cianfa72 said:
Which is the natural embedding of the cube in ##\mathbb R^3##, how is it defined?
You get the same situation with the absolute value function ##M=\{(x,|x|)\,|\,-1<x<1\} \subseteq \mathbb{R}^2.## The cube has only a lot more cases to define it.
##M\subseteq \mathbb{R}^2## is a (continuous) embedded manifold, ##D=\{(x, 1-\sqrt{1-x^2})\,|\,-1<x<1\}\subseteq \mathbb{R}^2## is a smooth embedded manifold. The point is that going from one chart to the manifold and back from the manifold on another chart where they overlap has to be a smooth real function. You can do this with the circle ##D## at ##(0,0)## but not with the lines ##M## at ##(0,0)## for two different charts containing the origin. In this case, you cannot even define differentiability on one chart containing the origin. Smooth in this context actually means a ##C^\infty## compatible atlas. ##D## has tangents everywhere, ##M## has not. The definition of tangents needs either an embedding that provides natural charts, or it requires a ##C^1## compatible atlas. In this case, ##D## has even a ##C^\infty ## compatible atlas and is thus a smooth manifold.
Instead of speaking of a smooth chart (which only makes sense for smooth chart mappings on their overlapping area), it would be better to speak of smooth manifolds, or even better better, to speak of a ##C^\infty ## compatible atlas.
The basic idea behind manifolds is simple: We open the atlas, choose a chart, do whatever we want to do in that real or complex Euclidean space the chart represents, take our result, and go back to the manifold. What we do on the chart is a calculation in local coordinates. If what we wanted to do is infinitely often differentiating and that is compatible with the overlappings of our atlas, then we call the manifold smooth. Minkowski spaces make no difference. All they require is to deal with an additional sign.
It is not by chance that the wording is the same as having a street map on your lap navigating in an unfamiliar region.