Nilpotent structures of oriented neutral vector bundles

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 hyperKähler 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 hyperKähler structure of $E$ if and only if there exists an $H$-nilpotent structure of $(E,h,\nabla )$.

arXiv admin note: text overlap with arXiv:2305.14741