Abstract <p>We derive differential identities in the domain of Einstein nonsymmetric geometry. We introduce a local form of the nonsymmetric linear connection of a totally skew-symmetric torsion, which has been constructed and published in a global form in a previous work. The local form of this connection is expressed in terms of the symmetric and skew-symmetric parts of the nonsymmetric metric tensor as well as their derivatives. We apply the Dolan–McCrea variational scheme using a nonsymmetric metric tensor <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12267_2025_5261_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_{\mu\nu}\)</EquationSource>  </InlineEquation>. We demonstrate our rigorous proof via analyzing the generalized second-order metric tensor <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12267_2025_5261_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_{\mu\nu}\)</EquationSource>  </InlineEquation> to its symmetric and skew-symmetric parts, respectively. The derived differential identities can be split into two differential identities such that one identity is expressed in terms of the symmetric part of <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12267_2025_5261_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_{\mu\nu}\)</EquationSource>  </InlineEquation>, and the second one in terms of the skew part of <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="12267_2025_5261_Article_IEq1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="29" /> </InlineMediaObject> <EquationSource Format="TEX">\(G_{\mu\nu}\)</EquationSource>  </InlineEquation>. One of the demonstrated differential identities can be considered as a generalization of the second Bianchi identity. This identity is reduced to the conventional Riemannian one in the case of using the Ricci scalar.</p>

Differential Identities in Einstein Nonsymmetric Geometry

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发表日期：

2025年6月11日

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M. I. Wanas,
Samah Nabil,
Nouran E. Abdelhamid,
Kyrillos ElAbd

摘要

Abstract

We derive differential identities in the domain of Einstein nonsymmetric geometry. We introduce a local form of the nonsymmetric linear connection of a totally skew-symmetric torsion, which has been constructed and published in a global form in a previous work. The local form of this connection is expressed in terms of the symmetric and skew-symmetric parts of the nonsymmetric metric tensor as well as their derivatives. We apply the Dolan–McCrea variational scheme using a nonsymmetric metric tensor \(G_{\mu\nu}\) . We demonstrate our rigorous proof via analyzing the generalized second-order metric tensor \(G_{\mu\nu}\) to its symmetric and skew-symmetric parts, respectively. The derived differential identities can be split into two differential identities such that one identity is expressed in terms of the symmetric part of \(G_{\mu\nu}\) , and the second one in terms of the skew part of \(G_{\mu\nu}\) . One of the demonstrated differential identities can be considered as a generalization of the second Bianchi identity. This identity is reduced to the conventional Riemannian one in the case of using the Ricci scalar.