2025/05/22 by Cocos, Mihail
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Morphological variations and asymmetry
paper · pdf · doi:10.48550/arxiv.2506.14778
openalex publication_date 2025/05/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We establish that any affine manifold (M,∇) endowed with a parallel volume form ω, admits, in any conformal class of Riemannian metrics, a representative H for which ∇ is the Levi-Civita connection. This provides a constructive proof that such manifolds are necessarily complete, generalizing the "if" direction of Markus' conjecture \citemarkus1962. 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 ∇-parallel, extending results of \citegoldman1982 and \citebenzecri1955 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 \citefried1980.