Does non-Euclidean geometry can be always immersed in Euclidean of dimension D+1?
This is probably very basic question, but I'm just trying to understand why do you need to consider sometimes very complicated non-euclidean geometries, as for example surface of the Earth (2D), while you can look at the picture from simpler point of view where Earth is just immersed in the 3D Euclidean space.
Although frequently the inspiration behind non-euclidean geometries, as you've noticed, comes from objects in euclidean geometry, it is far from true that all geometries can be embedded in euclidean space. Take, for example, the Fano plane, which is a projective geometry that has only 7 points and seven lines, where each lines contains three points and each point is contained withing three lines.