= Solution
If two simple closed <geodesics> were disjoint on a sphere-like surface, they would bound an <annulus>. The <Gauss-Bonnet theorem> applied to it gives $\int_AK\,dA+\int_{\partial A}k_gds=2\pi\chi(A)=0$. Both boundary <geodesic> curvatures vanish, but positive <Gaussian curvature> makes the area integral strictly positive, a contradiction. Hence \b[the two <geodesics> must intersect].
The assertion fails for nonzero constant signed <geodesic curvature>. On the unit round sphere a latitude of colatitude $\theta$, oriented in increasing longitude, has $k_g=\cot\theta$. The latitude of colatitude $\pi-\theta$ has the opposite value; reversing its orientation gives the same value as the first. Choosing $0<\theta<\pi/2$ produces two disjoint smooth simple closed curves with the same positive nonzero curvature. Reversing both handles negative values, and $\theta$ can be chosen for any specified nonzero $\lambda$. Thus \b[constant nonzero <geodesic curvature> does not force intersection].
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