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$f(T)$ Cosmology with Nonzero Curvature

We investigate exact and analytic solutions in $f\left( T\right) $ gravity within the context of a Friedmann--Lema\^ıtre--Robertson--Walker background space with nonzero spatial curvature. For the power law theory $f\left( T\right) =T^{n}$ we find that the field equations admit an exact solution with a linear scalar factor for negative and positive spatial curvature. That Milne-like solution is asymptotic behaviour for the scale factor near the initial singularity for the model $f\left( T\right) =T+f_{0}T^{n}-2Λ$. The analytic solution for that specific theory is presented in terms of Painlevé Series for $n>1$. Moreover, from the\ value of the resonances of the Painlevé Series we conclude that the Milne-like solution is always unstable while for large values of the indepedent parameter, the field equations provide an expanding universe with a de Sitter expansion of a positive cosmological constant. Finally, the presence of the cosmological term $Λ$ in the studied $f\left( T\right) $ model plays no role in the general behavior of the cosmological solution and the universe immerge in a de Sitter expansion either when the cosmological constant term $Λ$ in the $f\left( T\right) $ model vanishes.