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

Let $(M,g,J,\omega)$ be an almost Hermitian manifold. Given an automorphism $\psi\in \mathrm{Aut}(TM)$, the existing structure can be twisted to obtain a new almost Hermitian manifold $(M,g^\psi,J^\psi,\omega^\psi)$. In the current paper, we study these $\psi$-twisted almost Hermitian structures with particular emphasis on questions regarding the integrability of $J^\psi$ and the Riemannian geometry of $g^\psi$. 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 K\"{a}hler structure on the $6$-sphere where we prove a nonintegrability result for the class of $g$-Codazzi maps.