Curves on sphere of constant curvature

Question

Here is the beginning of my question:

At first, I want to prove $\int \frac{\tau}{\kappa}ds$=0 for a closed curve on sphere. I used the criterion for spherical curves $${(\frac{1}{\kappa })^2} + {[\frac{1}{\tau }(\frac{1}{\kappa })']^2} = {a^2}$$. But there is a gap if $\kappa$ is constant. In this case if $\int \tau ds=0$ it will be okay. Then I wonder:

1. Is there a spherical curve of constant curvature (whether closed or not) except a circle?

2. For closed spherical curve, does $\int \tau ds=0$ always hold?

Any help will be appreciated.

Answer 1

So what you need to prove is this: If $\kappa$ is constant for a closed curve lying on a sphere, then the curve is the intersection of a plane with the sphere. (Comment: If $\kappa$ is constant, so then is geodesic curvature $\kappa_g$.) So, yes, the curve is a circle and is planar, so $\tau=0$. (In general, you should be able to prove that for any closed curve $C$ lying on a sphere, $\int_C \tau\,ds \equiv 0\pmod{2\pi}$.)