This paper generalizes Einstein's theorem. It is shown that under the transformation T Λ : U i k ℓ → U ¯ i k ℓ ≡ U i k ℓ + δ i ℓ Λ k − δ k ℓ Λ i , curvature tensor S k ℓ m i ( U ) , Ricci tensor S i k ( U ) , and scalar curvature S ( U ) are all invariant, where Λ = Λ j d x 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 k ℓ m i ( U ) (or Ricci tensor S i k ( U ) ) invariant T V : U i k ℓ → U ¯ i k ℓ ≡ U i k ℓ + V i k ℓ must be T Λ transformation, where V (its components are V i k ℓ ) is a second order differentiable covariant tensor field with vector value.