Analog of selfduality in dimension nine
Anna Fino, Pawel Nurowski

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 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.
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