Suppose that a 3-D surface has the property that $|k_1|\leq 1$ and $|k_2|\leq 1$ everywhere, where $k_1$ and $k_2$ are the principal curvatures. Prove or disprove that the curvature $k$ of a curve on that surface also satisfies $|k|\leq 1$.
I don't think so. Take, for example, a plane, which has principal curvatures zero. Pick a small circle, with radius $R$ smaller than $1$ in that plane. The circle has curvature $1/R>1$.