Is a connected compact Riemannian manifold of dimension 1 unique?

Question

The tiles says almost everything. It is known that a connected compact topological manifold of dimension 1 is isomorphic to $S^1$. What if we replace "topological" by "riemannian"?

Answer 1

If you are asking as abstract Riemannian manifolds, all closed one dimensional manifolds of the same length are isometric to each other. This has little to do with any planned placement of the curve (well, loop) in a plane or 3-space, but that is the nature of abstract manifolds. There is no such thing as intrinsic curvature for one-dimensional Riemannian manifolds. The isometry, meanwhile, comes from arc-length parametrization; you take some smooth curve that goes around the whole thing, then reparametrize to get unit speed.

As soon as you get to surfaces, there is the influence of (intrinsic) curvature.