Problem set 12

Integration on manifold (2 dim case)¶

Problem 1¶

Let $M\subset \mathbb{R}^3$ be a surface, let $\mathbf{x}(u,v)=\begin{pmatrix}x_1(u,v)\\x_2(u,v)\\x_3(u,v)\end{pmatrix}$ be parametrization on $U$ and $\mathbf{x}(s,t)=\begin{pmatrix}x_1(s,t)\\x_2(s,t)\\x_3(s,t)\end{pmatrix}$ be parametrization on $V$.

Let $\mathbf{x}_u = \begin{pmatrix}(x_1)_u\\ (x_2)_u\\ (x_3)_u\end{pmatrix}$, where $(x_i)_u$ means derivative $\frac{\partial x_i}{\partial u}(u,v)$, and $\mathbf{x}_v = \begin{pmatrix} (x_1)_v \\ (x_2)_v \\ (x_3)_v \end{pmatrix}$.

(1) Let $\begin{pmatrix} u(s,t)\\v(s,t)\end{pmatrix}$ be change of coordinate, and $J:=\begin{pmatrix} \frac{\partial u}{\partial s}& \frac{\partial u}{\partial t}\\ \frac{\partial v}{\partial s} & \frac{\partial v}{\partial t}\end{pmatrix}$ be its Jacobi matrix. Verify that $\begin{pmatrix}\mathbf{x}_u \\ \mathbf{x}_v\end{pmatrix} = J \begin{pmatrix} \mathbf{x}_s \\ \mathbf{x_t}\end{pmatrix}$.

Let $E = \langle \mathbf{x}_u,\mathbf{x}_v\rangle$ be square length of $\mathbf{x}_u$, $G = \langle \mathbf{x}_v,\mathbf{x}_v\rangle$ be square length of $\mathbf{x}_v$, and $F = \langle \mathbf{x}_u,\mathbf{x}_v\rangle$. Let $g = \begin{pmatrix}E&F\\F&G\end{pmatrix}$.

(2) Let $\tilde E = \langle \mathbf{x}_s,\mathbf{x}_s\rangle$, $\tilde{G} = \langle \mathbf{x}_t,\mathbf{x}_t\rangle$ and $\tilde{F} = \langle \mathbf{x}_s,\mathbf{x}_t \rangle$ be the corresponding data on $V$, and $\tilde{g}:=\begin{pmatrix}\tilde{E}&\tilde{F}\\ \tilde{F} &\tilde{G} \end{pmatrix}$. Show that $\tilde g = J^{T}g J$.

(3) Consider the two form $\sqrt{\det(g)}du\wedge dv$. Assume $\det J> 0$. Show that $\sqrt{\det g}\;du\wedge dv = \sqrt{\det \tilde{g}}\;ds\wedge dt$.

If one can always choose parametrizations such that $\det J>0$ for every change of coordinate, we call such $M$ oriented. We always assume a surface is oriented. The above exercise indicates that one can define a positive measure $dV$ on $M$, which is given on any parametrization $U$ by $\sqrt{\det g}\;du\wedge dv$. This is well defined on $M$ because on any coordinate it is of the same form by (3).

Let $M$ be a compact surface, let $(U_i)_{i=1}^N$ be its coordinate charts. On each $U_i$ we parametrize $M$ by $\mathbf{x}(u_i,v_i)$. Let $dV$ be the area measure on $M$. Now we define the integral $\int_{M}f\;dV$ for $f\in \mathscr{C}^{\infty}(M)$.

Choose partition of unity $(\rho_i)_{i=1}^N$ subordinate to $(U_i)_{i=1}^N$. We know that $\rho_i f$ is supported on $U_i$. Then we define

. One can check that this does not depend on choice of partition of unity.

Smoothing operators¶

The statements are given on surface, the same argument will work on compact smooth manifolds with integration defined. If you are really uncomfortable with integration on surfaces, you can replace all “$M$” by an open set $U$ in $\mathbb{R}^n$.

Let $M$ be a compact smooth surface. Let $K\in \mathscr{C}^{\infty}(M\times M)$. Define $T_K:\mathscr{C}^{\infty}(M)\to \mathscr{C}^{\infty}(M)$ given by $T_K f(x) = \int_{M}K(x,y)f(y)dV(y)$.

Problem 2¶

(1) Let $T_{K_1},T_{K_2}$ be smoothing operators. Show that $T_{K_1}\circ T_{K_2}$ is a smoothing operator. In other words, $T_{K_1}\circ T_{K_2} = T_{K'}$ for some $K'\in \mathscr{C}^{\infty}(M\times M)$.

Denote $K'$ in (1) by $K_1\circ K_2$.

(2) Let $\mathrm{Vol(M)} =\int_{M}dV$ be the volume (area) of $M$. Show that if $\max_{(x,y)\in M\times M}{|K(x,y)|}<\frac{\epsilon}{V}$ for some $0<\epsilon < 1$ then $I-T_K$ is invertible.

Differential operator on manifold¶

Problem 3¶

Let $U,V$ be open sets in $\mathbb{R}^2$. $\varphi:V\to U$ be a diffeomorphism (bijective, smooth, $\det$(Jacobi matrix)$\neq 0$). Let $f:U\to \mathbb{R}$ be a smooth function. Then $f\circ \varphi$ is a smooth function on $V$, we denote this by $\varphi^* f$. Since φ is a smooth diffeo, we can do samething for $\varphi^{-1}$, so that $\varphi^*$ and $(\varphi^{-1})^*$ give bijection $\mathscr{C^{\infty}}(U)$ and $\mathscr{C}^{\infty}(V)$.

(1) Check that $(\varphi^{-1})^*D_j^{\alpha}\varphi = ((\varphi^{-1})^*D_j\varphi))^{\alpha}$.

(2) Let $P$ be a differential operator with degree $m$ on $V$, we know that $(\varphi^{-1})^*P\varphi^*$ is a differential operator on $U$. The idea is, for $f\in \mathscr{C}^{\infty}(U)$, we first “pull back” $f$ to a smooth function on $V$, apply $P$, then “pull back” back to $U$. Show that $((\varphi)^{-1})^{*}P \varphi^*$ is also degree $m$.

Hint: Write $Pf = \sum_{|\alpha| \leq m}a_{\alpha}(s,t)D^{\alpha}f(s,t)$. Write $\varphi:V\to U$ by $\begin{pmatrix}u\\v\end{pmatrix} = \varphi(\begin{pmatrix}s\\t\end{pmatrix}) = \begin{pmatrix} u(s,t)\\v(s,t)\end{pmatrix}$. Then (1) can be verified directly, and (2) reduce to check the case $P = D_k$. By the chain rule, we have $D_s\varphi^*f(s,t) = D_s(f(u(s,t),v(s,t))) = \frac{\partial u}{\partial s}D_u f(u(s,t),v(s,t))+\frac{\partial v}{\partial s}D_vf(u(s,t),v(s,t)) = \frac{\partial u}{\partial s}\varphi^*D_u f + \frac{\partial v}{\partial s}\varphi^* D_v f$.

Problem 4¶

Let $f,g\in \mathscr{C}_c^{\infty}(U)$, complex valued. Let $\int_{U}f\bar{g}\;dm$ be the inner product. Let $P = \sum_{|\alpha|\leq m}a_\alpha(x)D^{\alpha}$, define its adjoint $P^*$ by requiring $(Pf,g) = (f,P^*g)$. Calculate $P^*$ using integration by parts. The $P^*$ is called the formal adjoint of $P$. Note that here it only applies on test functions.