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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.

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    By the Whitney Embedding Theorems, any smooth $n$ dimensional manifold can be embedded in Euclidean space of dimension at most $2n$. So provided the space you are considering is a smooth manifold, yes. There are geometries and spaces that are not. – Simon S Apr 18 '15 at 14:50
  • You may find you encourage more helpful answers if you are more specific about what sorts of non-Euclidean geometries you are considering. If you just mean those with constant curvature, the answer is yes, as spheres ($K>0$), planes ($K=0$), and one sheet of the hyperboloid of two sheets ($K<0$). On the other hand, if you count the flat torus as non-Euclidean, the lowest-dimensional curvature-preserving embedding is in $\mathbb{R}^4$ as $(R\sin{u},R\cos{u},r\sin{v},r\cos{v})$ or similar. – Chappers Apr 18 '15 at 15:11

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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.

Archaick
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