Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections

We study a class of affine manifolds equipped with a flat affine connection $\mathrm{\nabla }$ and a global Riemannian metric $g$ that is diagonal in local affine coordinates. These structures are closely related to \emph{Hessian manifolds}, where the metric locally arises as the Hessian of a smooth potential. For example, the Hopf manifold $({\mathbb{R}}^{n+1}\setminus \{0\})\mathrm{/}⟨x\mapsto 2x⟩$ with metric $g=({\sum }_i{x}_i^2{)}^{-1}{\sum }_{i}d{x}_i^2$ admits a proper deformation of $\mathrm{\nabla }$ into its Levi-Civita connection. By Theorem 2.3 in \cite{cocos2025}, such deformations force the Euler characteristic to vanish, providing evidence for Chern’s conjecture. The geometry of these manifolds is reminiscent of the work of Yau on affine and Hessian structures \cite{cheng_yau_1986}.