Given a metric of a torus can we measure its thickness?

Question

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?

Answer 1

The problem is that the thickness of the torus is related to its external curvature, whereas a metric only gives information about its intrinsic curvature.

Think of a sheet of paper: when it is flat, its metric is clearly zero. Now warp it into a cylinder. This in fact does not change its metric, indeed any transformation that does not crease the paper cannot change its intrinsic metric (practical formulation: an ant crawling on the curved sheet of paper cannot determine its curvature, e.g. if the ant developed geometry it would find the Euclidean axioms are a good description of its universe). The only curvature that changes is the extrinsic one, which is the one determining the thickness or circumference of our cylinder.

Note: the ant cannot tell the extrinsic curvature from local measurements, however the global topology does change. So by walking all away across the cylinder it might find itself back to its starting point. If it measures the minimal length (using its local metric) that it needs to walk to loop back, it can in fact deduce the circumference of the cylinder. It would work similarly for your torus. There however cannot be a simple formulation in terms of the local metric (as I tried to explain above), but so local metric + topology can give you some information :)

Answer 2

Please asks just one question. For the first one:

The thickness of a 2 torus does not enter in. It is simply a 2D manifold, it has length and diameter, and only the outside surface matters. At least classically. And by the way a 2 torus is a flat space. It is one of a number of possible compact flat spaces in 2D.

See https://en.m.wikipedia.org/wiki/Flat_manifold

Now, if you want to consider it embedded as a solid but semi-thin torus in a 3D manifold, again classically, then thickness counts. Perhaps up I want to calculate things that way and if you think a 2D limit as thickness goes to zero might answer something do it that way. You could try doing some small perturbation on thickness and calculate.

Maybe it is different in string theory, or More likely branes. You need to be more specific.