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

Abstract

This paper generalizes Einstein's theorem. It is shown that under the transformation TΛ:Uikℓ→U¯ikℓ≡Uikℓ+δiℓΛk−δkℓΛi, curvature tensor Skℓmi(U), Ricci tensor Sik(U), and scalar curvature S(U) are all invariant, where Λ=Λjdxj is a closed 1-differential form on an n-dimensional manifold M.It is still shown that for arbitrary U, the transformation that makes curvature tensor Skℓmi(U) (or Ricci tensor Sik(U)) invariant TV:Uikℓ→U¯ikℓ≡Uikℓ+Vikℓ must be TΛ transformation, where V (its components are Vikℓ) is a second order differentiable covariant tensor field with vector value.