Averaging geometrical objects on a differentiable manifold

TL;DR

This paper develops a rigorous, covariant method for averaging tensor fields on differentiable manifolds using the Weitzenböck connection, emphasizing the importance of frames and connections in the process.

Contribution

It introduces a mathematically precise averaging framework for tensor fields on manifolds, utilizing the Weitzenböck connection for parallel transport.

Findings

A covariant averaging procedure for tensor fields is formulated.

Frames and connections are identified as key geometrical objects in averaging.

The framework ensures exact and fully covariant averaging on manifolds.

Abstract

We construct a framework within which a mathematically precise, fully covariant, and exact averaging procedure for tensor fields on a manifold can be formulated. In particular, we introduce the Weitzenb\"ock connection for parallel transport and argue that, within the context of averaging, frames and connections are the natural geometrical objects on the manifold.