Recently I decided to go back to the absolute basics and answer "why do we need differential geometry"? This is what I came up with:
One of the properties of smooth $n$-manifold $M$ is:
- $M$ is locally Euclidean of dimension n.
Now, let $(M,g)$ be $n$-dimensional Riemannan manifold. The previous property can be formulate in following way:
$\forall \ x\in U\subset M,\quad \exists \ \text{ a homeomorphism }\phi:U\rightarrow\tilde{U}\subset(\mathbb{R}^n,\delta) $,
such that $\tilde{x}=\phi(x)$ is a local coordinate map (for $x\in M$ and $\tilde{x}\in \mathbb{R^n}$).
However, if we want to $\phi$ be a (local) diffeomorphism, the curvature tensor must vanish.
Is that it? That is basically why we need a "new" differential operators (e.g. covariant derivative, etc.)? Because (by definition) we have "only" homeomorphism rather than diffeomorphism?