Suppose that $f\colon \mathbb R^n\to \mathbb R$ is a $C^1$ function such that $\nabla f(x)\ne 0$ for all $x\in \mathbb R^n$. Consider the initial value problem $$ \begin{cases} \dot{u}=-\frac{\nabla f(u)}{|\nabla f(u)|}, \\ u(0)=x,\end{cases}$$ and let $\Phi(t, x):=u(t).$
Question. Let $a\in\mathbb R$. Is it true that $f$ is constant on $\Phi(t, \{ f=a\})$? Assume that $t$ is sufficiently small.
Here, of course, $\Phi(t, \{f=a\})$ denotes the set $\{\Phi(t, x)\ :\ f(x)=a\}$.
This question might contain the answer in a differential geometric language. However, it is interesting to see an elementary answer.