Constant Geodesic Curvature Curve

CGC Foliation

Let $M$ be a $2$-dimensional Riemannian manifold. Then the CGC foliation of $M$ is a foliation such that each leaf is a ($1$-dimensional) curve with constant geodesic curvature.

CGC ${\text{O}}_2$ Action

The CGC ${\text{O}}_2$ action is a smooth group action on a smooth Riemannian $2$-manifold $(M,g)$, say $\varphi :{\text{O}}_n\times M\to M,(k,p)\mapsto {\varphi }_k(p)$, such that $l({\varphi }_k(p))=\frac{{\alpha }_k}{2\pi }\cdot l(p)$ where $l(p)$ denotes the arc length of $p$ in the CGC leaf, and $\alpha :{\text{O}}_2\to S^1\times \{1,-1\}$ is the canonical chart for the Lie group ${\text{O}}_2$.

Remark

Geometrically, it means that we find the CGC leaf containing $p\in M$, then if $k\in {\mathrm{SO}}_2$, then we move $p$ a distance of $\frac{{\alpha }_k}{2\pi }\cdot l(p)$ along the leaf; otherwise, $k$ is in ${\text{O}}_2$ but not in ${\text{SO}}_2$, then go in the opposite direction.

Theorem

Suppose $(M,g)$ is a Riemannian $2$-manifold. Then $g$ is rotationally-invariant with respect to some ${\mathrm{O}}_2$ group action if and only if it is rotationally-invariant with respect to the CGC ${\text{O}}_2$ action.