Twists, Codazzi Tensors, and the $6$-sphere

Abstract

Let $(M,g,J,\ddot{I}‰)$ be an almost Hermitian manifold. Given an automorphism $\ddot{I}ˆ\in \mathrm{Aut}(TM)$, the existing structure can be twisted to obtain a new almost Hermitian manifold $(M,g^{\ddot{I}}ˆ,J^{\ddot{I}}ˆ,\ddot{I}{‰}^{\ddot{I}}ˆ)$. In the current paper, we study these $\ddot{I}ˆ$-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of $J^{\ddot{I}}ˆ$ and the Riemannian geometry of $g^{\ddot{I}}ˆ$. By studying the latter, we identity a certain class of $\mathrm{Aut}(TM)$ with nice transformation properties. We call these automorphisms $g$-\textit{Codazzi maps} because of their close relationship with Codazzi tensors. The aforementioned results are ultimately applied to the standard nearly Kähler structure on the $6$-sphere where we prove a nonintegrability result for the class of $g$-Codazzi maps.

28 pages