The Levi-Civita tensor noncovariance and curvature in the pseudotensors space

TL;DR

This paper reveals that the conventional covariant derivative of the Levi-Civita tensor is not truly covariant and introduces a modified curvature concept in pseudotensors space, leading to new geometric insights.

Contribution

It demonstrates the noncovariance of the Levi-Civita tensor's covariant derivative and proposes a modified curvature tensor in pseudotensors space, along with new geometric constructs.

Findings

Conventional covariant derivative of Levi-Civita tensor is not covariant.

A modified curvature tensor in pseudotensors space is introduced.

New vector constructs from metric and Levi-Civita density offer geometric insights.

Abstract

It is shown that conventional "covariant" derivative of the Levi-Civita tensor is not really covariant. Adding compensative terms, it is possible to make it covariant and to be equal to zero. Then one can be introduced a curvature in the pseudotensors space. There appears a curvature tensor which is dissimilar to ordinary one by covariant term including the Levi-Civita density derivatives hence to be equal zero. This term is a little bit similar to Weylean one in the Weyl curvature tensor. There has been attempted to find a curvature measure in the combined (tensor plus pseudotensor) tensors space. Besides, there has been constructed some vector from the metric and the Levi-Civita density which gives new opportunities in geometry.