Is there any "realistic" metric of type $\mathrm{(A)dS}_2 \times \mathcal{T}_2$ in General Relativity? After studying the Bertotti-Robinson metric, which describes a $\mathrm{AdS}_2 \times \mathcal{S}_2$ universe, I was wondering about other kind of closed topologies with holes, like $\mathrm{(A)dS}_2 \times \mathcal{T}_2$, where $\mathrm{(A)dS_2}$ is a deSitter (or Anti-deSitter) spacetime in 2 dimensions, while $\mathcal{T}_2$ is the curved torus.
Are there any known metrics of that type, with realistic matter distribution (i.e. no exotic matter)?


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*Of course, any Lorentzian metric is a solution to Einstein's equation, but it usually involve some exotic or unrealistic matter.  I'm not interested in that kind of matter!

 A: Quite likely, there are no such solutions, if by “realistic” matter we assume matter satisfying various energy conditions. However, if we include negative cosmological constant under the “reasonable” matter, such solutions could be found for a variety of additional matter content, with Maxwell field being the most obvious. 
Solutions of Einstein equations with the $AdS_2$ factor often emerge as a near horizon geometry of extremal black holes (with degenerate horizons). See e.g. here on the universality of $AdS_2$ factor. Consequently, the existence or non-existence of solutions is guaranteed by appropriate theorems of black hole existence/uniqueness/no-hair  for a given matter content. 
Spacetimes with topology $AdS_2 \times \mathcal{T}_2$ must be the near horizon limit of extremal topological black holes. If we assume the existence of at least one Killing vector on the torus, then the theorem prohibiting the existence of black holes in 2+1 dimensions with the matter satisfying dominant energy condition would apply to this situation and there would be no solutions. 
If there is a negative cosmological constant, then solutions do exist. For example taking the near horizon limit of the metric from this paper (eq. (2)) would produce such solution with negative cosmological constant and Maxwell field.
