Calibrated associative and Cayley embeddings

TL;DR

This paper proves new embedding theorems for 3- and 4-manifolds into G_2- and Spin(7)-manifolds using Cartan-Kahler theory and real algebraic methods, expanding understanding of special geometric structures.

Contribution

It introduces two novel isometric embedding theorems for manifolds into G_2- and Spin(7)-manifolds, utilizing advanced algebraic and differential geometric techniques.

Findings

Embedded 3-manifolds as associative submanifolds in G_2-manifolds

Embedded 4-manifolds as Cayley submanifolds in Spin(7)-manifolds

Derived Bochner's theorem via real algebraic tools

Abstract

Using the Cartan-Kahler theory, and results on real algebraic structures, we prove two embedding theorems. First, the interior of a smooth, compact 3-manifold may be isometrically embedded into a G_2-manifold as an associative submanifold. Second, the interior of a smooth, compact 4-manifold K, whose double has a trivial bundle of self-dual 2-forms, may be isometrically embedded into a Spin(7)-manifold as a Cayley submanifold. Along the way, we also show that Bochner's Theorem on real analytic approximation of smooth differential forms, can be obtained using real algebraic tools developed by Akbulut and King.