Nilpotent structures of oriented neutral vector bundles

TL;DR

This paper explores the conditions under which an oriented neutral vector bundle admits a neutral hyperK"ahler structure, focusing on nilpotent structures and their relation to specific Lie subgroups.

Contribution

It introduces the concept of H-nilpotent structures and establishes their equivalence with the existence of neutral hyperK"ahler structures on the bundle.

Findings

Existence of complex and paracomplex structures related to H-nilpotent structures.

Characterization of neutral hyperK"ahler structures via nilpotent structures.

Connection between Lie subgroup properties and geometric structures.

Abstract

In this paper, we study nilpotent structures of an oriented vector bundle $E$ of rank $4n$ with a neutral metric $h$ and an $h$-connection $\nabla$. We define $H$-nilpotent structures of $(E, h, \nabla )$ for a Lie subgroup $H$ of $SO(2n, 2n)$ related to neutral hyperK\"{a}hler structures. We observe that there exist a complex structure $I$ and paracomplex structures $J_1$, $J_2$ of $E$ such that $h$, $\nabla$, $I$, $J_1$, $J_2$ form a neutral hyperK\"{a}hler structure of $E$ if and only if there exists an $H$-nilpotent structure of $(E, h, \nabla )$.