Definition of a Cartesian coordinate system

Solution 1:

In the mathematical literature, the term "Cartesian coordinates" is used most frequently to refer simply to the standard coordinate functions on $\mathbb R^n$, namely the functions $x^1,\dots,x^n\colon \mathbb R^n\to \mathbb R$ defined by $x^i(a^1,\dots,a^n) = a^i$. Somewhat less frequently, I've also seen the term used to refer to any coordinate system on $\mathbb R^n$ obtained by composing the standard coordinates with a rigid motion, which can also be characterized as those coordinates for which the standard coordinate vectors $\partial/\partial x^1,\dots,\partial/\partial x^n$ are orthonormal.

The point is that it only makes sense to talk about "Cartesian coordinates" on $\mathbb R^n$ itself, or on an open subset of $\mathbb R^n$. On an arbitrary smooth manifold, the term has no meaning. Of course, on any smooth manifold $M$, each point has a neighborhood $U$ on which we can find a smooth coordinate chart, and such a chart allows us to identify each point $p\in U$ with its coordinate values $(x^1(p),\dots,x^n(p))\in\mathbb R^n$, and thus to temporarily identify $U$ with an open subset of $\mathbb R^n$; but we would not call these coordinates "Cartesian coordinates on $M$."

If your manifold $M$ is endowed with a Riemannian metric $g$, then there is more that can be said. For example, one could ask whether it's possible to find a coordinate chart in which the given Riemannian metric has the same coordinate expression as the Euclidean metric: $g= (dx^1)^2 + \dots + (dx^n)^2$. If this is the case, then geodesics and distances within this coordinate neighborhood are given by the same formulas as they are in Euclidean space; but that might not hold true elsewhere on the manifold. I think this might be the question you're getting at in your last paragraph, although I would not call these "Cartesian coordinates" because they don't have an open subset of $\mathbb R^n$ as their domain. Off the top of my head, I don't know of any standard nomenclature for such coordinates, but it wouldn't be inconsistent to call them "Euclidean coordinates" or "flat coordinates."

It's a basic theorem of Riemannian geometry that it is impossible to find such coordinates unless the curvature tensor of the Riemannian metric is identically zero on the open subset $U$. You'll find a proof of this fact in virtually any book on Riemannian geometry, such as my Riemannian Manifolds: An Introduction to Curvature (Theorem 7.3). If you want a treatment that doesn't use so much of the machinery of Riemannian manifolds, my Introduction to Smooth Manifolds has a proof that it's impossible to find Euclidean coordinates for the ordinary $2$-sphere in $\mathbb R^3$ (Proposition 13.19 and Corollary 13.20).