Parallel calibrations and minimal submanifolds

Given a parallel calibration $\varphi \in {\mathrm{\Omega }}^p(M)$ on a Riemannian manifold $M$, I prove that the $\varphi$–critical submanifolds with nonzero critical value are minimal submanifolds. I also show that the $\varphi$–critical submanifolds are precisely the integral manifolds of a ${\mathcal{C}}^{\mathrm{\infty }}(M)$–linear subspace $\text{sP}\subset {\mathrm{\Omega }}^p(M)$. In particular, the calibrated submanifolds are necessarily integral submanifolds of the system. (Examples of parallel calibrations include the special Lagrangian calibration on Calabi-Yau manifolds, (co)associative calibrations on $G_2$–manifolds, and the Cayley calibration on $\text{tSpin}(7)$–manifolds.)