Manifold #7. Differentiable Manifolds매니폴드 #7. 미분 가능 매니폴드

These are my notes on XylyXylyX’s What is a Manifold? lectures. This post covers lesson 7 of the playlist, Differentiable Manifolds. Add one condition to the topological manifold from the previous post and it becomes a differentiable manifold. Every differentiable manifold is a topological manifold, but the converse does not hold.

Topological manifolds, reviewed

A topological manifold is a topological space $(X, \mathcal{T}_X)$ with an atlas $\mathcal{A}$ on it. An atlas is a collection of charts, and a chart is a pair of a neighborhood $U_i$ and a homeomorphism $\gamma_i : U_i \to \R^d$. Both $\gamma_i$ and $\gamma_i^{-1}$ are continuous, the neighborhoods cover all of $X$, and every chart goes to Euclidean space of the same dimension $d$. The $\R^d$ that each chart lands in is a different copy.

The $d$-tuple $\gamma_1(P)$ where $\gamma_1$ places a point $P$ is called the coordinates of $P$ in this chart. Coordinates only make sense once we fix a chart. A point in another chart $U_2$ has entirely different coordinates in a different $\R^d$. What these charts do is carry what happens on the topological manifold into Euclidean space, where we can use multivariable calculus. From a physicist’s point of view the manifold is reality or a model of reality, and a chart is a road from it to a model we can compute in. So far, though, there is not a word about differentiation.

Charts overlap

To talk about differentiation, we need to look where two charts are present together. For two charts $(U_1, \gamma_1)$ and $(U_2, \gamma_2)$, write their intersection as $U_{12} = U_1 \cap U_2$. As we saw in the stereographic projection, charts often overlap, and the overlap can be very severe. The two chart regions were identical except for one point each.

There are less severe ways. If we cover a circle by splitting it into several arcs, only neighboring arcs overlap a little. But however we split it, the overlap itself cannot be avoided. Neighborhoods are open sets and do not contain their boundaries, so to cover a boundary point, another neighborhood must surround it.

Transition functions

A point $P$ in $U_{12}$ has coordinates in one $\R^d$ through $\gamma_1$ and in another through $\gamma_2$. A function that takes the coordinates on one side to the coordinates on the other is built like this. Start from the point $\gamma_1(P)$, go back to $P$ with $\gamma_1^{-1}$, and send that point with $\gamma_2$.

This is a transition function. If the composition symbol looks unfamiliar, read the whole expression as the name of one function. The domain of the function on the right is the domain of the whole, and the range of the function on the left is the range of the whole. The lecture calls it a function from $\R^d$ to $\R^d$, and strictly the domain need not be all of $\R^d$, and it is the open subset $\gamma_1(U_{12})$ of $\R^d$, which is all we need to remember. (This supplement was not covered in the lecture.) There is also the opposite direction, $\gamma_1 \circ \gamma_2^{-1}$. A transition function is a coordinate transformation that turns the coordinates of one chart into those of another, and the functions in the two directions are inverses of each other.

Take the stereographic projection with chart $\gamma_1$ shot from the south pole and chart $\gamma_2$ shot from the north pole. Start from a point of one plane, find the point on the sphere with $\gamma_1^{-1}$, and apply $\gamma_2$ to it to get a point of the other plane. The transition function skips this detour and goes straight from plane to plane. In the cross-section of the circle we can write it as a formula. From the coordinates $u = x/(1+y)$ and $v = x/(1-y)$ of the last post we have $uv = 1$, so the transition function is $v = 1/u$. (This formula was added by me. The lecture only says the function can be built with angles and trigonometry and appears in books, without writing a formula.)

Since this function is a multivariable function from $\R^d$ to $\R^d$, we can do calculus on it. For any topological manifold, a transition function can always be built and is always continuous, because $\gamma_1^{-1}$ and $\gamma_2$ are both continuous and so is their composition. A continuous function is $C^0$.

Q. Why is a transition function of a topological manifold always continuous?
A. By the definition of a chart, $\gamma_1^{-1}$ and $\gamma_2$ are both continuous. The composition of two continuous functions is continuous, so on a topological manifold it is automatically $C^0$. Differentiability beyond that is not guaranteed by being a topological manifold.

Differentiable manifolds

A transition function can be better than continuous. A function is $C^k$ if it can be differentiated $k$ times and its $k$-th derivative is continuous. $C^1$ can be differentiated once, and $C^2$ twice. A function that can be differentiated for any $k$ is $C^\infty$, and such a function is called smooth. The definition used in the lecture is this. If every transition function of the atlas is $C^\infty$, we have a differentiable manifold. If the transition functions are only $C^k$, it is a $C^k$ manifold, and if only $C^0$ it is just a topological manifold. Other books define it slightly differently, but not very differently.

This condition is an extra requirement on top of a topological manifold. A topological manifold alone may have a transition function with a cusp. There the function is continuous but has no derivative. A space with such a transition function is not a differentiable manifold. What is new is a requirement on the relation between charts, and the space stays the same.

Q. What does a differentiable manifold add to a topological manifold?
A. The space stays as it is, and a condition on the transition functions of the atlas is added. Every transition function must be smooth. Being a topological manifold guarantees that transition functions are continuous but not that they are differentiable.

Coordinate functions

Sometimes we want to look at just one component of the coordinates of a point. A chart gives a $d$-tuple, and the function that picks out its $i$-th component is called a coordinate function. A transition function for a single coordinate can be built the same way, and one coordinate in the other chart usually depends on all the coordinates of the first chart. The same smoothness is still required. When we speak of a chart we normally mean the function that gives the whole tuple, and coordinate functions become important again when we handle tangent spaces.

Compatibility

We define the compatibility of two charts. For charts $(U, \gamma)$ and $(V, \varphi)$, there are two ways to be compatible. First, if $U \cap V = \emptyset$ they are compatible, since with no overlap there is no place to conflict. Second, if they overlap, both transition functions $\gamma \circ \varphi^{-1}$ and $\varphi \circ \gamma^{-1}$ must be smooth. Compatibility is a property of pairs of charts.

Now look at a whole atlas. If all the charts in an atlas are compatible with each other, it is an atlas of a differentiable manifold. This is why this expression is often used to define a differentiable manifold: a topological manifold with an atlas whose charts are all compatible with each other is a differentiable manifold.

Merging two atlases

An interesting question comes up here. Say the same topological manifold has atlases $\mathcal{A}$ and $\mathcal{B}$, and inside each one all charts are compatible. Form the union $\mathcal{C} = \mathcal{A} \cup \mathcal{B}$. It is still an atlas, since each atlas covers the space. But whether it is an atlas of a differentiable manifold is a different matter. If even one pair of a chart from $\mathcal{A}$ and a chart from $\mathcal{B}$ is incompatible, $\mathcal{C}$ is not an atlas of a differentiable manifold. In that case $\mathcal{A}$ and $\mathcal{B}$ give different differentiable structures.

This can happen even in dimension 1. On the line $\R$, consider the atlas with the single chart $u \mapsto u$ and the atlas with the single chart $u \mapsto w = u^3$. Each has one chart, so there is no problem inside it. But the transition function $u \mapsto u^3$ is smooth, while the opposite one, $w \mapsto \sqrt[3]{w}$, has infinite slope at $w = 0$ and is not differentiable. The two are incompatible, so the union is not an atlas of a differentiable manifold. The toy below shows the transition functions of both pairs. The two charts of the stereographic projection join smoothly everywhere, and the cube chart breaks down at 0 in the backward direction. (The cube chart example was added for understanding and was not covered in the lecture.)

Q. The cube chart $w = u^3$ is a fine chart in itself, so why is it incompatible?
A. A chart only needs to be a homeomorphism, and $u \mapsto u^3$ is one-to-one and continuous in both directions, so it meets that requirement. Compatibility is decided by whether the transition function with another chart is smooth in both directions, and the backward one, $w \mapsto \sqrt[3]{w}$, is not differentiable at $0$, so they are not compatible.

Maximal atlases and differentiable structures

Here the lecture introduces concepts we will meet in other books. An atlas that contains as many charts as possible while staying compatible with an atlas and all its charts is a maximal atlas. Merging other compatible atlases does not make it bigger. Each differentiable structure has one maximal atlas. If we try to put $\mathcal{B}$ above into the maximal atlas built from the structure of $\mathcal{A}$, an incompatible pair turns up somewhere, so it cannot go in.

The lecture explains that how many different differentiable structures one topological manifold can have is very deep mathematics. It reports that dimensions 1, 2, and 3 have no such problem, that in dimension 5 and higher each topological manifold has a finite number of differentiable structures, and that in dimension 4 there can be infinitely many. As an example it names the 7-dimensional sphere, saying that the fact that it has several differentiable structures is the story of Milnor spheres. For the count, the lecture only says it is something like 15 from memory, without confidence.

There are places where we should not take the lecture at face value, so let us point them out. Even in dimension 1, the union of two atlases can be incompatible, as with the cube chart above. What is true in dimensions 1, 2, and 3 is that the differentiable structures a topological manifold can have are essentially one, unique up to diffeomorphism, and the claim that the union is always compatible is false. The finiteness of differentiable structures in dimension 5 and higher holds for compact manifolds. I also could not confirm the number 15. (This paragraph was added by me and was not covered in the lecture.) The lecture says maximal atlases and differentiable structures do not come up at the level of physics in general relativity, and suggests we move on without much worry.

Summary

A differentiable manifold is a topological space with an atlas in which all charts are compatible with each other. Compatible means the transition functions are differentiable and smooth. We covered only the smooth case here. Coordinates have meaning only inside a chart, and transition functions are the rule for changing coordinates, which is the base for what comes next. The next post deals with smooth functions between manifolds.