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

Let $(M,g,J,\omega )$ be an almost Hermitian manifold. Given an automorphism $\psi \in \mathrm{A}\mathrm{u}\mathrm{t}(TM)$, 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 $\mathrm{A}\mathrm{u}\mathrm{t}(TM)$ with nice transformation properties. We call these automorphisms $g$-$\mathit{\text{Codazzi}}\mathit{\text{maps}}$ 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.