15.2 Differentiable maps and curves

中文

Let $M$ be a differentiable manifold of dimension $n$. A map $f:M\to \mathbb{R}$ is said to be differentiable at a point $p\in M$ if for some coordinate chart $(U,\varphi ;x^i)$ at $p$ the function $\hat{f}=f\circ {\varphi }^{-1}:\varphi (U)\to \mathbb{R}$ is differentiable at $\varphi (p)=\mathbf{x}(p)=\left(x^1(p),x^2(p),\dots ,x^n(p)\right)$ This definition is independent of the choice of chart at $p,$ for if $(U^{\prime },{\varphi }^{\prime })$ is a second chart $a$ $p$ that is compatible with $(U,\varphi )$ , then

$${\hat{f}}^{\prime }=f\circ {\varphi }^{-1}=\hat{f}\circ \varphi \circ {\varphi }^{-1}$$

is $C^{\infty }$ since it is a differentiable function of a differentiable function. We denote by ${\mathcal{F}}_p(M)$ the set of all real-valued functions on $M$ that are differentiable at $p\in M$

Given an open set $V\subseteq M$ , a real-valued function $f:M\to \mathbb{R}$ is said to be differentiable or smooth between manifolds on $V$ if it is differentiable at every point $p\in V$ . Clearly, the function need only be defined on the open subset $V$ for this definition. We will denote the set of all real-valued functions on $M$ that are differentiable on an open subset $V$ by the symbol $\mathcal{F}(V)$ . Since the sum $f+g$ and product $fg$ of any pair of differentiable functions $f$ and $g$ are differentiable functions, $\mathcal{F}(V)$ is a ring. Furthermore, $\mathcal{F}(V)$ is closed with respect to taking linear combinations $f+ag$ where $a\in \mathbb{R}$ and is therefore also a real vector space that at the same time is a commutative algebra with respect to multiplication of functions $fg$ . All functions in ${\mathcal{F}}_p(M)$ are differentiable on some open neighbourhood $V$ of the poin $p\in M$

Exercise

Show that ${\mathcal{F}}_p(M)$ is a real commutative algebra with respect to multiplication of functions.

If $M$ and $N$ are differentiable manifolds, dimensions $m$ and $n$ respectively, then a map $\alpha :M\to N$ is differentiable at $p\in M$ if for any pair of coordinate charts $(U,\varphi ;x^i)$ ) and $(V,\psi ;y^a)$ covering $p$ and $\alpha (p)$ respectively, its coordinate representation

$$\hat{\alpha }=\psi \circ \alpha \circ {\varphi }^{-1}:\varphi (U)\to \psi (V)$$

is differentiable at $\varphi (p)$ . As for differentiable real-valued functions this definition is independent of the choice of charts. The map $\hat{\alpha }$ is represented by $n$ differentiable real-valued functions

$$y^a={\alpha }^a(x^1,x^2,\dots ,x^m)(a=1,2,\dots ,n),$$

where ${\alpha }^a={\mathbf{pr}}_a\circ \hat{\alpha }$

A diffeomorphism is a map $\alpha ;M\to N$ that is one-to-one and both $\alpha$ and ${\alpha }^{-1}:N\to M$ are differentiable. Two manifolds $M$ and $N$ are said to be diffeomorphic, written $M\cong N_{:}$ if there exists a diffeomorphism $\alpha :M\to N;$ ; the dimensions of the two manifolds must of course be equal, $m=n$ . It is a curious and difficult fact that there exist topological manifolds with more than one inequivalent differentiable structure.

A smooth parametrized curve on an $n$-dimensional manifold $M$ is a differentiable map $\gamma :(a,b)M$ from an open interval $(a,b)\subseteq \mathbb{R}$ of the real line into $M$. The curve is said to pass through $p$ at $t=t_0\mathrm{if}\gamma (t_0)=p$ , where $a<t_0<b$ . Note that a parametrized curve consists of a map, not the image points $\gamma (t)\in M$ . Changing the parameter from $t$ to $t^{\prime }=f(t)$ , where $f:\mathbb{R}\to \mathbb{R}$ is a monotone differentiable function and $a^{\prime }=f(a)<t^{\prime }<b^{\prime }=f(b)$ , changes the parametrized curve to ${\gamma }^{\prime }=\gamma \circ f$ , but has no effect on the image points in $M$. Given a chart $(U,\varphi ;x^i)$ at $p,$ , the inverse image of the open set $U$ is an open subset ${\gamma }^{-1}(U)\subseteq \mathbb{R}$ . Let $(a_1,b_1)$ be the connected component of this set that contains the real number $t_0$ such that $p=\gamma (t_0)$ . The ‘coordinate representation’ of the parametrized curve $\gamma$ induced by this chart is the smooth curve $\hat{\gamma }=\varphi \circ \gamma :(a_1,b_1)\to {\mathbb{R}}^n$ , described by $n$ real-valued functions $x^i={\gamma }^i(t)$ where ${\gamma }^i={\mathsf{pr}}_i\circ \varphi \circ \gamma$ . We often write this simply as $x^i=x^i(t)$ when there is no danger of any misunderstanding (see Fig. 15.3). In another chart $(U^{\prime };x^{\prime j})$ the $n$ functions representing the curve change to $x^{\prime j}={\gamma }^{\prime j}(t)=x^{\prime j}(\gamma (t))$ . Assuming compatible charts, these new functions representing the curve are again smooth, although it is possible that the parameter range $(a^{\prime },b^{\prime })$ is altered.

Figure 15.3 Parametrized curve on a differentiable manifold

Problems

Problem 15.5

Let $R$ be the manifold consisting of $R$ with differentiable structure generated by the chart $(\mathbb{R};y=x^3)$ . Show that the identity map ${\mathrm{id}}_{\mathbb{R}}:{\mathbb{R}}^{\prime }\to \mathbb{R}$ is a differentiable homeomorphism, which is not a diffeomorphism.

Problem 15.6

Show that the set ofreal $m$ $m$ matrices $M(m,n;\mathbb{R})$ ) is a manifold ofdimension $mn$. Show that the matrix multiplication map $M(m,k;\mathbb{R})\times M(k,n;\mathbb{R})\to M(m,n;\mathbb{R})$ is differentiable.