Anisotropic spacetimes in $f(T,B)$ theory I: Bianchi I Universe

That is the first part of a series of studies on analyzing the higher-order teleparallel theory of gravity known as the $f(T,B)$-theory. This work attempts to understand how the anisotropic spacetimes are involved in the $f(T,B)$-theory. In this work, we review the previous analysis of $f(T,B)$-gravity, and we investigate the global dynamics in the case of a locally rotational Bianchi I background geometry. We focus in the case of $f(T,B)=T+F\left(B\right)$ theory and we determine the criteria where the $f(T,B)$-theory solves the homogeneity problem. Finally, the integrability properties for the field equations are investigated by applying the Painlevé analysis. The analytic solution is expressed by a right Painlevé expansion.