International Journal of Mathematics and Mathematical Sciences
Volume 6 (1983), Issue 4, Pages 727-736
doi:10.1155/S0161171283000629
Some invariant theorems on geometry of Einstein non-symmetric field theory
Shu-Lin Liu1
and Sen-Lin Xu2
1Institute of Mathematics, The Academy of Sciences of China, China
2Department of Mathematics, University of Science and Technology of China, China
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.