What does tell me this manifold is a circle?

Question

I was thinking about this: suppose we want to define an atlas on, for example, a circle $S^1$ to stay easy. Let's take the atlas made by the four charts

$$(\{x>0\}, x);\ (\{x<0\}, x);\ (\{y>0\}, y);\ (\{y<0\}, y)$$

In few words: the left, right, north and south arcs with their respective projections on the axis as local coordinates.

When I take a point on $S^1$ and I calculate the tangent space, since it's the vector space of the derivatives on the point and it has, as a basis, the derivatives with respect upon the local coordinates, if I take, for example, the right arc the coordinate function of which is $x$, a vector over the tangent space has an expression like

$$b\frac{\partial}{\partial x}$$

and $b$ runs all over $\mathbb{R}$.

Now the question: where is the information which tells me the manifold is actually a circle and not, say, an ellipse or something else?

Where is in the usual sense, the slope of the straight line (that is the tangent space)?

Is it seen only from the transition maps between different charts, or even from here?

Answer 1

When working with manifolds it is import to keep in mind the different layers of structure that you can put onto it.

The basic setup is a topological space $X$ which has the property of being a manifold (i.e. second countable, Hausdorff and locally euclidean) to which you can add more and more structure. Examples of such layers are:

• A differentiable structure $\mathcal{A}$.
• A Riemannian metric $g$.
• An embedding $ f \colon X\rightarrow \mathbb{R}^m$.

With each structure comes a set of properties that you can ask your manifold to have, e.g.

• Given $\mathcal{A}$, is the manifold parallelizable? Does it admit a complex structure? Is it a Lie-group?
• Given $g$, what intrinsic curvature does it have, what is its volume?
• Given $f$, what is the extrinsic curvature, is it a minimal submanifold, how do the tangent spaces lie in the tangent spaces of $\mathbb{R}^m$?

The questions you are asking, namely what the slope of the tangent lines or whether it is a circle or an ellipse are all properties that you can only talk about once you have embedded the manifold into euclidean space. Only specifying an atlas as you did is simply not enough structure to talk about these things.

Answer 2

Now the question: where is the information which tells me the manifold is actually a circle and not, say, an ellipse or something else?

There is no such information. Indeed, a circle and an ellipse are "the same" as smooth manifolds (that is, they are diffeomorphic).

A smooth manifold structure is much more like a topology than a rigid geometric structure. It tells you what it means for a map on your space to be differentiable (like how a topology tells you what it means for a map to be continuous), but not really anything else (it doesn't even tell you what the derivative of a map is numerically, just whether it is differentiable). It doesn't tell you how to measure distances or angles or curvature or anything like that. To get such "geometric" data, you need more structure than just a smooth manifold structure, such as a Riemannian metric.