By the way, do you know if it is possible to get a quantity which can in certain case obtained only by describing a surface extrinsically : z=f(x,y), instead of intrinsically : metric g(x,y) (does not need the embedding space with z)....or the opposite way. The same question tackled me about hyperbolical Minkowskian space : how is it possible to make a negative metric coefficient, with an extrinsic description of the space-time, because the diagonal metric coefficient are defined as gn=<xn|xn> which can be only negative is the scalar product in the embedding space is itself not positive definite, hence non-euclidean...
Second question : let give a curve (1-dimensional variety). If it is described intrinsically, how do you know you have followed a knotted path or not. Indeed, if the extrinsic description of the curve is given, it is clear that, embedding a curve in a 2D space does not allow knots (because you have to cross the curve itself, which is but allowed in 3 (and upper dimensions)