F}({\Phi }_{\beta }{|}_p)\circ \mathcal{F}({\Phi }_{\alpha }{|}_p{)}^{-1}\circ {\iota }^{-1}$$

As $\mathcal{F}$ is a covariant functor, $\mathcal{F}({\Phi }_{\beta }{|}_p)\circ \mathcal{F}({\Phi }_{\alpha }{|}_p{)}^{-1}=\mathcal{F}({\Phi }_{\beta }{|}_p)\circ \mathcal{F}(({\Phi }_{\alpha }{|}_p{)}^{-1})=\mathcal{F}({\Phi }_{\beta }{|}_p\circ {\Phi }_{\alpha }{|}_p^{-1})$, which is $\mathcal{F}$ applied to the original transition $\mathcal{F}({\tau }_{\beta \alpha }(p))$. Thus, it suffices to show the following function

$${\tau }_{\beta \alpha }^{\prime }(p)=\iota \circ \mathcal{F}({\tau }_{\beta \alpha }(p))\circ {\iota }^{-1}(p)=C_{\iota }(\mathcal{F}({\tau }_{\beta \alpha }(p)))$$

is a smooth function of $p$. This follows from the smoothness of ${\tau }_{\beta \alpha }$, $\mathcal{F}$, and $C_{\iota }$. Thus, by the smooth manifold chart lemma, $\mathcal{F}(E)$ has a topology and smooth structure making it into a valid vector bundle over $M$.

Problem 2 (E11-3). Let ${\text{Vec}}_1$ be the category of finite-dimensional vector spaces and linear isomorphisms as above. Define a functor $\mathcal{F}:{\text{Vec}}_1\to {\text{Vec}}_1$ by setting $\mathcal{F}(V)=V^{\ast }$ for a vector space $V$, and $\mathcal{F}(A)=(A^{-1}{)}^{\ast }$ for an isomorphism $A$. Show that $\mathcal{F}$ is a smooth covariant functor, and that for every $M$, $\mathcal{F}(TM)$ and $T^{\ast }M$ are canonically smoothly isomorphic vector bundles.

Proof) $\mathcal{F}$ is clearly a covariant functor. For any vector space $V$, fix a basis $(v_i)$ and use the basis $({\lambda }_i)$ of $V^{\ast }$ where ${\lambda }_i$ is such that ${\lambda }_i{v}_J={\delta }_i^j$. In these coordinates, $\mathcal{F}:GL(V)\to GL(V^{\ast })$ is such that $\mathcal{F}(A){\lambda }_i(v_j)=(A^{-1}{)}^{\ast }{\lambda }_i(v_j)={\lambda }_i(A^{-1}{v}_j)=A_{ij}^{-1}$; thus, $\mathcal{F}(A)=(A^{-1}{)}^T$ (in standard column vector interpretation), which is smooth. Therefore, $\mathcal{F}$ is a smooth covariant functor.

Now consider any smooth atlas $(U_{\alpha })$ of $M$. Using the canonical coordinates $({\lambda }_i)$ of $({\mathbb{R}}^m{)}^{\ast }$, $U_{\alpha }$ is a trivializing cover for both $\mathcal{F}(TM)$ and $T^{\ast }M$. Consider two charts $U_{\alpha }$ and $U_{\beta }$ with transition function $\Phi :U_{\alpha }\cap U_{\beta }\to GL(\mathbb{R},m)$. Then, $T^{\ast }M$ has transition maps ${\tau }_{\beta \alpha }(p)=(\Phi (p{)}^{-1}{)}^T$ while $\mathcal{F}(TM)$ has transition maps $\iota \circ \mathcal{F}({\tau }_{\beta \alpha }(p))\circ {\iota }^{-1}(p)=\iota \circ (\Phi (p{)}^{-1}{)}^{\ast }\circ {\iota }^{-1}(p)=(\Phi (p{)}^{-1}{)}^T$. Thus, $\mathcal{F}(TM)$ and $T^{\ast }M$ have the same transition maps on a trivializing cover, making them smoothly isomorphic bundles.

Remark: The canonical isomorphism between $T^{\ast }M$ and $\mathcal{F}(TM)$ is in fact the desired isomorphism, as can be easily verified.

Problem 3 (E11-4). Let $M$ be a smooth manifold with or without boundary and $p$ be a point of $M$. Let ${\ell }_p$ denote the subspace of $C^{\infty }(M)$ consisting of smooth functions that vanish at $p$, and let ${\ell }_p^2$ be the subspace of ${\ell }_p$ spanned by functions of the form $fg$ for some $f,g\in {\ell }_p$. (a) Show that $f\in {\ell }_p^2$ if and only if in any smooth local coordinates, its first-order Taylor polynomial at $p$ is zero. (Because of this, a function in ${\ell }_p^2$ is said to vanish to second order) (b) Define a map $\Phi :{\ell }_p\to T_p^{\ast }M$ by setting $\Phi (f)=d{f}_p$. Show that the restriction of $\Phi$ to ${\ell }_p^2$ is zero, and that $\Phi$ descends to a vector space isomorphism from ${\ell }_p/{\ell }_p^2$ to $T_p^{\ast }M$.

Proof) (a) It is trivial that if $f\in {\ell }_p^2$ it’s first-order Taylor polynomial vanishes; the converse is the main content of this problem. In any coordinates (of a convex chart centered at $p$), $f$ is then equal to the Taylor remainder

$$f(x)=\sum _{I:|I|=2}{x}^I{\int }_0^1(1-t){\partial }_{I}f(tx)dt$$

$f$ is then the finite sum of terms of the form

$$x^i{x}^j{\int }_0^1(1-t){\partial }_i{\partial }_{j}f(tx)dt$$

which is the multiple of two smooth functions ($x^i$ and $x^j{\int }_0^1(1-t){\partial }_i{\partial }_{j}f(tx)dt$) which both vanish at $p$. Denote these functions $f_1$ and $f_2$; we have shown that there is some regular coordinate ball $U$ on which $f=f_1{f}_2$.

We need some extra tricks to define $f_1$ and $f_2$ on the entirety of $M$. Take a bump function $\psi$ such that $\psi =1$ on some neighborhood of $p$ and $\text{supp}(\psi )\subset U$. Decompose $f$ into $f={\psi }^{2}f+(1-{\psi }^2)f$; ${\psi }^{2}f=(\psi f_1)(\psi f_2)$ which is a product of two smooth functions defined over all of $M$ that both vanish at $p$. $(1-{\psi }^2)f$ is also such a product as both $1-{\psi }^2$ and $f$ are smooth functions that vanish at $p$. Thus, $f\in {\ell }_p^2$.

(b) To show $\Phi ({\ell }_p^2)=0$, it suffices to show $d(fg{)}_p=0$ if $f,g\in C^{\infty }(M)$ vanish at $p$. This follows from the fact that all first-partial derivatives of $fg$ vanish at $p$. Furthermore, if $d{f}_p=0$, its first-partials must all vanish, meaning its first-order Taylor polynomial vanishes. By (a), this implies $f\in {\ell }_p^2$; therefore, $\Phi$ descends to a vector space isomorphism from ${\ell }_p/{\ell }_p^2$ to $T_p^{\ast }M$, as desired. Surjectivity is trivial as, in any smooth chart centered at $p$, $d{x}^i$ spans $T_p^{\ast }M$, so one simply smoothly extends the coordinate functions.

Problem 4 (E11-5). For any smooth manifold $M$, show that $T^{\ast }M$ is a trivial vector bundle if and only if $TM$ is trivial.

Proof) $TM$ is trivial iff there exists a global smooth frame $(E_1,E_2,\dots ,E_m)$, which exists iff there is a global smooth coframe $({ϵ}_1,{ϵ}_2,\dots ,{ϵ}_m)$ (simply take the dual frame). Again, this is true iff $T^{\ast }M$ is trivial.