# What curves lying on a sphere have constant geodesic curvature?

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 ?

• Do you have any ideas as to which are the curves you are looking for? If you can find them then an approach might be to appeal to the theory of ODEs and to say that (given initial conditions) a curve on the sphere is determined by its geodesic curvature. Apr 19, 2021 at 8:30
• I think are all circles. Apr 19, 2021 at 9:49

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 $$\gamma$$.