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?

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    $\begingroup$ in fact "closed paths" in $M$ $\endgroup$ – janmarqz Jan 16 '14 at 4:05
  • $\begingroup$ Yes, it is closed.@janmarqz $\endgroup$ – gaoxinge Jan 16 '14 at 4:06
  • $\begingroup$ What about differentiability? A path is piecewise smooth, and this is important in defining the multiplication of paths by composition. $\endgroup$ – user121697 Jan 16 '14 at 4:20

No, in order to define the fundamental group you have to consider all based loops, not only simple ones (which is the usual terminology for connected 1d submanifolds). In order to convince yourself in this start by considering fundamental group of the circle. Taking powers of a simple loop results in loops which cannot be made simple via homotopy. Similar problems persist in dimensions 2 (powers still cannot be made simple in general) and even 3 (in a way). In dimensions 4 and higher you can define fundamental group via isotopy classes of based simple loops only, but it is very unnatural.

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