Derivative of the quotient map $\mathbb R \to \mathbb R / T\mathbb{Z}$

Question

We consider the quotient space $\mathbb R / T\mathbb{Z}$ and the quotient map $\pi:\mathbb{R} \to \mathbb R / T\mathbb{Z}\ $ defined by $\pi(t):= t\bmod T:=t+T\mathbb{Z}$. In a journal i read that $\frac{d}{dt} \pi(t) =1$. Is it true ? If yes, how can i prove it? Thanks.

Answer 1

I will use $[\cdots]$ for the class of... and $\lfloor\cdots\rfloor$ for integer part (max integer $\le\cdots$).

Easy case: $t_0\in(0,T)$. Let be $V = \{[t]:t\in(0,T)\}$, nhod of $t_0$ in the quotient. A chart $\varphi:V\longrightarrow(0,T)$ is $$[t]\mapsto t - \lfloor t\rfloor.$$

Less esay case: $t_0 = 0$. Let be $W = \{[t]:t\in(-T/2,T/2)\}$ nhod of $0$ in the quotient. Now, a chart $\psi:W\longrightarrow(0,T)$ is $$[t]\mapsto t + T/2 - \lfloor t + T/2\rfloor.$$

With this definitions, we can write the compositions: $${\Bbb R}\longrightarrow V\longrightarrow (0,T),$$ $$ t\longmapsto\pi(t) = [t]\longmapsto\varphi\circ\pi(t) = \varphi([t]) = t - \lfloor t\rfloor. $$

$${\Bbb R}\longrightarrow W\longrightarrow (0,T),$$ $$ t\longmapsto\pi(t) = [t]\longmapsto\psi\circ\pi(t) = \psi([t]) = t + T/2 - \lfloor t + T/2\rfloor. $$ And now observe that the terms $\lfloor t\rfloor$ and $T/2 - \lfloor t + T/2\rfloor$ are locally constant.