I'm interested in generalizations the following well-known theorem of Fáry and Milnor.

**Theorem.** (Fáry-Milnor) If a simple closed curve $\gamma \subset \mathbb{R}^3$ is knotted, then the total curvature of $\gamma$ is greater than $4\pi$.

I have found some sources that prove essentially the same result when $\mathbb{R}^3$ is equipped with a non-positively curved metric (e.g. the usual hyperbolic metric), maybe satisfying some extra hypotheses (for example, see [1,2]). I'm interested in the opposite situation.

**Question.** Is there a version of the Fáry-Milnor theorem for positively curved metrics on $S^3$?

It seems that one could derive *some* inequality from the usual one using contraction maps to and from Euclidean space and estimating the resulting effect on the total curvature along $\gamma$. However, this would depend strongly on the metrics/contraction maps involved. I would really prefer something metric independent if possible.

[1] F. Brickell and C. C. Hsiung. *The total absolute curvature of closed curves in Riemannian manifolds.*

[2] C. Schmitz. *The Theorem of Fáry and Milnor for Hadamard Manifolds*