SOME INVARIANT THEOREMS ON GEOMETRY OF EINSTEIN NON-SYMMETRIC FIELD THEORY

SHU-LIN LIU ； SEN-LIN XU

《International Journal of Mathematics and Mathematical Sciences》 1983卷 (1983/12) Pp. 727-736

https://doi.org/10.1155/S0161171283000629

摘要

This paper generalizes Einstein's theorem. It is shown that under the transformation T_Λ : U_(ik)^ℓ → U_(ik)^ℓ ≡ U_(ik)^ℓ + δ_i^ℓΛ_k - δ_k^ℓΛ_i, curvature tensor S_(kℓm)^i(U), Ricci tensor S_(ik)(U), and scalar curvature S(U) are all invariant, where Λ = Λ_jdx^j 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 S_(kℓm)^i(U) (or Ricci tensor S_(ik)(U)) invariant T_V : U_(ik)^ℓ → U_(ik)^ℓ ≡ U_(ik)^ℓ + V_(ik)^ℓ must be T_Λ transformation, where V (its components are V_(ik)^ℓ) is a second order differentiable covariant tensor field with vector value.

關鍵字

Einstein non-symmetric field ； Einstein theorem ； curvature tensor ； Ricci tensor ； scalar curvature ； T_Λ transformation ； T_v transformation