Calibrated submanifolds with flat normal bundles

TL;DR

This paper proves that calibrated submanifolds with flat normal bundles in Euclidean space are necessarily planes, and extends this result to certain conditions in Riemannian manifolds, showing they are totally geodesic.

Contribution

It establishes a new rigidity result for calibrated submanifolds with flat normal bundles and commuting shape operators, characterizing them as totally geodesic.

Findings

Calibrated submanifolds with flat normal bundles are planes in Euclidean space.

In Riemannian manifolds with parallel calibration, commuting shape operators imply the submanifold is totally geodesic.

The paper generalizes classical results on minimal submanifolds to calibrated settings.

Abstract

We show that submanifolds of Euclidean space which are calibrated by a constant-coefficient differential form and have flat normal bundles are planes. In fact, in a Riemannian manifold equipped with a parallel calibration, a calibrated submanifold subject to the condition that all shape operators commute is totally geodesic.