Some invariant theorems on geometry of Einstein non-symmetric field theory (Q792640)

Let \(\Lambda =\Lambda_ jdx^ j\) be a closed 1-differential form on a manifold M (of di\(m\geq 2)\). It is proved that under the transformation \(T_{\Lambda}:U^{\ell}_{ik}\to \bar U^{\ell}_{ik}\equiv U^{\ell}_{ik}+\delta^{\ell}_ i\Lambda_ k-\delta^{\ell}_ k\Lambda_ i\) the curvature tensor \(S^ i_{k\ell m}\), the Ricci tensor \(S_{ik}\) and the scalar curvature S of U are invariant. Therefore if \(T_{\Omega}:U^{\ell}_{ik}\to \bar U^{\ell}_{ik}\equiv u^{\ell}_{ik}+\delta^{\ell}_ i\Omega_ k-\delta^{\ell}_ k\Omega_ i\) makes the curvature tensor \(S^ i_{k\ell m}\) (or Ricci tensor \(S_{ik})\) of some U invariant, \(T_{\Omega}\) must be \(T_{\Lambda}\) where \(\Omega =\Omega_ jdx^ j\) is a 1-differential form. But a necessary and sufficient condition that transformation \(T_{\Omega}\) makes the scalar curvature S of some U invariant is: \(g^{ik}\Omega_{k,i}-g^{ik}\Omega_{i,k}=0\) using these results and some other invariance theorems on the curvature tensor \(S^ i_{k\ell m}\), Ricci tensor \(S_{ik}\) and scalar curvature S of U, the authors generalize Einstein's transformation given by: \(T_{\lambda}:U^{\ell}_{ik}\to \bar U^{\ell}_{ik}=U^{\ell}_{ik}+\delta^{\ell}_ i\lambda,k- \delta^{\ell}_ k\lambda,i\) and they give converse theorems for arbitrary n (\(\geq 2)\). All theorems in this paper are proved by straightforward calculations.