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

Liu Shu-Lin1 and Xu Sen-Lin2,3

1Institute of Mathematics, The Academy of Sciences of China, China
2Department of Mathematics, University of Science and Technology of China, China
3Department of Mathematics, Princeton University, 08544, New Jersey, USA

Received 30 August 1982

Copyright © 1983 Liu Shu-Lin and Xu Sen-Lin. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.