Extrinsic geometry of calibrated submanifolds

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 Kähler, 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 $\iota \text{\hspace{0.17em}⁣}:L\to M$ 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 $\mathrm{\nabla }$ on $TL$ and $D$ on $NL$, respectively, is second order information. We characterize the conditions imposed on the extrinsic geometric data $(A,\mathrm{\nabla },D)$ when the Riemannian immersion $\iota \text{\hspace{0.17em}⁣}:L\to M$ 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 ${\mathbb{R}}^4$.