Analog of selfduality in dimension nine

TL;DR

This paper introduces a novel nine-dimensional Riemannian geometry analogous to selfduality in four dimensions, characterized by a special 4-form and related to a specific group representation, with examples of homogeneous structures.

Contribution

It defines a new geometric structure in nine dimensions linked to a 4-form and explores its properties, extending the concept of selfduality beyond four dimensions.

Findings

Constructed locally homogeneous examples of the new geometry.

Established the relation to a 9-dimensional irreducible representation of SO(3)×SO(3).

Analyzed metric connections with totally antisymmetric torsion preserving the 4-form.

Abstract

We introduce a type of Riemannian geometry in nine dimensions, which can be viewed as the counterpart of selfduality in four dimensions. This geometry is related to a 9-dimensional irreducible representation of ${\bf SO}(3) \times {\bf SO} (3)$ and it turns out to be defined by a differential 4-form. Structures admitting a metric connection with totally antisymmetric torsion and preserving the 4-form are studied in detail, producing locally homogeneous examples which can be viewed as analogs of self-dual 4-manifolds in dimension nine.