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

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

2006-08-09

arxiv.org/abs/math/0608230v1

Mathematics

Differential Geometry

43 pages

Scientific paper

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.

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