This post was translated from Korean by LLM (Kimi). The translation may contain errors or awkward sentences. The Korean original is the source of truth.
In topology we defined the notion of a topological manifold; in this series we study differentiable manifolds, and in particular smooth manifolds.
Notation
We will frequently deal with \(m\)-dimensional coordinate systems, so we fix the following notation. For \(\mathbb{R}^m\), we denote the \(i\)-th projection \(\pr_i\) by \(r^i\). Similarly, for any set \(X\) and function \(f:X\rightarrow\mathbb{R}^m\), the \(i\)-th component function of \(f\) is defined by the formula \(f^i=r^i\circ f\).
Now let \(f\) be a function from \(\mathbb{R}^m\) to \(\mathbb{R}\). Then we define the partial derivative of \(f\) with respect to the \(i\)-th coordinate by the formula
\[\frac{\partial}{\partial r^i}\bigg|_t f=\frac{\partial f}{\partial r^i}\bigg|_t=\lim_{h\rightarrow 0}\frac{f(t^1,\ldots, t^{i-1}, t^i+h, t^{i+1},\ldots, t^m)-f(t^1,\ldots, t^m)}{h}\]As in the notation above, following [Lee] we write the \(i\)-th coordinate as \(x^i\) rather than \(x_i\).
If all the component functions are \(k\) times differentiable and the resulting derivatives are continuous, we call \(f\) a \(C^k\) function. For example, saying that a function \(f:\mathbb{R}^2\rightarrow\mathbb{R}\) is \(C^2\) means that the following partial derivatives
\[\frac{\partial^2 f}{\partial x^2},\quad\frac{\partial^2 f}{\partial x\partial y},\quad\frac{\partial^2 f}{\partial y\partial x},\quad\frac{\partial^2 f}{\partial y^2}\]all exist and are continuous. If \(f\) is \(C^k\) for every natural number \(k\), we call it \(C^\infty\).
Differentiable Manifolds
Unlike a general topological space, a topological manifold looks locally like \(\mathbb{R}^n\), so we can carry over the notion of differentiation defined there to \(M\). The reason this is possible is that differentiability is essentially a local property.
Definition 1 Let a topological manifold \(M\) be given. For \(0\leq k\leq\infty\), we say that coordinate charts \((U,\varphi)\) and \((V,\psi)\) are \(C^k\)-compatible if the two transition maps
\[\psi\circ\varphi^{-1}:\varphi(U\cap V)\rightarrow\psi(U\cap V),\qquad\varphi\circ\psi^{-1}:\psi(U\cap V)\rightarrow\varphi(U\cap V)\]are both \(C^k\). A collection \(\mathcal{A}=\{(U_\lambda, \varphi_\lambda)\}_{\lambda\in\Lambda}\) of \(C^k\)-compatible charts satisfying \(M=\bigcup U_\lambda\) is called a \(C^k\)-atlas.
Among the \(C^k\)-atlases defined on \(M\), one that is maximal with respect to inclusion is called a \(C^k\)-differentiable structure, and in this case \(M\) is called a \(C^k\)-differentiable manifold. In the special case \(k=\infty\), this structure is called a smooth differentiable manifold, or more simply a differentiable manifold.
The reason we think of a maximal atlas as giving a differentiable structure is that it is quite possible for two non-maximal atlases to define essentially the same differentiable structure. For example, \(\mathbb{R}\) admits the following \(C^\infty\)-atlas
\[\mathcal{A}=\{(\mathbb{R}, \id_\mathbb{R})\}\]but also another atlas
\[\mathcal{A}'=\{((-\infty, 1), \id_{(-\infty, 1)}), ((-1, \infty),\id_{(-1,\infty)})\}\]However, as we shall verify in Proposition 3, given any atlas there is a uniquely determined maximal atlas containing it, so in an essential sense this is not a serious difference.
Meanwhile, to understand an object in mathematics it suffices to understand the functions defined on it. Henceforth, all manifolds are understood to be smooth differentiable manifolds.
Definition 2 Consider a manifold \(M\) and a point \(p\in M\). A function \(f\) defined on some open neighborhood of \(p\) is said to be \(C^\infty\) at \(p\) if there exists a coordinate chart \((U,\varphi)\) containing \(p\) such that the function \(f\circ\varphi^{-1}:U'\rightarrow\mathbb{R}\) is \(C^\infty\) at the point \(\varphi(p)\).
Suppose another coordinate chart \((V,\psi)\) is defined on another open neighborhood of \(p\). If \(f\circ\varphi^{-1}\) were \(C^\infty\) at \(\varphi(p)\) but \(f\circ\psi^{-1}\) were not \(C^\infty\) at \(\psi(p)\), this definition would be ill-defined. However, on \(\psi(U\cap V)\) we have
\[f\circ\psi^{-1}=(f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1})\]and therefore \(f\circ\psi^{-1}\) is \(C^\infty\) at \(\psi(p)\). By a similar argument one can prove the following.
Proposition 3 Let a \(C^k\)-atlas \(\mathcal{A}\) be given on a topological manifold \(M\). Then there exists a unique maximal \(C^k\)-atlas containing \(\mathcal{A}\). Hence any \(C^k\)-atlas \(\mathcal{A}\) defines a unique \(C^k\)-differentiable structure on \(M\).
Proof
Define \(\mathcal{A}'\) by the formula
\[\mathcal{A}'=\{(V,\psi)\mid\psi\circ\varphi_\lambda^{-1}, \varphi_\lambda\circ\psi^{-1}\text{ are $C^k$ for all $\varphi_\lambda\in\mathcal{A}$}\}\]Then \(\mathcal{A}'\) contains \(\mathcal{A}\), and therefore covers \(M\) by coordinate charts. Moreover, if \((V,\psi)\) and \((V',\psi')\) are elements of \(\mathcal{A}'\) and \(V\cap V'\neq\emptyset\), then the transition map
\[\psi'\circ\psi^{-1}:\psi(V\cap V')\rightarrow\psi'(V\cap V')\]is \(C^k\). For any \(p\in\psi(V\cap V')\), pick \((U,\varphi)\in\mathcal{A}\) with \(p\in U\); then on \(U\cap V\cap V'\) we have
\[\psi'\circ\psi^{-1}=(\psi'\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1})\]so that \(\psi'\circ\psi^{-1}\) is \(C^k\) at the point \(p\). Since \(p\) was chosen arbitrarily, this shows that \(\psi'\circ\psi^{-1}\) is \(C^k\). Of course, by interchanging the roles of \((V,\psi)\) and \((V',\psi')\) one can show that the transition map in the opposite direction is also \(C^k\).
Obviously \(\mathcal{A}'\) is a maximal \(C^k\)-atlas by definition, and its uniqueness is easily verified.
Example 4 Consider two atlases on the set of real numbers \(\mathbb{R}\):
\[\mathcal{A}_1=\{(\mathbb{R},\id_\mathbb{R})\},\qquad \mathcal{A}_2=\{(\mathbb{R}, x\mapsto x^3)\}\]Since these consist of a single chart, they are obviously \(C^\infty\). By the preceding Proposition 3, there exists a differentiable structure containing each of them. But they are not equal. This is because the two charts \((\mathbb{R},\id_\mathbb{R})\) and \((\mathbb{R}, x\mapsto x^3)\) are not \(C^\infty\)-compatible. (\(x\mapsto x^3\) is a \(C^\infty\) function, but its inverse \(x\mapsto x^{1/3}\) is not.)
Nevertheless, although the two atlases in Example 4 do not give the same differentiable structure, they give diffeomorphic differentiable structures.
Smooth partition of unity
We showed that a continuous partition of unity exists on any topological manifold; however, when dealing with differentiable manifolds a merely continuous partition of unity is of little use. For example, if we multiply an arbitrary \(C^\infty\) function by a partition of unity that is merely continuous, the differentiability of the function is immediately weakened.
Therefore we need to construct a smooth partition of unity, and for this it suffices to prove the following lemma.
Lemma 5 (\(C^\infty\) Urysohn lemma) Let real numbers \(a'<a<b<b'\) be given. Then there exists a \(C^\infty\) function \(\psi:\mathbb{R}\rightarrow[0,1]\) that is equal to \(1\) on \([a,b]\) and equal to \(0\) outside \((a',b')\).
Proof
We may assume without loss of generality that \(a'=-2\), \(a=-1\), \(b=1\), and \(b'=2\). First define a function \(f\) by
\[f(t)=\begin{cases}e^{-1/t}&t>0\\0&t\leq 0\end{cases}\]In particular, \(f\) is always non-negative and is \(C^\infty\). Now define
\[g(t)=\frac{f(t)}{f(t)+f(1-t)}\]Then \(g\) is likewise always non-negative, its values are always at most \(1\), and in particular \(g(t)\equiv 1\) for \(t\geq 1\) and \(g(t)\equiv 0\) for \(t\leq 0\). Therefore, defining \(\psi\) by the formula
\[\psi(t)=g(t+2)g(2-t)\]gives the desired function.
Using the above \(C^\infty\) Urysohn lemma instead of the ordinary Urysohn lemma, one can construct a smooth partition of unity on a differentiable manifold.
References
[Lee] John M. Lee. Introduction to Smooth Manifolds, Graduate texts in mathematics, Springer, 2012
[War] Frank W. Warner. Foundations of Differentiable Manifolds and Lie Groups, Graduate texts in mathematics, Springer, 2013
댓글남기기