It's well known that there is no intrinsic geometry of curves and that the only connected $1$-manifolds are $\mathbb{R}$ and $\mathbb{S}^1$. In other words $R \equiv 0$ (the curvature tensor) for any $1$-dimensional smooth manifold. I've been thinking about this and naturally I've come to the conclusion that if I were a point on the circle and my friend was a point on the real line, we'd have no way to tell where we are (not without leaving our worlds and talking about ambient spaces) by any geometrical properties: that is, there is no geometrical intrinsic property from $\mathbb{R}$ or $\mathbb{S}^1$ that can distinguish between them. This is kind of intuitive: no matter how I smoothly deform a rope, the distance of points on the rope will always remain the same (I think this means any diffeomorphism between two $1$-dimensional Riemannian manifolds must be an isometry - is that right?).
But there is one obvious experiment that an inhabitant of the circle could perform to conclude they're not in the real line: if they start walking forward, they're eventually gonna come back to the same place they started the walk. This obviously doesn't happen on the real line. So this must be a topological property. What is it? I think it must have something to do with the fundamental group, because it's obviously not compactness, connectedness or anything I can tell straight away. So what's the name for this property?
One property is compactness! The circle is compact but the real line is not.
The one you’re thinking of, though, is that the real line is simply-connected but the circle is not.
It’s not true that every map between one-manifolds is an isometry. Just take for instance $f(x)=2x$ which maps the real line to itself but clearly distorts distances.