The terminology of curve, path and one dimensional manifold is always in the textbooks about topology, differential manifold and riemannian geometry. The definition of path and one dimensional manifold is very clear. However, a curve is always have two meanings.
There are some interesting relationships between them.
Fact. When we learn the curve in $\mathbb R^n$, we always assume curve is $r:I\rightarrow \mathbb R^n$ and $I$ is an interval.
This is same of a path. I think that when we say a curve in $\mathbb R^n$ or in other differential manifold $M$, we also can mean an one dimensional manifold in $\mathbb R^n$ or $M$. However, one dimensional manifold is homeomorphism to $I$ or $\mathbb S$. That is to say, we can parameterize it and if it is homeomorphism to $\mathbb S$, we can assume $r(a)=r(b)$.
So the question is when we compute the fundamental group of a differential manifold $\pi_1(M)$, can we consider the one dimensional manifold on $M$?
The computation of $\pi_1(M)$ relies on the classification of the path under homotopy. Is that to say, we just need to consider the classification of one dimensional manifold under homotopy?