What I mean is, if you have a metric of a torus $T_2$, and you want to distinguish between say very thin stringy tori and very thick tori with the same surface area, is there a nice formula for this such as an integral over certain curvature tensors?
For example a very thin torus could be very stringy spread out all over the place but have the same surface area of a fat torus.
Also I wonder if the same invariant could distinguish between a smooth torus and one with lots of 'spikes'.
I'm thinking the stringy and spiky tori would have lots of places where the curvature is very high. So my guess would be if you integrated over the sum of the scalar curvature squared? Would this work?
smoothness $ \propto 1/\int R^2 dx^2 $ ???
But I know in 2 dimensions R can be written in terms of g.
My second question is more physics based and that is, in M-Theory do we just care about the surface area of a membrane world-sheet when computing its amplitude or are other quantities considered, such as the "smoothness" as talked about above?
For example in analogy to string theory they say the energy of a membrane is proportional to its surface area. But is this just a guess? Why can't it also rely on other things like it's smoothness? (Which has no counterpart in string theory without branes).
Also, I was thinking if the amplitude is higher for smoother membranes it would mean the amplitude would be higher for a single sphere rather than two smaller spheres with the same total surface area. Could this explain why the Universe is not split in two?
Is this related at all to the physics of soap bubbles?