Manifold #8. Diffeomorphisms매니폴드 #8. 미분동형사상

These are my notes on XylyXylyX’s What is a Manifold? lectures. This post covers lesson 8 of the playlist, Diffeomorphisms. The earlier material is in the previous post.

The gap this lecture fills

Let us recall the structure built so far. Give a set $X$ a topology $\mathcal{T}_X$ and put an atlas on it, and it becomes a topological manifold. A chart in the atlas is a pair of an open region $U$ of $X$ and a map $\gamma$, and $\gamma$ sends $U$ into a Euclidean space $\R^d$. Compositions such as $\varphi \circ \gamma^{-1}$ on the overlap of two charts are already guaranteed to be continuous. If we add the condition that these compositions are smooth, that is, infinitely differentiable, we get a differentiable manifold. From now on, when we say atlas we mean such a smooth atlas.

There is one habit that matters most in this structure. Whenever differentiation comes up on a manifold, read it as moving to $\R^d$ through charts and differentiating a multivariable function from $\R^m$ to $\R^n$. The manifold $X$ is just a set of points, so there is no subtraction of points and no meaning of limits, and derivatives exist only in Euclidean space. Today we use this habit to reread curves, coordinates, and functions between manifolds in turn.

A curve is a function into the manifold

A curve is a function that sends a parameter $\lambda$ on the real line to a point of the manifold.

It is also common to define it on just an open interval or a closed interval $[0, 1]$, but we ignore that difference for now. A curve may cross the regions of several charts, just as any set on a manifold does not fit into a single chart.

There is a mistake beginners easily make when finding the coordinates of a curve. It is to write $(a, b) = c(\lambda)$. The value $c(\lambda)$ is a point of $X$, not a pair of numbers. The pair of numbers appears only after going through a chart.

The domain of a composition is the domain of the right-hand function $c$, and its codomain is that of the left-hand function $\gamma$, so $\gamma \circ c : \R \to \R^2$. This expression makes sense only when $c(\lambda_0)$ lies in $U$. If the same point also lies in the region of another chart $\varphi$, then $\varphi \circ c$ gives another pair of numbers, and the transition function connects the two pairs.

Differentiating a curve works the same way. Many books write $c'$, but what is really differentiated is $(\gamma \circ c)'$. The codomain of $c$ is a manifold, where differentiation is not defined, and $\gamma \circ c$ is a function from $\R$ to $\R^2$, which we can differentiate. When a textbook quietly drops $\gamma$ and gives the whole composition one name, that is a convenient shorthand. When learning it for the first time, it is good to write out which function stands behind the shorthand.

Q. How do we say that a curve $c$ is continuous or differentiable?
A. We define it by $\gamma \circ c$ being continuous or differentiable. Since the charts fit together smoothly, checking in any chart gives the same conclusion.

Coordinate functions

The value a chart $\gamma$ gives at a point $P$ is a bundle of $d$ numbers. The function that computes the number in each slot separately is called a coordinate function. For $d = 2$ it looks like this.

The map $\gamma$ bundles the coordinate functions into an ordered pair, and each coordinate function sends one point to one real number. The $x^0, x^1, \dots$ in general relativity textbooks are exactly these. Numbering the first coordinate from 0 is only a notation following that convention.

New coordinates are functions of old coordinates

Textbooks contain a sentence like this: “The new coordinates $x'^0, x'^1$ are functions of the old coordinates $x^0, x^1$.” Let us follow how this sentence comes out of the chart structure. Call the chart giving the old coordinates $(U, \gamma)$, the chart giving the new coordinates $(V, \varphi)$, and the new coordinate functions $x'^0, x'^1 : V \to \R$. The two coordinate systems can be used together only on $U \cap V$.

The old coordinates are the image $\gamma(P) = (x^0, x^1)$ of a point $P \in U \cap V$ under $\gamma$. To get one new coordinate from this, take the following steps.

1. Go back to $P$ with $\gamma^{-1}$. That the inverse exists is guaranteed by the definition of a chart.
2. Going forward with $\varphi$ would give an ordered pair, but we want a single number, so in place of $\varphi$ we apply the coordinate function $x'^0$ of $\varphi$.

So writing one new coordinate as a function of the old coordinates is the following composition.

The textbook uses the same letter $x'^0$ for the value on the left and for the function on the right. A number and a function share a name. To write the old coordinates as functions of the new ones, use $x^0 \circ \varphi^{-1}$.

The transition function $\varphi \circ \gamma^{-1}$ sends an ordered pair to an ordered pair, and the textbook notation writes its output one slot at a time. In other words, $x'^j \circ \gamma^{-1}$ is the $j$-th component of the transition function. They are the same information written in two ways.

Q. In the textbook’s $x'^0(x^0, x^1)$, are the $x'^0$ before and after the parentheses the same object?
A. The lecture points this out. The same letter in the textbook means a coordinate value when it stands alone, and when arguments are attached it means the coordinate function $x'^0 \circ \gamma^{-1}$. The form is abbreviated, so it confuses at first, but remembering the order of the composition makes it readable.

Coordinate lines are a grid pushed back

We can also see in a picture how coordinates lie on the manifold. In one chart of $\R^2$, draw the horizontal lines joining points with the same value of $x^0$ and the vertical lines joining points with the same value of $x^1$. Pushing these lines back with $\gamma^{-1}$ lays them over $U$. The points on one line share the same $x^0$ value, and the coordinates of a point are the names of the two lines through it. If another chart gives polar coordinates $(r, \theta)$, the set of points with the same $r$ is a family of concentric circles, and pushing it back lays another curved grid over the manifold. The winding coordinate lines in textbooks are grids pushed back by inverse maps in this way. The grid is a set of labels attached to points, and the manifold itself stays as it was.

Smooth functions between manifolds

Now take two manifolds. $X$ has a topology $\mathcal{T}_X$ and an atlas $\mathcal{A}$, $Y$ has a topology $\mathcal{T}_Y$ and an atlas $\mathcal{B}$, and both are differentiable manifolds. Let the dimensions be $\dim X = d$ and $\dim Y = b$. We want to ask whether a function $F : X \to Y$, sending points to points, is smooth. $F$ goes from one topological space to another, so we cannot differentiate it directly. So we attach charts on both sides and turn it into a function between Euclidean spaces. Take a chart $(U, \gamma)$ of $X$ and a chart $(V, \varphi)$ of $Y$, and the following composition is a function from $\R^d$ to $\R^b$.

Reading from the right, we return from coordinates to the point ($\gamma^{-1}$), go to $Y$ with $F$, and get coordinates again with the chart $\varphi$. If this composition is smooth for every pair of charts where it is defined, $F$ is called a smooth function. $F$ itself hides between the two chart regions, and the differentiation happens in the Euclidean spaces made by the charts on either side.

Let me add one thing. The reason the result does not depend on which pair of charts we check is that the atlas is smooth. If we use another chart $\gamma'$, then $\varphi \circ F \circ \gamma'^{-1} = (\varphi \circ F \circ \gamma^{-1}) \circ (\gamma \circ \gamma'^{-1})$, and the composition of two smooth functions is smooth. (This explanation is added to aid understanding and was not covered in the lecture.)

Diffeomorphisms

A smooth function need not be one-to-one or onto. If all three of the following conditions hold, $F$ is called a diffeomorphism.

1. $F$ is one-to-one and onto. That is, it is a bijection.
2. $F$ is smooth.
3. The inverse $F^{-1} : Y \to X$ is also smooth. In terms of charts, $\gamma \circ F^{-1} \circ \varphi^{-1} : \R^b \to \R^d$ is smooth.

A homeomorphism was a map that is a bijection, continuous, and has a continuous inverse. A diffeomorphism replaces continuous with differentiable, and since differentiable implies continuous, it is a stronger condition than a homeomorphism. It also differs in that it can be stated only between manifolds with a differentiable structure. If there is even one diffeomorphism connecting two manifolds, the two spaces are diffeomorphic. Even if we cannot construct that function explicitly, it is enough that its existence is proved. Asking whether two spaces are diffeomorphic means constructing such a function or proving its existence in some other way.

Let us confirm with an example that the last condition cannot be dropped. $F(x) = x^3$ is a smooth bijection from $\R$ to $\R$ with a continuous inverse $\sqrt[3]{x}$, so it is a homeomorphism. But $\sqrt[3]{x}$ is not differentiable at $x = 0$, so it is not a diffeomorphism. On the other hand, $x \mapsto e^x$ is a diffeomorphism from $\R$ to the positive reals $(0, \infty)$. In one dimension the slope of the inverse is $(F^{-1})'(F(x)) = 1 / F'(x)$, so if $F'$ is zero at some point, the inverse is not differentiable there. (This explanation is added to aid understanding and was not covered in the lecture.)

In the toy below, pick a function and move the point. The graph of $f$ and the graph of its inverse are drawn side by side, and being one-to-one, being onto, and the differentiability of the inverse are checked in turn to tell whether it is a diffeomorphism. (This toy is added to aid understanding and was not covered in the lecture.)

Q. If $F$ is a smooth bijection, is it a diffeomorphism?
A. No. The inverse must be smooth as well. $x^3$ is a counterexample. In the toy above, pick $x^3$ and move to $x_0 = 0$, and you can see the tangent of the inverse stand vertically.

Summary and the next lecture

Today we collected the following. A curve is a function from the real line into the manifold, and its coordinates and derivative are defined through a chart. The coordinate functions, and the textbook phrase “new coordinates are functions of old coordinates”, are components of the transition function. A function between manifolds is made smooth or not by moving it from $\R^d$ to $\R^b$ with charts. A diffeomorphism is a smooth bijection with a smooth inverse. Coordinate expressions such as $x^0 \circ c$ of a curve will be used a lot in the next lecture. In the next lecture we use curves to define the tangent space at a point of a manifold.