I am studying manifold theory ( smooth manifold). As I study the subject, I feel more and more confused. I am not yet at chapter dealing with deferential forms and integration. I can understand that we might have weird spaces out there where we need to see how our usual calculus carries on to them, but let me restrict my question to two manifolds, the sphere, and the tours.
If the sphere, say $S^2$, is a subset of $R^3$, why do not we just use our integration methods on R^3, restricted to this sphere. Why should we define all these stuff?. What would fails on $S^2$ ?.
Similar case, I have a question and not sure if my question is even valid, How can I see the difference between $R^n$ as a flat Euclidean space and S^n as curved space. How is the underlying geometry different.
continuing the last question, what is the deep difference between Euclidean geometry and Euclidean geometry.
Thanks in Advance.
Firstly one would want to develop the machinery for a general manifold so that one can see what is essential and not dependent on being just R^n. Secondly the properties of the metric emerge when studying a general situation. The most important non-Euclidean example is of course in General Relativity, where the metric changes from +++ in R^3 to -+++ in R^4.
Curvature is defined in general using the Riemann curvature tensor. Of course the constant non-zero curvature of S^2 is different from that of R^3, where it is 0.