What is a manifold?

A Manifold is a space on which one can do calculus. This is what I usually say when asked about manifolds. In this post, I want to start explaining what this phrase means.

What is a space?

Most people have some idea what space is. We live inside it after all. That said, it is very hard to give a precise definition of the term space. In mathematics, instead of trying to define space, we just try and describe some of its properties. The first property: A space has a set whose elements we call points. For example, we could talk about a space with points \(\{ {\rm apples}, {\rm oranges} \}\) or the space with points \(\{ z \in \mathbb{C} : \lvert z \lvert = 1 \}\). The first space may seem a little strange, but the second one should be more familiar. It is the circle, a 1 dimensional space:

There is not much that you can do mathematically with just a set of points. We need to describe our spaces more precisely. There are many ways to do this. If you are a probability theorist, you could interpret your points as possible outcomes of an experiment and equip the set of points with a probability measure. If you are an algebraic topologist, you could write down a simplicial set whose connected components are the points in your space. If you like algebraic geometry, you could write down a system of polynomial equations whose solutions are the points in your space. In order to describe spaces on which we can do calculus, we need to introduce observables.

Observables

The primary way in which humans understand the objects around them is by interpreting the light which bounces off them. If you stand still and look at a stationary object, your brain associates a color (which we identify with its frequency as an electromagnetic wave) to each point on the surface of the object. If we think of the surface of our object as a space, then color defines a partial real valued function on this space (it is only a partial function because we can't see the back of the object!) The second property of spaces: A space has an algebra of observables which are partial functions from the set of points into a field. By algebra, all I mean is that you can add observables and multiply observables. Often the field is just the real numbers. In many branches of mathematics, the set of points is suppressed and all focus is directed at the algebra of observables. It is important to note that every observable is a partial function from points into the real numbers but not every partial function from the points to the real numbers is an observable. Usually observables are required to have nice properties which allow us to study them mathematically. For example, in algebraic geometry, all the observables are required to be polynomial functions and in differential geometry all the observables are required to be smooth functions. We shall always require our observables to be continuous. For this to even make sense, our set of points needs to be equipped with a topology. From now on, whenever we talk about the set of points of a space, it comes equipped with a topology and whenever we talk about observables, they are continuous.

The sheaf of observables?!????

Mathematicians often like to invent fancy words when they are talking or writing. These fancy words are what allow us to keep all the mathematics inside our head organized. This is a form of abstraction, and it one of the reasons why mathematics seems so formidable to the uninitiated. The sheaf of observables is an example of this. Take a space and write \(X\) for the set of points. If \(U \subseteq X\) is an open set then we define \(\mathcal{O}_X(U)\) to be the algebra of continuous functions \(U \to \mathbb{R}\). We call \(\mathcal{O}_X(U)\) the observables defined on \(U\). The gadget \(\mathcal{O}_X\) is called the sheaf of observables. It is kind of like a function because it takes open subsets of \(X\) to algebras but there is more structure. If \(U \subseteq V\) are open sets in \(X\), then we have an algebra homomorphism \(\mathcal{O}_X(V) \to \mathcal{O}_X(U)\) defined by restriction. If you want to be fancy, you can say that \(\mathcal{O}_X\) is a contravariant functor from the category of open sets in \(X\) into the category of commutative algebras.

Manifolds, Finally!

Now we can finally explain what a manifold is. Take a space \(X\) (we are going to start committing the standard abuse of notation which identifies a manifold with its set of points). We call \(X\) an \(n\) dimensional manifold if:

  1. each point \(x \in X\) has an open neighborhood \(x \in U \subseteq X\) with \(n\) observables \(x_1,\dots,x_n\) such that \((x_1,\dots,x_n) : U \to \mathbb{R}^n\) is injective with image an open subset of \(\mathbb{R}^n\). We call \(x_1,\dots,x_n\) coordinates on \(U\).
  2. If \(x_1,\dots,x_n\) are coordinates on \(U\), \(y_1,\dots,y_n\) are coordinates on \(V\) and \(U \cap V \not= \emptyset\) then \(x_i(y_1,\dots,y_n)\) and \(y_j(x_1,\dots,x_n)\) are smooth functions.

If \(f:X \to \mathbb{R}\) is a function, then we say that \(f\) is smooth if \(f(x_1,\dots,x_n) : U \to \mathbb{R}\) is smooth for every choice of coordinate chart. Since changing coordinates is smooth, in order to check that \(f\) is smooth, you only need to check that \(f(x_1,\dots,x_n)\) is smooth in one set of coordinates around each point. We can define a smooth map \(F : X \to Y\) between manifolds. For each coordinate chart \(V \subseteq Y\) and each coordinate chart \(U \subseteq F^{-1}(V)\), \(F : U \to V\) is smooth in coordinates.

Examples

Manifolds are everywhere, so lets see some examples.

  1. Take the circle \(S^1\) which is depicted above. Write \(N = (0,1)\) and \(S = (0,-1)\). Then define \[\alpha : S^1 \backslash \{ N \} \to (\pi/2,5 \pi/2)\] \[\beta : S^1 \backslash \{ S \} \to (-\pi/2,3\pi/2)\] to be the functions which take a point on the circle to the corresponding angle. Then \[\alpha = \beta + 2 \pi\] so \(\alpha\) is a smooth function of \(\beta\).
  2. Let \(V\) be a vector space over \(\mathbb{R}\). Choose a basis \(x_1,\dots,x_n\) for \(V^*\). Then \(x_1,\dots,x_n\) is a coordinate chart which covers \(V\). We call \(\{x_i\}\) a linear coordinate chart.
  3. Let \(V\) be a 3-dimensional vector space and let \(\mathbb{P}(V)\) be the set of lines in \(V\) which pass through the origin. Let \(X_0,X_1,X_2\) be a basis for \(V^*\). The functions \(X_i\) are not well defined on \(\mathbb{P}(V)\), but the ratios \(X_i/X_j\) are. The functions \(X_1/X_0, X_2/X_0\) are a coordinate chart on \(\{L : X_0(L) \not= 0 \}\). Note that even though \(X_0\) is not a well defined function on \(\mathbb{P}(V)\), the condition \(X_0(L) = 0\) is well defined. Similarly, \(X_0/X_1, X_2/X_1\) are coordinates on \(\{ L : X_1(L) \not= 0\}\) and \(X_0/X_2, X_1/X_2\) are coordinates on \(\{L : X_2(L) \not= 0 \}\). Therefore we can cover \(\mathbb{P}(V)\) with three coordinate charts. Changing coordinates is easy: \[ X_i/X_j = \frac{1}{X_j/X_i}\]
  4. Define \({\rm GL}_r(\mathbb{R})\) to be the set of \(r \times r\) invertible matricies. A matrix is invertible iff its determinant is nonzero, therefore \({\rm GL}_r(\mathbb{R}) \subseteq {\rm Mat}_{r \times r}(\mathbb{R}) \cong \mathbb{R}^{r^2}\) is an open subset. We have a coordinate chart on \({\rm GL}_r(\mathbb{R})\) given by the restriction of the matrix coefficients.

In later posts we will see many more examples. We have defined manifolds and seen our first examples (the circle, euclidean space and the projective plane), but we still haven't explained why we can do calculus on a manifold. We will start to see why in the next post about tangent spaces.

Exercises

I will always try and include some exercises in these posts.

  1. Let \(L\) be the set of all lines in \(\mathbb{R}^3\) (not just those passing through the origin). Prove that \(L\) is a manifold.
  2. For any vector space \(V\), prove that \(\mathbb{P}(V)\) is a manifold.
  3. (harder) Prove that every manifold has a well defined dimension. hint.
  4. (harder) Prove that the only two 1-dimensional manifolds are \(\mathbb{R}\) and \(S^1\).