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

TL;DR

This paper introduces a mollifier smoothing technique for tensor fields on differentiable manifolds, enabling the extension of Riemannian geometric concepts to non-regular manifolds with applications to curvature measures.

Contribution

It defines a mollifier smoothing for tensor fields that preserves key geometric structures and facilitates generalizations in non-regular Riemannian geometry.

Findings

Convergence of Levi-Civita connection and curvature tensors under smoothing

Extension of Lipschitz-Killing curvature measures to non-regular manifolds

Provides a foundation for analyzing non-regular Riemannian structures

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.