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

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.