A Riemannian Characterization of Compact Affine Manifolds with Parallel Volume

TL;DR

This paper proves that compact affine manifolds with a parallel volume form are necessarily complete and Riemannian-flat, providing a constructive method to find a compatible metric and extending previous results to higher dimensions.

Contribution

It introduces a new Riemannian characterization of affine manifolds with parallel volume, showing they are intrinsically flat and providing explicit construction methods.

Findings

Such manifolds are necessarily complete.

They admit a Riemannian metric making the affine connection Levi-Civita.

The structure is intrinsically Riemannian-flat.

Abstract

We establish that any affine manifold $(M,\nabla)$ endowed with a parallel volume form $\omega,$ admits, in any conformal class of Riemannian metrics, a representative $H$ for which $\nabla$ is the Levi-Civita connection. This provides a constructive proof that such manifolds are necessarily complete, generalizing the "if" direction of Markus' conjecture \cite{markus1962}. Moreover, our result demonstrates that these structures are intrinsically Riemannian-flat, a stronger conclusion than the affine completeness asserted by Markus. The metric $H$ arises naturally from the Hessian of volume-normalized distance functions and is shown to be globally smooth and $\nabla$-parallel, extending results of \cite{goldman1982} and \cite{benzecri1955} to higher dimensions with additional geometric structure. The construction proceeds through three technically novel steps: (1) local parallel metric normalization using the given volume form, (2) explicit Hessian calculations in adapted coordinates, and (3) gluing via affine transition maps that preserve the volumetric geometry. This approach reveals an unexpected rigidity in flat affine manifolds with compatible volume that goes beyond the topological constraints studied in \cite{fried1980}.