Differential Equations on Manifolds (12/09/2025)

This post is based upon my reading of the section: Differential Equations on Manifolds (Arnold’s ODE).

Part 1: What is a differentiable manifold?

Goal: we first start by giving a few definitions; we then take a look at a few representative examples.

Definition of a differentiable manifold: it is a set M with a differentiable manifold structure on it. This structure is introduce on M if an atlas consisting of charts are consistently prescribed.

Definition of chart: let W be a subset of M and $\phi: M \rightarrow U \subset \mathbb{R}^m$ be bijective and continuous. Then $(W, \phi)$ is a chart.

Definition of consistent charts: Let $(W_i, \phi_i), (W_j, \phi_j)$ be two charts on M. Denote $U_i = \phi_i(W_i), U_j = \phi_j(W_j)$ , and $\phi_i(W_i \cap W_j) = U_{ij}, \phi_j(W_i \cap W_j) = U_{ji}$ . Then, we say that the two charts are consistent if:

$U_ij, U_ji$ are open (they can be empty, i.e. when $W_i \cap W_j = \emptyset$ ) (~~does this guarantee that $\phi_i, \phi_j$ are homeomorphisms? In class, we required $\phi_i, \phi_j$ to be homeomorphisms so this open condition naturally follows~~. Note: right now, there is no notion of ‘open’ or ‘closed’ sets on M because we do not have any notion of ‘distance’ on M.)
$\phi_ij = \phi_j(\phi_i^{-1}), \phi_ji = \phi_i(\phi_j^{-1})$ are diffeomorphisms between $U_{ij}, U_{ji}$ (these are called the transition maps).

Interpretation: to study a manifold, we need to ‘pull’ it back to the vector space $\mathbb{R}^m$ (or $\mathbb{C}^m$ as we will see later) through some mapping. So it is essentially a study of subsets of $\mathbb{R}^m$ along with a corresponding mapping. This map is what endows the image with the structure that $\mathbb{R}^m$ has.

So now the question is what do we mean by a ‘differentiable’ manifold? Does it even make sense to say a ‘chart’ is differentiable? If not, what is the criterion that makes a manifold ‘differentiable’?

Turns out, the ‘differentiable’ comes from the transition map. The differentiable class of the transition map determines what type of manifold we are dealing with.

If the transition maps are smooth, then our manifold is smooth.

If the transition maps are $C^0$ (i.e. homeomorphisms), then our manifold is a topological manifold.

Interpretation: the reason the smoothness of our manifold is uniquely determined by the smoothness of our transition maps is because the ‘differentiable structure’ that we have given to our manifold arises from the charts. So to study the structure, we study the relations between the charts. And it makes perfect sense to look at the differentiability of transition maps as they are maps from $\mathbb{R}^m \rightarrow \mathbb{R}^m$ (or $\mathbb{C}^m \rightarrow \mathbb{C}^m$ ).

Definition of atlas: an atlas is a collection of charts on M that satisfies:

any pair of charts are consistent
any point $x \in M$ belongs to the domain of at least one chart (i.e. the set $W_i$ covers the entire M)

Definition of equivalent atlases: 2 atlases on M are equivalent if their union is an atlas (really, this just means that any chart on the first atlas is consistent with any chart on the second).

We call an equivalence class of atlases a “differentiable manifold structure” on M.

Question/Investigation: It would be interesting to find a set M such that it can be endowed with more than one differentiable manifold structure. Also, how can we “count” the number of differentiable manifold structure that M has?

Important: we impose two conditions on our definition of a manifold to avoid pathological objects.

Condition 1: Hausdorff condition: any 2 points $x, y \in M$ have disjoint neighborhoods. This is equivalent to saying there exist either 2 disjoint charts that contain x and y respectively, or there exists a chart that contains both x and y (why? because suppose we have 2 charts in which one contains x and another contains y, since x and y belong to disjoint neighborhoods (i.e. $N_{\epsilon_x}(x), N_{\epsilon_y}(y)$ , we can find $\epsilon_x, \epsilon_y$ small enough such that these two neighborhoods are contained in the respective charts and are disjoint. Now, by restricting the charts to these two neighborhoods, we have obtained two charts that are disjoint and contain x and y, respectively).

What happens if we do not require this condition?

We give an example in $\mathbb{R}^2$ . Consider the two lines given by the x- and y-axis. We now construct the set obtained by these two lines by identifying points with equal negative coordinates x and y as one point. Notice that we have TWO origins, yet they do not belong in disjoint open neighborhoods.

Condition 2: second-countability, i.e. there exists an atlas of M consisting of at most countably many charts.

So now, let’s look at some examples:

Example 1: $S^2$ defined by $x_1^2 + x_2^2 + x_3^2 = 1$ in $\mathbb{R}^3$ .

By stereographic projection, we see that this sphere can be covered with an atlas of 2 charts. Arnold also pointed out that a family of tangent circles on the sphere passing through the North Pole N gets mapped to a family of tangent circles on one chart and a family of parallel lines on the other.

The map for chart 1 is given by:

$\phi_1: (x_1, x_2, x_3) \rightarrow (\frac{x_1}{x_3-1}, \frac{x_2}{x_3-1})$

and its inverse is given by:

$\phi_1^{-1}: (X, Y) \rightarrow (\frac{2X}{1+X^2+Y^2}, \frac{2Y}{1+X^2+Y^2}, \frac{-1+X^2+Y^2}{1+X^2+Y^2})$ .

The map for chart 2 is given by:

$\phi_2: (x_1, x_2, x_3) \rightarrow (\frac{x_1}{1+x_3}, \frac{x_2}{1+x_3})$

and the inverse is given by:

$\phi_2^{-1}: (X, Y) \rightarrow (\frac{2X}{1+X^2+Y^2}, \frac{2Y}{1+X^2+Y^2}, \frac{1-(X^2+Y^2)}{1+X^2+Y^2})$ .

The transition map from chart 1 to 2 is given by:

$\phi_2(\phi_1^{-1}): (X, Y) \rightarrow (\frac{X}{X^2 + Y^2}, \frac{Y}{X^2 + Y^2})$ , which is a diffeomorphism from $\mathbb{R}^2 \backslash \{(0, 0)\}$ to itself. Hence, our 2 charts are consistent.

Example 2: a torus in $\mathbb{R}^3$

Without loss of generality, we fix an origin on the torus and set up angular coordinates. We can consider the atlas consisting of 4 charts (each chart’s mapping is the identity map) whose domains on the torus are (Note: each point on the torus can now be uniquely determined, relative to our ‘origin’, using the two angular components $\phi, \psi$ ):

$0 < \phi < 2\pi, 0 < \psi < 2\pi$ (both inequalities MUST be strict since we want the image of the intersections to be open sets)

$0 < \phi < 2\pi, - \pi < \psi < \pi$

$- \pi < \phi < \pi, 0 < \psi < 2\pi$

$- \pi < \phi < \pi, -\pi < \psi < \pi$ .

The figure above shows the 4 images of our domain on $\mathbb{R}^2$ with $\phi, \psi$ as our coordinates.

Example 3: $\mathbb{RP}^2$

I have attached screenshots of my notes showing the explicit formulas for the mappings corresponding to the 3 charts (each of whose domain is a copy of the affine plane, $\mathbb{R}^2$ ).

Since both $y_1 \neq 0, z_1 \neq 0$ , it follows that the two maps are diffeomorphisms.

An interesting picture from Arnold’s ODE shows how the diffeomorphism $\phi_{0,1}$ maps plane (Arnold used the tiles to better show where each point gets mapped to):

We start with a ‘square’ with the section of the $y_1$ -axis removed. Under this map, the points close to $y_1$ axis gets mapped to points far away from the origin. And those that are originally far from the origin gets mapped to points close to the origin (this is due to the fact that we have $\frac{1}{y_1}$ in both components).

Differential Equations on Manifolds (12/09/2025)

Share this:

Leave a Reply Cancel reply