Affine-Orthogonal Manifolds and Deformation to Levi-Civita Connections

TL;DR

The work addresses deforming flat affine connections compatible with a diagonal global metric toward Levi-Civita connections, using a one-parameter path $\nabla^t=(1-t)\nabla+tD$ with diagonal $g^t$ to ensure $\nabla^t g^t=0$ along the path. In local affine coordinates, the Christoffel symbols satisfy $\Gamma^k_{ij}(\nabla^t)=\Gamma^k_{ij}(g^t)$, establishing a proper deformation from $\nabla$ to the Levi-Civita connection $D$. This deformation implies the vanishing of the Euler characteristic $\chi(M)=0$ for compact affine-orthogonal manifolds, aligning with Chern-type conjectures. The paper further generalizes to quasi-metric connections on vector bundles, defines a globally well-defined Euler form via the Pfaffian of the curvature, and provides explicit torus and Hopf-manifold examples that reveal obstructions to metric deformation and non-local-metrizability in broader settings.

Abstract

We study a class of affine manifolds equipped with a flat affine connection $\nabla$ 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 $(\mathbb{R}^{n+1}\setminus \{0\}) / \langle x \mapsto 2x \rangle$ with metric $g = (\sum_i x_i^2)^{-1} \sum_i dx_i^2$ admits a proper deformation of $\nabla$ 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}.

Theorems & Definitions (15)

• Example 1.1: Hopf manifold with diagonal metric
• Example 1.2: One-Parameter Family of Non-Equivalent Flat Connections on $\mathbb{T}^n$
• Theorem 2.1
• proof
• Theorem 2.2
• proof
• Definition 3.1
• Remark 3.1
• Remark 3.2
• Example 3.1