Lagrangian-type submanifolds of Spin(7) manifolds and their deformations

TL;DR

This paper introduces L-submanifolds in Spin(7) manifolds and characterizes their deformation space as the space of closed 3-forms, extending previous work on similar structures in G2 manifolds.

Contribution

It defines L-submanifolds in Spin(7) manifolds and describes their deformation space, expanding the understanding of Lagrangian-type submanifolds in special holonomy geometries.

Findings

Deformation space of L-submanifolds is identified with closed 3-forms.

Introduces a new class of Lagrangian-type submanifolds in Spin(7) manifolds.

Extends deformation theory from G2 to Spin(7) contexts.

Abstract

In an earlier paper we showed that the space of deformations of a smooth, compact, orientable Harvey-Lawson submanifold HL in a G2 manifold M can be identified with the direct sum of the space of smooth functions and closed 2-forms on HL. In that paper, we also introduced a new class of Lagrangian-type 4-dimensional submanifolds inside G2 manifolds, called them RS submanifolds, and proved that the space of deformations of a smooth, compact, orientable RS submanifold in a G2 manifold M can be identified with closed 3-forms on RS. In this short note, we define a new class of Lagrangian-type 4-dimensional submanifolds inside Spin(7) manifolds, which we call L-submanifolds. We show that the space of deformations of a smooth, compact, orientable L-submanifold in a Spin(7) manifold N can be identified with the space of closed 3-forms on L.