Chapter 1: Smooth Manifolds

天地有正气，杂然赋流形。 ——文天祥《正气歌》

You can read the LaTeX document online (for the latest updated chapters) from the link: https://gaomj.cn/pdfjs/web/viewer.html?file=Manifold.pdf

Chapter 1: Smooth Manifolds Manifolds are fundamental objects in modern mathematics. They are not only very important in geometry, topology, analysis, but also very useful in applied mathematics and theoretical physics. We can picture them as generalizations of smooth curves and surfaces, and do differentiation and integration on them as on curves and surfaces. There are many subfields in math based on smooth manifolds, such as Riemannian geometry, complex geometry and symplectic geometry. In physics, relativity models spacetime as a 4-dimensional manifold with a certain geometric structure.

The purpose of this note is to study some basic properties of smooth manifolds. Much of work in this note is learned from John M. Lee's excellent book Introduction to Smooth Manifolds.

Contents
Contents
 1.  Smooth manifolds
 2.  Smooth maps

1. Smooth manifolds Smooth manifolds are some spaces which "locally looks like \mathbb R^n". For example, the surface of the Earth looks locally flat, and we can hardly find out in daily life that this surface globally looks like a sphere. In this example, a 2-dimensional sphere locally looks like \mathbb R^2, and it can be treated as a 2-dimensional manifold.

We often treat the sphere S^2 as a subset of \mathbb R^3. This is helpful for our intuition, but it has more advantages if we can study manifolds abstractly, without reference to an ambient space. The structure of the ambient space often have nothing to do with the problem at hand. For example, relativity models spacetime as a 4-dimensional manifold, but if we think there is a larger space containing it we will find it difficult to interpret and the theory becomes more complicated. Therefore we appreciate the theory that studies manifold itself, and in order to achieve this goal we have to do a lot of work.

Mathematically, what is a manifold? We need a topology space first, and for the development of calculus over it, we then need some structure of smoothness. We now formalize the concept of something "locally looks like \mathbb R^n": a topological space is said to be locally Euclidean of dimension n if each point of it has a neighborhood that is homeomorphic to an open set of \mathbb R^n. We can also say it is locally homeomorphic to \mathbb R^n for this definition.

It is not difficult to see that the definition is equivalent if we replace the requirement "an open set of \mathbb R^n" with "an open ball in \mathbb R^n", or with "the \mathbb R^n itself". For example, for a topological space M, if each point p of it has a neighborhood U which is homeomorphic to \mathbb R^n, then M is locally Euclidean.

In fact this is not the full definition of topological manifold. Usually we need two more conditions: First M should be a Hausdorff space, that is, for every pair of p,q\in M\, (p\neq q), there are neighborhoods U,V containing p,q respectively such that U\cap V=\varnothing; Second M should be second-countable, that is, there exists a countable basis for M. In this note we will always consider such spaces, so they are omitted from the definition.

It is said that with some advanced techniques we can prove the dimension is a topological invariant: An n-dimensional topological manifold can be homeomorphic to an m-dimensional manifold iff m=n. Thus every topological manifold has a unique dimension.

Now we consider to define a smooth structure on a topological manifold. Recall that for a function f:U\to V, where U\subseteq \mathbb R^n and V\subseteq\mathbb R^m are open sets, we call f a smooth function when f\in C^\infty, i.e., f is infinitely differentiable. When f is bijective and the reverse f^{-1} is also smooth, we call f a diffeomorphism. What about a smooth function on a manifold M? It is natural to require f\circ\varphi^{-1}:\widehat U\to\mathbb R is smooth, but we need to ensure that this smoothness is independent of the choice of chart.

Obviously an atlas always exists. For two charts (U _ i,\varphi _ i),(U _ j,\varphi _ j), the composite map \varphi _ j\circ\varphi _ i^{-1}:\varphi _ i(U _ i\cap U _ j)\to\varphi _ j(U _ i\cap U _ j) is a homeomorphism, and it is called the transition map from \varphi _ i to \varphi _ j.

We can define a smooth structure on M by giving a smooth atlas. Then we can tell if a function f:M\to\mathbb R is smooth by checking the smoothness of f\circ\varphi _ i^{-1} for all i. Sometimes different smooth atlases give the same smooth structure, in the sense that they determine the same smooth functions. We can define a smooth structure as an equivalence class of smooth atlases. Alternatively, we can define in the following way:

Any chart in a maximal smooth atlas of the smooth manifold is called a smooth chart, and the corresponding coordinate map \varphi is called a smooth coordinate map (the smoothness of a map will be discussed later). Sometimes we will simply say a manifold to indicate a smooth manifold. This convention is also adopted for charts, coordinate maps.

It is not so easy to construct using a maximal atlas. However we actually only need to specify some charts for the following reason: Every smooth atlas \mathcal A for M is contained in a unique maximal smooth atlas, called the smooth structure determined by \mathcal A.

This proposition is not difficult to prove. Let \overline{\mathcal A} be the set of all charts that are smoothly compatible with every chart in \mathcal A. Then, it can be proved that \overline{\mathcal A} is a smooth atlas. Now, any chart that is smoothly compatible with every chart in \overline{\mathcal A} must be smoothly compatible with every chart in \mathcal A, so it is already in \overline{\mathcal A}. Thus \overline{\mathcal A} is maximal. If \mathcal B is any other smooth atlas containing \mathcal A that is maximal, then \mathcal B\subseteq\overline{\mathcal A}. We must have \mathcal B\subseteq\overline{\mathcal A}.

It is also not difficult to see: two atlases for M determine the same smooth structure iff their union is a smooth atlas.

As long as we work "in the chart \varphi" we can pretend we are working in \mathbb R^n, just as we could pretend we live on \mathbb R^2.
2. Smooth maps The smooth structure enables us to define smooth functions on manifolds and smooth maps between manifolds.

The smoothness of f at a point is defined in such a way that it is independent of the chart (U,\varphi). If f is smooth at p for (U,\varphi), and (V,\psi) is another chart containing p, then on \psi(U\cap V), f\circ\psi=(f\circ\varphi^{-1})\circ(\varphi\circ\psi^{-1}) is smooth at \psi(p).

The function \widehat f:\varphi(U)\to\mathbb R^k defined by \widehat f=f\circ\varphi^{-1} is called the coordinate representation of f.

This definition indicates that a smooth map F is necessarily continuous, which is easy to prove.

Again, we define the smoothness in a way such that it is independent of the choice of charts. If F is smooth at p\in M and (U,\varphi) is a chart containing p in M, (V,\psi) is a chart containing F(p) in N, it can be shown that \psi\circ F\circ\varphi^{-1} is smooth at \varphi(p), analogous exactly to the case of a smooth function at a point p. Then we can prove the following Proposition 8 straightforward.

The function
\widehat{F} =\psi\circ F\circ\varphi^{-1}: \varphi(U\cap F^{-1}(V))\to\psi(V)\subseteq\mathbb R^kis called coordinate representation of F with respect to the given coordinates.

A diffeomorphism of manifolds is a smooth bijective map F:M\to N whose inverse is also smooth. Naturally we have

We prove it from Proposition 8. By definition \varphi is a homeomorphism, so it suffices to check the smoothness of \varphi and \varphi^{-1}. We can take an open subset of a manifold as a submanifold. Let U\subseteq M be an open subset of M, and \mathcal A _ U be the collection of smooth charts for M such that the domain is contained in U. For every point p in M there is a chart (V,\varphi) containing p. Then (V\cap U, \varphi _ {V\cap U}) is a chart in \mathcal A _ U containing p. Such charts form an atlas on U, and it is easy to see this is a smooth atlas. Thus U is itself an open submanifold of M.

Now to check the smoothness of \varphi where (U,\varphi) is a chart for M, we use the atlas \{(U,\varphi)\} with a single chart on U and the atlas \{(\varphi(U),\mathbf{1} _ {\varphi(U)})\} with a single chart on \varphi(U). We have \mathbf 1 _ {\varphi(U)}\circ \varphi\circ \varphi^{-1}=\mathbf 1 _ {\varphi(U)}, which is clearly smooth. By Proposition 8, \varphi is smooth. The smoothness of \varphi^{-1} is similar.

Let (U _ i,\varphi _ i) be a smooth chart in the maximal atlas of M. Then \varphi,\varphi^{-1} are both smooth. We have known the composite of smooth maps is also smooth, so F\circ\varphi^{-1} and \varphi\circ F^{-1} are smooth. The transition map is therefore a diffeomorphism, indicating (U,F) is compatible with the maximal atlas. By the maximality, the chart (U,F) is already obtained in it.

Notation

Finally in this chapter, we take a look at an important notational convention. Suppose we have n terms x _ 1y _ 1,\dots,x _ ny _ n and we want to add them together. A useful notation is \sum _ {i=1}^{n}x _ iy _ i:=x _ 1y _ 1+\dots+x _ ny _ n. However this is often abbreviated further:
x^iy _ i:=\sum _ {i=1}^{n}x^iy _ i.The subscript of x has been replaced by the superscript to facilitate the abbreviation, which is called Einstein summation convention. If in a term there is an index in the superscript and subscript simultaneously, then this term is an abbreviation of a summation (over all possible values of the index).