The space of rough Riemannian metrics

2025/07/14 by Lashi Bandara, Bandara, Lashi, A. Hassan +1
Computer Science · Mathematics · #53C23 #58A05 #58D17 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Metric Geometry (math.MG) #Morphological variations and asymmetry #Topological and Geometric Data Analysis

paper · pdf · doi:10.48550/arxiv.2507.10274

openalex publication_date 2025/07/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

On a smooth connected manifold, we consider all possible locally elliptic and locally bounded measurable coefficient Riemannian metrics called rough Riemannian metrics. We equip this set with an extended metric which is connected if and only if the manifold is compact. We prove that this is a complete length extended metric space and the components on which the distance is finite are path-connected. Moreover, we identify the closure of smooth metrics in this space to be continuous metrics.

Citations

• Geometric singularities and Hodge theory
• Quadratic estimates for degenerate elliptic systems on manifolds with lower Ricci curvature bounds and boundary value problems
• Heat kernels and regularity for rough metrics on smooth manifolds
• A Flow Tangent to the Ricci Flow via Heat Kernels and Mass Transport

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