curved space vs linear space with curved basis

Question

What is actually the difference between a curved space and an euclidean space represented in curvilinear basis?

Answer 1

I guess what you have in mind is something what mathematicians have in mind when they are talking about Riemannian, Lorentzian or pseudo-Riemannian manifolds.

In all these three cases you are looking at spaces which are locally (in the sense of differential topology, which amounts to saying indistinguishable up to diffeomorphism) looking like Euclidean space (these objects are called differentiable manifolds). You can describe any of them up to diffeomorphism and will always get the same object with a different (local) description. This is what a coordinate change amounts to.

By looking at that kind of object you are already ignoring the vector space structure to some extent. To deal with this you think of this kind of spaces as having a vector space attached to them at each point (think of a two dimensional surface in $\mathbb{R}^3$ with a tangent plane attached in each point. This is almost always a good picture to start with). That vector space can be thought of as a local linear approximation of the manifold.

To get back to your question, you now start to introduce an additional structure, called a metric (Riemannian, Lorentzian, pseudo-Riemannian). There are two aspects to this: first, it is defined as a bilinear form $g$ on each tangent space (positive definite in the Riemannian, with signature $1$ in the Lorentzian case and with general signature in the pseudo Riemannian case, which, in principle, covers the other two). This allows you define a length of tangent vectors $v$ in each point $p$, as the square root of $g_p(v,v)$ (at least in case where this is a positive expression).

With a length attached to tangent vectors you can define the length of curves, by integrating the lengths of their tangent vectors along the curves. Again this is easy in the positive definite case and needs additional definitions and thought in the general case.

Once you have a defintion of length of curves you can define the distance between points (in the positive definite case: the $\inf$ of lenghts joining two given points). This is the second aspect.

Now with a definition of distance between points you can try to compare your new object with Euclidean space (or Lorentz space in the case of metrics with signature $-1$)

An example is, again, a two manifold in three space. The scalar product on the tangent space is simply inherited from the ambient space.

As you can visualize if you compare a two dimensional sphere in three space, say, with it's tangent plane Euclidean three space, you will easily see that they are not isometric, but that the lengths on the sphere and the tangent plane are comparable in radial direction (with respect to the point where you have the tangent plane is attached) but differ along distance circles. Also the usual Euclidean rules for triangles (like angles sum up to 180 degree) are no longer true on the sphere. It's basically this difference in local distance which is measured by what is called curvature. It's easy to visualize for two surfaces in three spaces (they bend), but rather hard in general.

So (assuming I got your question correctly) the difference between a curved space and Euclidean space represented in curvilinear coordinates is that you look at objects with an additional structure (a metric, which induces a definition of length) which is not necessarily preserved under diffeomorphisms (and consequently not under change of coordinates).

(I hope this isn't more confusing than helpful ;-)