= Solution
A simple closed curve on the sphere bounds two topological disks. Equal areas give area $2\pi$ for each, since the unit sphere has area $4\pi$. Apply <Gauss-Bonnet theorem> to either disk $D$, with $K=1$ and $\chi(D)=1$:
$$
2\pi+\int_\gamma k_g\,ds=2\pi,\qquad\int_\gamma k_g\,ds=0.
$$
A continuous <geodesic curvature> that were strictly positive everywhere, or strictly negative everywhere, would have a correspondingly strict sign for this integral. Therefore \b[neither everywhere strictly positive nor everywhere strictly negative <geodesic curvature> is possible]. Orientation reversal changes its sign but not this conclusion.
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