Are a circle and an ellipse isometric as smooth manifolds?

Question

It is a well known result that all compact $1$-dimensional smooth manifolds are diffeomorphic to a circle. I'm wondering if the result still holds if we replace the word diffeomorphic with isometric. The group of isometries of a circle $S^1$ is given by $\text{ISO}(S^1)\cong O(2)$, while it seems to me that the group of isometries of an ellipse (of non-zero eccentricity) is generated by the reflections on the two axis and the rotation by a straight angle. Since the two groups are not isomorphic, it follows that a circle and an ellipse are, in general, not isometric. Hence not all compact $1$-dimensional manifolds are isometric. Am I correct in my conclusions?

Answer 1

Parametrize the curve by arc-length, assuming it is embedded isometrically in some euclidean space. This will give you local isometries between any two Riemannian $1$-folds, and it will extend to a global isometry exactly when the global length is the same.

This is roughly like saying that if you fix a point of the curve and consider the exponential map from that point, it will always give you an isometry between $\mathbb{R}$ quotiented by some discrete set $a\cdot\mathbb{Z}$, $a>0$, and your curve. The only invariant is this number $a$, the length of the curve.