  We provide new results and new proofs of results about the torsion of curves
in $\mathbb{R}^3$. Let $\gamma$ be a smooth curve in $\mathbb{R}^3$ that is the
graph over a simple closed curve in $\mathbb{R}^2$ with positive curvature. We
give a new proof that if $\gamma$ has nonnegative (or nonpositive) torsion,
then $\gamma$ has zero torsion and hence lies in a plane. Additionally, we
prove the new result that a simple closed plane curve, without any assumption
on its curvature, cannot be perturbed to a closed space curve of constant
nonzero torsion. We also prove similar statements for curves in Lorentzian
$\mathbb{R}^{2,1}$ which are related to important open questions about time
flat surfaces in spacetimes and mass in general relativity.
