How could singularities on two different manifolds be treated as connected, if it makes sense to do so? Could such a treatment describe a possibility that two manifolds of different dimensions and perhaps different properties could be connected through a common singularity? More specifically perhaps, could two singularities on the same manifold be connected in such a manner? By describing geometric flows on those manifolds, is it interesting to study the propagation of singularities and perhaps two singularities coalescing?
Are there manifolds (perhaps among infinite-dimensional varieties), or some topological spaces equipped with really simple geometry, that are a surface on their own, kind of like a ball without the interior, if that makes any sense?
What kind of ideas describe "interactions" (a bit sloppy word) between manifolds?
A differentiable k-manifold is defined with a family of injective, k-times continuously differentiable maps.
https://en.wikipedia.org/wiki/Differentiable_manifold
This perhaps goes against the definition, but if one of the maps to be included were k-1 times continuously differentiable but not k times how would one deal with it, with a constraint that the k-1 times continuously differentiable map is replaced by a k-times continuously differentiable map by a lifting, equivalence relation or "some way to get around it"? What are some "ways to get around", and do they help in addressing singularities of some sort? Are such ideas sensible in algebraic geometry or elsewhere?
Like the existence of the Riemannian metric on a smooth manifold, what are other existence theorems for metrics on k-manifolds?
Existence of Riemannian Metric
My motivation comes from the Einstein Rosen bridge
https://pdfs.semanticscholar.org/17bb/a85d420d075700cf3e7fef6d9934f5aa9ec3.pdf
and from the ER = EPR conjecture
https://en.wikipedia.org/wiki/ER%3DEPR
in trying to understand "how can two parts of spacetime or two different spacetimes be connected or interact via (apparent?) non-locality? and trying to understand how one could describe two different spacetimes possibly interacting or not being able to interact", which is probably better asked in physics stack exchange. And so, I want to explore the mathematical structure that could underly such a description which leads to ideas of how manifolds (those equipped with geometric flows among other properties could be interesting, those with some sorts of associated singularity) can interact?