Differentiable manifolds are a generalization of the local geometry of Euclidean space. In fact, every differentiable manifold of dimension $m$ is locally diffeomorphic to the Euclidean space of the same dimension.
On the other hand, in general, curves and surfaces (and other fractal objects of higher dimension) that can be embedded in a Euclidean space in general, do not admit a simple differentiable structure, so they cannot be treated as differentiable manifolds.
I wonder if anyone has defined a kind of "fractal variety/manifold" that considers fractal geometries that are not explicitly constructions within a Euclidean space and in some sense can be analyzed intrinsically as are differentiable manifolds.
