Extrinsic geometry of calibrated submanifolds

Abstract

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 \(\iota :L \rightarrow M\) 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 \(\iota :L \rightarrow M\) 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 \({\mathbb {R}}^4\).