Covariant derivative of a vector field along a curve in relation to a affine conection

Question

I'm reading doCarmo Riemannian Geometry and in the chapter with affine connections is this proposition: In c.) says if $V$ is the restriction to $c$ of some vector field $Y$ defined on $M$. Isn't true that any vector field along a curve is the restriction of a vector field defined on $M$? If not, a counter exemple will help and also, when this property is true?

Answer 1

Let $\gamma:I\rightarrow M$ be a smooth curve that is only an immersion in the sense that it is self-intersecting. For some $t_1,t_2\in I$ let $\gamma(t_1)=\gamma(t_2)$.

Let $Y:I\rightarrow TM$ be a vector field along $\gamma$ with $Y(t_1)\neq Y(t_2)$. $Y$ is still well defined, because it is single-valued on $I$, however it is easy to see that this vector field along $\gamma$ has no extension to an open set around $\gamma(I)$, because any such extension would necessarily be multivalued at $\gamma(t_1)=\gamma(t_2)$.

However, since every immersion is locally an embedding, if you restrict your attention to a small neighborhood $(t-\epsilon,t+\epsilon)$ around an arbitrary $t\in I$, the vector field along $\gamma$ will always be locally extendible.

EDIT: I am not sure in this, and I certainly cannot prove it by myself, but I think that if $\gamma$ is an embedding, then any vector field along it is extendible to $M$.