# Differential geometry in hyperspace

After many years away from maths, I have recently started to dive into a subject I only ever learned the rudiments of: differential geometry, particularly helped by a brilliant series of lectures on youtube (Lectures on Geometrical Anatomy of Theoretical Physics if you wish to check it out). This has set me thinking, though; a manifold is of course a topological space, and I wonder if there has been any work on exploring the possibility of somehow "lifting" at least parts of the differentiable structure into in a hyperspace over the manifold. I haven't been able to find anything, so far - but then, what would one search for?

• I have read the definition of hyperspace in topology. I was not familiar with it. However, I have this remark. Before talking about a differentiable structure, the topological space should first be a topological manifold, so that it must be second countable, Hausdorff and locally Euclidean. I have a feeling that hyperspaces over a manifold do not satisfy these conditions in general. I hope this helps. May 18, 2017 at 8:12
• Good points - so the question is under which conditions would that be? Another, potentially more interesting question, I think, is whether any of these could be relaxed somewhat, without making the construction meaningless or unworkable. One of the remarkable things about quantum mechanics is that there seems to be a certain degree of non-locality involved (like 'action at a distance') - could that be modeled in terms of the separation axions? May 19, 2017 at 9:33
• Regarding modelling non-locality in QM, it is perhaps better to create another post in the Physics section (expect to get a variety of different answers). May 19, 2017 at 10:38
• Oh, I don't know about talking to physicists; their main focus tends to be on the physics, and what I am interested in at the moment, is more the mathematical framework. I have felt quite frustrated over the years when trying to move beyond "QM as dogma" in discussions with physicists, but I feel strongly that it must be possible to device a mathematical framework from which the peculiarities of QM (and GR) can be derived rather than assumed a priory. May 23, 2017 at 10:37
• There is a number of mathematical formulations of QM, which are by now "classical". However, from what I understand, you are interested in understanding non-locality (or rather "deriving" it mathematically) within a Mathematical framework. I am not an expert on that, but since the EPR paper came out, there has been tons of papers written on the subject; in particular, the Bell inequalities are related to such discussions. Have some discussion with a Mathematically inclined Physicist. I am curious as to what they might say (and then maybe post it here, with their permission of course). May 24, 2017 at 8:45