How to identify an Euclidean space?

Question

How do I determine that a certain space is Euclidean space or not? Are Spherical and Cylindrical Coordinates Euclidean too?
This question may be be elementary but I need to understand this.

Answer 1

The curvature of a space (or spacetime) is described by the Riemann curvature tensor. For a Euclidean space this tensor will be zero regardless of the coordinates you choose.

For a Euclidean space the distance between two points is calculated using the metric, and in Cartesian coordinates this is simply:

$$ ds^2 = dx^2 + dy^2 + dz^2 $$

which we all remember as Pythagoras' theorem (in three dimensions). If you choose polar coordinates instead the metric becomes:

$$ ds^2 = dr^2 + r^2d\theta^2 + r^2\sin^2\theta d\phi^2 $$

And it isn't obvious (at least not to me) that this second form of the metric describes a flat space. However if you calculate the Riemann tensor from the metric then in both cases you will find the result is zero.

Answer 2

The curvature of a space (or spacetime), which in particular may be expressed through (non-zero) values of certain curvature invariants, is derived from distances (or at least: distance ratios) between sufficiently many elements of the space under consideration; or suitably generalized distances (such as values of spacetime intervals $s^2$) between events in spacetime. This involves the comparison of triangles (in terms of ratios of the lengths of their sides, or their corresponding heights); or more generally, the evaluation of the values $\kappa$ for which certain Gram determinants, involving distances are vanishing:

$$ 0 = \begin{vmatrix} 1 & \text{Cos}[ \, \sqrt{\kappa} \, AB \, ] & \text{Cos}[ \, \sqrt{\kappa} \, AC \, ] & \dots \cr \text{Cos}[ \, \sqrt{\kappa} \, BA \, ] & 1 & \text{Cos}[ \, \sqrt{\kappa} \, BC \, ] & \dots \cr \text{Cos}[ \, \sqrt{\kappa} \, CA \, ] & \text{Cos}[ \, \sqrt{\kappa} \, CB \, ] & 1 & \dots \cr \vdots & \vdots & \vdots & \ddots \end{vmatrix}.$$

For a flat space (which, for historical reasons is a.k.a. Euclidean space) or flat spacetime (Minkowski space) the vanishing of the above Gram determinates in the limit of vanishing curvature (expressed as vanishing value $\kappa$) simplify to the vanishing of Cayley-Menger determinants in terms of the relevant distances:

$$0 = \begin{vmatrix} 0 & (AB)^2 & (AC)^2 & \dots & 1 \cr (BA)^2 & 0 & (BC)^2 & \dots & 1 \cr (CA)^2 & (CB)^2 & 0 & \dots & 1 \cr \vdots & \vdots & \vdots & \ddots & 1 \cr 1 & 1 & 1 & \dots & 0 \end{vmatrix};$$

cmp. MTW, Box 13.1: "Metric distilled from distances".

These relations are obviously and manifestly independent of whether and which coordinates might be assigned to the elements of the space (or spacetime, resp.).

(If coordinates are assigned to the elements, i.e. points, or events, of flat space or of flat spacetime, resp., then the above general coordinate-free relations induce certain more or less simple formulas for the expression of distance values; such as the Pythagorean formula (of Euclidean distance), for Cartesian coordinates assigned to flat space, or the Minkowski product for Minkowski coordinates assigned to flat spacetime.)