What curves lying on a sphere have constant geodesic curvature?

Question

My question is:

Which curves in the sphere have a constant geodesic curvature.

I find that if $\displaystyle{k_{g} = constant}$ then $\displaystyle{k_{g} = \pm \kappa \sin{\psi} = constant}$, where $\displaystyle{\kappa}$ is the curvature of curve $\displaystyle{\boldsymbol{\gamma}}$ and $\displaystyle{\psi}$ is the angle between the unit vector on the sphere $\displaystyle{\boldsymbol{N}}$ and the principal vertical vector $\displaystyle{\boldsymbol{N}_{\boldsymbol{\gamma}}}$ of the curve $\displaystyle{\boldsymbol{\gamma}}$. Also, we know that $\displaystyle{\boldsymbol{N} \cdot \boldsymbol{N}_{\boldsymbol{\gamma}} = \cos{\psi}}$. We can assume that $\displaystyle{\boldsymbol{\gamma}}$ is a unit-speed curve of the sphere.

How could I continue afterwards to solve the problem ?

Answer 1

From Wikipedia:

If $\gamma$ lies on $M$, the geodesic curvature is the norm of the projection of the covariant derivative $DT/ds$ on the tangent space to the submanifold.

The ODE $DT/ds=k_g$ should specify all solutions with geodesic curvature $k_g$, where $T$ is the unit tangent to the curve $\gamma$.