The title is a quote from p.256 of Wilhelm Klingenberg's 1995
*Riemannian Geometry* (Google Books link):
Every surface homeomorphic to a sphere $\mathbb{S}^2$ has three distinct, simple, closed geodesics.
Another source is p.466 of Berger's *A Panoramic view of Riemannian Geometry* (Springer link).

My question is:

What is known about simple, closed curves of

constant, non-zerogeodesic curvature? Are there always three, simple closed curves for every constant $k_g$, on a surface homeomorphic to $\mathbb{S}^2$ ?

**Update**. macbeth noted that this question was posed on MO earlier and adequately answered:
"Curves of constant curvature on S^2."