Is a piecewise smooth curve contained in a single chart?

Question

If $M$ is a smooth manifold of dimension $\ge 2$, and $\gamma:[a,b]\to M$ is a piecewise $C^1$ curve, is the image of $\gamma$ contained in the domain of a single chart of $M$? It should be emphasized that the domain of $\gamma$ should be compact. In the case when $\gamma$ has no self-intersections, this is true by virtue of a tubular neighborhood theorem. In the case when self-intersections are allowed, I'm not sure what to do because the tube self-intersects.

Answer 1

Let $M$ be the two-dimensional torus. It is well-known that the complete graph $K_5$ can be embedded into $M$. Also, we can achieve that the image of $\gamma$ is this $K_5$. If $U\subset \Bbb R^2$ and a chart $U\to M$ contains the image of $\gamma$, we obtain an embedding of $K_5$ into the plane, which is absurd.