Let $S$ be a smooth surface embedded in $\mathbb{R}^3$. When are (some of) the principal lines of curvature geodesics on $S$? Perhaps on the ellipsoid below, the (blue) central cycle, a max principal line, is a geodesic? And perhaps the (red) min principal line connecting the two umbilical points is a geodesic?

^{ Image from Jorge Sotomayor.1 }

Is there any $S$ all of whose principal lines of curvature are geodesics?

^{1}Sotomayor, Jorge. "Historical Comments on Monge's Ellipsoid and the Configuration of Lines of Curvature on Surfaces Immersed in ${\mathbb R}^ 3$." arXiv Abstract (2004).

*São Paulo Journal of Mathematical Sciences*2, 1 (2008), 99–143.