Curve with intersection manifold

Question

Consider a smooth manifold $M$ and a smooth curve $c \, : \, I \, \rightarrow \, M$ with $I$ an open interval of $\mathbb{R}$. Assume that there exist $t_0,t_1 \in I$ with $t_0 < t_1$ such that $c(t_0) = c(t_1)$. I am wondering whether the image $c(I)$ defines a submanifold of $M$. I guess the answer is no but I am not completely convinced.

Answer 1

In general the answer will be no, e.g. when the intersection is transversal, that is when $c^\prime(t_0)$ and $c^\prime(t_1)$ are linear independent. Then the point of selfintersection will not conform with the definition of a manifold (it looks like the center of a figure eight).

There are, however, cases, where the image of $c$ may well be an embedded manifold, e.g. when $c$ is a smooth periodic parametrization of a circle like submanifold of $M$.