  Weak pion production off the nucleon at low energies has been systematically
investigated in manifestly relativistic baryon chiral perturbation theory with
explicit inclusion of the $\Delta$(1232) resonance. Most of the involved
low-energy constants have been previously determined in other processes such as
pion-nucleon elastic scattering and electromagnetic pion production off the
nucleon. For numerical estimates, the few remaining constants are set to be of
natural size. As a result, the total cross sections for single pion production
on neutrons and protons, induced either by neutrino or antineutrino, are
predicted. Our results are consistent with the scarce existing experimental
data except in the $\nu_\mu n\to \mu^-n\pi^+$ channel, where higher-order
contributions might still be significant. The $\Delta$ resonance mechanisms
lead to sizeable contributions in all channels, especially in $\nu_\mu p\to
\mu^- p\pi^+$, even though the considered energies are close to the production
threshold. The present study provides a well founded low-energy benchmark for
phenomenological models aimed at the description of weak pion production
processes in the broad kinematic range of interest for current and future
neutrino-oscillation experiments.
