compact surface with two non-intersecting geodesics

Question

I need to find an example of a compact geometric surface M such that

1. Gaussian curvature $K>=0$

2. M is diffeomorphic to a sphere

3. M has two simply closed geodesics (smoothly closed loops) that never meet.

How to find such M? What is its geometric structure???

I only know that M must have Euler characteristic 2 since it looks like a sphere.

Answer 1

Take a bounded cylinder of radius $r$ and glue on two hemispheres of radius $r$ on to each end of the cylinder. Any two simple closed geodesics on the cylinder will not intersect and they are certainly still simple closed geodesics after gluing on the hemispheres which makes the surface diffeomorphic to a sphere. It's clear that the Gaussian curvature of this surface is then greater than or equal to zero.