We very well know how to calculate curvature in gravitation. But this time i just need a physical explanation to this question on my mind: "In order to describe n dimensional space with constant curvature why do we need to go to n+1 dimensional flat space? Why don't we use just n-dimensional spherical coordinates instead?" I know this is about the curvatures of hypersurfaces and every hypersurface is an n-dimensional manifold embedded in an n+1 dimensional space. So there may be some mathematical difficulties in calculation. But isn't there any other methods to describe n dimensional space with constant curvature?