I plan to study general relativity in the next few months and for now I keep gathering information about it. I read here and there that Penrose and Hawking proved that general relativity as we know it entails the existence of a singularity inside a black hole. I would like to know if this follows from an assumption of a Riemaniann structure where spacetime may lose it. Laurent Nottale in his books explains quantum effects by lack of differentiability of spacetime, so are there models in the literature where the supposed singularity is described as a "smooth fractal" (loose translation of French "fractale lisse")? Like in the visualization of Nash isometric embedding theorem, that is a $C^{1}$ (and not $C^2$) manifold with infinitely many corrugations?

I plan to study general relativity in the next few months and for now I keep gathering information about it. I read here and there that Penrose and Hawking proved that general relativity as we know it entails the existence of a singularity inside a black hole. I would like to know if this follows from an assumption of a Riemaniann structure where spacetime may lose it. Laurent Nottale in his books explains quantum effects by lack of differentiability of spacetime, so are there models in the literature where the supposed singularity is described as a "smooth fractal" (loose translation of French "fractale lisse")? Like in the visualization of Nash isometric embedding theorem, that is a $C^{1}$ (and not $C^2$) manifold with infinitely many corrugations (see http://hevea-project.fr/ENPageToreDossierDePresse.html)?