I've been studying differential geometry for a little while now, but I've never properly justified to myself rigorously the need to consider other more general coordinate maps, other than Cartesian on a manifold with a metric defined on it (or indeed in general, without a metric). Am I correct in saying that the Cartesian coordinate system is a special kind of mapping which directly relates the intrinsic distance between two points on a manifold to the 'numerical' distance between their coordinates in $\mathbb{R}^{n}$. As, in general, a coordinate patch on a manifold will have a non-Euclidean geometry, although it will be possible to construct a one-to-one mapping such that these points can be labeled by coordinates in $\mathbb{R}^{n}$, it will not be possible to construct a map that preserves the intrinsic distance between two points in this patch such that it corresponds to the 'coordinate distance' between their corresponding coordinates in $\mathbb{R}^{n}$. In other words, although we will be able to construct a coordinate map, it will be impossible to construct a Cartesian coordinate map for this patch, which requires a Euclidean metric (apart from within a small neighbourhood around each point in this patch)?

N.B. By "Cartesian coordinate system" I mean the usual coordinate system used for $\mathbb{R}^{n}$ in which the geometry is flat (i.e. has a Euclidean metric)

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    $\begingroup$ What do you mean by "Cartesian coordinate system" on a differential manifold? $\endgroup$ – A.P. Jun 19 '15 at 18:39
  • $\begingroup$ @A.P. I mean the standard Cartesian coordinate map that one usually uses to map Euclidean space (with the Euclidean metric) to $\mathbb{R}^{n}$. This can be used on more general manifolds right? $\endgroup$ – Will Jun 19 '15 at 18:51
  • $\begingroup$ Not really. A coordinate system, or atlas, on a (real) manifold $M$ is a set $\{(U_{\alpha},\varphi_{\alpha})\}$ where $\{U_{\alpha}\}$ is an open covering of $M$ and $\varphi_{\alpha}: U_{\alpha} \to V_{\alpha}$ are homeomorphisms, with $V_{\alpha}$ an open set of $\Bbb{R}^n$. The definition of differential manifold has just a couple of extra conditions. Then you can use $\varphi_{\alpha}^{-1}$ to get local coordinates $x_1,\dotsc,x_n$ such that $\varphi_{\alpha}^{-1}(x_1,\dotsc,x_n) = m \in U_{\alpha}$, but that's it. $\endgroup$ – A.P. Jun 19 '15 at 19:13
  • $\begingroup$ @A.P. I was trying to motivate when one has a notion of distance (i.e. a metric) defined on a given manifold why one would ever choose to use coordinate systems other than Cartesian ones. I've read that in general it is not possible to construct such a map (i.e. a Cartesian one) on a curved manifold as the manifold will be non-Euclidean in any given coordinate patch. I was trying to heuristically motivate the idea from a physicists background, approaching differential geometry for use in general relativity?! $\endgroup$ – Will Jun 19 '15 at 19:23
  • $\begingroup$ You should clarify what you mean by "Cartesian coordinate system". If you are talking about the usual coordinate system on $\Bbb{R}^n$, then the only meaningful way to define a similar thing on a manifold is the one I just described. The only analogy I can give you is this one: in $\Bbb{R}^n$ every point is the intersection of $n$ orthogonal lines, and the Cartesian system associates a tuple of real numbers to each point by fixing a reference point and defining a length function on each of those lines. $\endgroup$ – A.P. Jun 19 '15 at 20:04

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