  Given a calibration $\alpha$ whose stabilizer acts transitively on the
Grassmanian of calibrated planes, we introduce a nontrivial Lie-theoretic
condition on $\alpha$, which we call compliancy, and show that this condition
holds for many interesting geometric calibrations, including K\"ahler, special
Lagrangian, associative, coassociative, and Cayley. We determine a sufficient
condition that ensures compliancy of $\alpha$, we completely characterize
compliancy in terms of properties of a natural involution determined by a
calibrated plane, and we relate compliancy to the geometry of the calibrated
Grassmanian.
  The condition that a Riemannian immersion $\iota \colon L \to M$ be
calibrated is a first order condition. By contrast, its extrinsic geometry,
given by the second fundamental form $A$ and the induced tangent and normal
connections $\nabla$ on $TL$ and $D$ on $NL$, respectively, is second order
information. We characterize the conditions imposed on the extrinsic geometric
data $(A, \nabla, D)$ when the Riemannian immersion $\iota \colon L \to M$ is
calibrated with respect to a calibration $\alpha$ on $M$ which is both parallel
and compliant. This motivate the definition of an infinitesimally calibrated
Riemannian immersion, generalizing the classical notion of a superminimal
surface in $\mathbb R^4$.
