I'm trying to analyze this article about area-preserving diffeomorphisms and don't quite understand a sentence.
4.1. Linear involutions. We start characterizing the linear involutions $R \! : \mathbb{T}^2 \rightarrow \mathbb T^2$ of the torus, induced by matrices $A$ in $\rm SL(2, \mathbb Z)$. After differentiating the equality $R^2 = Id_\mathbb{T^2}$ at any point of $\mathbb T^2$, we obtain $A^2 = Id_\mathbb{R^2}$.
This is from page 7.
How do they go from an operator defined on $\mathbb T^2$ to an operator defined on $\mathbb R^2$?
Just to be clear, a linear operator $R$ is called an involution, iff $R = R^{-1}$.
EDIT
Am I correct that you can represent an operator on a torus as an operator on the whole space $\rm mod$ something?