Calibrated associative and Cayley embeddings
Colleen Robles, Sema Salur

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.
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Taxonomy
TopicsAdvanced Algebra and Geometry · Advanced Topics in Algebra · Advanced Combinatorial Mathematics
