We call a point $x\in B$ smooth (or $\pi$-smooth) if there is a neighborhood $U$ of $x$ such that the fibration $\pi^{-1}(U)\to U$ is isomorphic to a fibration $\pi_{can}^{-1}(V)\to V$ for some open subset $V\subset{{\bf R}}^{n}$. Here the isomorphism $\pi^{-1}(U)\simeq\pi_{can}^{-1}(V)$ is taken in the category of $K$-analytic spaces while $U\simeq V$ is a homeomorphism.

In this case we will call $\pi$ (or the triple $(\pi^{-1}(U),\pi,U)$) an analytic torus fibration.