I need to find an exemple of closed curve that is $C^1$ but not $C^2$ in the dimension I want.
I thought of the curve $\gamma : [-1,1] \to \mathbb R^2$ defined by $\gamma(t)=\big(0,t^3\cdot \sin(\frac{1}{t})\big)$ and $\gamma(0)=(0,0)$.
I do not know if I am allowed to define it this way and if this segment is considered as a closed curve.
One example is the boundary of a half-disk in the upper half plane, i.e. a line segment from $(-1,0)$ to $(1,0)$ along the $x$-axis, and then a half circle from $(1,0)$ to $(-1,0)$, passing through the point $(0,1)$. Formally, you can define this curve as $$ \gamma(t) := \begin{cases} (2t+1,0) & t \in [-1,0]\\ (\cos(\pi t),\sin(\pi t)) & t \in [0,1] \end{cases}. $$ Intuitively, it is clear that $\gamma$ is $C^1$ but not $C^2$, because of the angles of $\gamma$ at the points $(\pm 1,0)$. Of course this is not rigorous, but I hope the idea makes sense.