Affine Isometric Embeddings and Rigidity

Abstract

The Pick cubic form is a fundamental invariant in the (equi)affine differential geometry of hypersurfaces. We study its role in the affine isometric embedding problem, using exterior differential systems (EDS). We give pointwise conditions on the Pick form under which an isometric embedding of a Riemannian manifold M3 into \(\mathbb{R}^4 \) is rigid. The role of the Pick form in the characteristic variety of the EDS leads us to write down examples of nonrigid isometric embeddings for a class of warped product M3's.