Mollifier Smoothing of tensor fields on differentiable manifolds and applications to Riemannian Geometry

2006/08/09 by Ryuichi Fukuoka, Fukuoka, Ryuichi · 1 citation
Mathematics · Physics and Astronomy · #41A35 #53A45 #53B20 (Secondary) #53B21 (Primary) #Advanced Differential Geometry Research #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Metric Geometry (math.MG) #Nonlinear Partial Differential Equations #math.DG #math.MG #msc:41A35 #msc:53A45 #msc:53B20 #msc:53B21

paper · pdf · doi:10.48550/arxiv.math/0608230

43 pages

arxiv created 2006/08/09 · openalex publication_date 2006/08/09 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Let M be a differentiable manifold. We say that a tensor field g defined on M is non-regular if g is in some local Lp space or if g is continuous. In this work we define a mollifier smoothing g_t of g that has the following feature: If g is a Riemannian metric of class C2, then the Levi-Civita connection and the Riemannian curvature tensor of g_t converges to the Levi-Civita connection and to the Riemannian curvature tensor of g respectively as t converges to zero. Therefore this mollifier smoothing is a good starting point in order to generalize objects of the classical Riemannian geometry to non-regular Riemannian manifolds. Finally we give some applications of this mollifier smoothing. In particular, we generalize the concept of Lipschitz-Killing curvature measure for some non-regular Riemannian manifolds.

Citations

• The Fractal Geometry of Nature
• Riemannian center of mass and mollifier smoothing

Cited by

• Allard's interior ε-Regularity Theorem in Alexandrov spaces

Related

• Local mollification of metrics with small curvature concentration
• The Levi-Civita tensor noncovariance and curvature in the pseudotensors space
• Nonlinear tensor distributions on Riemannian manifolds
• Smoothing metrics on closed Riemannian manifolds through the Ricci flow
• Operator-valued formulas for Riemannian Gradient and Hessian and families of tractable metrics