What is the best (in terms of effectively building understanding) direction from which to approach manifolds?

Question

Thedore Frankel's book The Geometry of Physics presents Manifolds right away in Chapter 1 in the following manner:

1. Introduce the Euclidean space $\mathbb{R}^N$ only as "the most important manifold".
2. A subset $M = M^n \subset \mathbb{R}^{n+r}$ is said to be an $n$-dimensional submanifold of $\mathbb{R}^{n+r}$ , if locally $M$ can be described by giving $r$ of the coordinates differentiably in terms of the $n$ remaining ones. This means that given $p \in M$, a neighborhood of $p$ on $M$can be described in some coordinate system $(x, y) = (x^1, . . . , x^n , y^1 , . . . , y^r )$ of $\mathbb{R}^{n+r}$ by $r$ differentiable functions $y^{\alpha} = f^{\alpha}(x^1,...,x^n)$, where $\alpha = 1, . .. r$

However, other books generally present manifolds much later, and in terms of the vocabulary of topology.

I find that Frankel's presentation doesn't provide me with any useful intuition, and it has a lot of rigorous loopholes (e.g. what is "differentiable"? differentiable where?, etc.).

Could you help either make his presentation more precise, or would you suggest that I take the route of other textbooks, and take the "long way around" to manifolds after doing a fair bit of topology/differential geometry before tackling manifolds?

Answer 1

This is a follow up on my comment above. This is by no means a complete answer and it should not really be treated as one.

The best way to gain intuition is to look at examples.

Let's look at the sphere, $S^2 = \{ (a,b,c) \in \mathbb{R}^3 : a^2 + b^2 + c^2 = 1\}$.

In this case, $M = S^2$. $n = 2$, and $S^2 \subset \mathbb{R}^{2+1}$.

Locally, $M$ can be described as a graph of a function. For instance, consider the subset of $M$ where $c>0$. Here, we can write $c = \sqrt{1-a^2 - b^2}$. Using the notation of Frankel's book:

$(x,y) = (x^1, x^2, y^1) = (a,b,c)$, with $c = y^1 = f^1(x^1,x^2) = f^1(a,b) = \sqrt{1-a^2 - b^2}$

Note that $f^1(a,b) = \sqrt{1-a^2-b^2}$ is a differentiable function on an open subset of $R^2$. (Namely, the set $\{(a,b)\in R^2: a^2+b^2 < 1\}$, which is precisely the set that maps into $c>0$ on $M$ under this map.)

The differentiability here is in the usual sense of being differentiable as a function of two variables.

What I have described here is one of the coordinate charts on $M$. See if you can find the rest of the coordinate chart. Note that it is very important that the patches cover all of $M$. (My chart only covers the points on $M$ with $c>0$. But what about the other points?)

EDIT: I just realized that this example is the second example in the said book, on pgs. 4-5.