TITLE: Global momentum of a closed universe
QUESTION [1 upvotes]: In a universe:

with 2 spatial dimensions
closed (that loop on itself)
with local positive and negative curvature
with no attraction and no repulsion
with particles randomly created by pair in the space, with opposite momentum
where this particles can collide

Can I say that the sum of all particles momentum is zero?

REPLY [0 votes]: I could be wrong but my answer is that there is no such spacetime possible that satisfies the Einstein Field Equations. Here is why. 
See Wikipedia at  https://en.m.wikipedia.org/wiki/(2%2B1)-dimensional_topological_gravity
Empty spacetime in 2+1 dimensions, with or without a cosmological constant, can only be one of three types: flat (i.e. Minkowski) if $\Lambda = 0$, or deStitter or anti deSitter, depending on the cosmological constant positive or negative. It has no gravitational propagating degrees of freedom, and is all defined by its topological structure, of which there are many varieties. It is closely related to Chern Simons theory, and a quantum theory for it can be solved. I am not a Chern Simons expert.
From the point of view of your question, it does not satisfy positive and negative curvature, it is either 0 for flat or positive or negative, constant in each case across the spacetime. 
I do not know if you can have patches one way and then another way, from what I've seen they have the same curvatures. But I don't know for sure. 
If flat it's got a lot of Killing vectors, i.e., symmetries, which are global. One is 2D spatial homogeneity, so momentum can be defined as constant in each of the Killing, i.e. symmetry, directions. 
Since closed I believe they have to be either flat (not simply connected), or deSitter. If flat momentum is conserved, as stated above. If deSitter it is actually maximally symmetric, and that includes spatially symmetric, so again you have conserved momentum. As with flat spacetime I am not sure of all the topological versions. 
So, in this spacetimes yes momentum would be conserved. 
But they would not satisfy: positive and negative curvature (in the same spacetime), unless you can join different curvature patches which I don't know if possible.
If you know the topological options maybe they can be constructed to satisfy your conditions. 
Since I am not sure I am not missing something in the topological options, I could just be off. Either way, these are the options I can think of, and somebody who knows more may correct me. Also either way, it is a very interesting question. In 4D since you do have dynamic gravitational degrees of freedom you could have all kinds of gravitational waves on tHe spacetime. 
BTW, I do understand that if you have matter or radiation in the 3D case, it's much more unrestricted.
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