We investigate the quasi-projective curvature tensor, a unified object that generalizes the Weyl projective, conharmonic, and M-projective tensors. Our analysis yields several key physical results. Quasi-projectively flat space-times are necessarily Einstein and, under generic conditions, have constant curvature, making them conformally flat and of Petrov type O (de Sitter, anti-de Sitter, or Minkowski). For perfect fluids, this flatness forces the dark energy equation of state <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$p = -\sigma $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mo>-</mo> <mi>σ</mi> </mrow> </math></EquationSource> </InlineEquation>, i.e., a cosmological constant. A divergence-free quasi-projective tensor implies the Ricci tensor is of Codazzi type, producing a Yang pure space in Gray’s subspaces <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\textbf{B}$</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="bold">B</mi> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$\textbf{B}'$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="bold">B</mi> <mo>′</mo> </msup> </math></EquationSource> </InlineEquation>. Also, we provide two examples to validate our results. In f(R)-gravity vacuum solutions, the Codazzi condition forces <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\nabla _j R$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="normal">∇</mi> <mi>j</mi> </msub> <mi>R</mi> </mrow> </math></EquationSource> </InlineEquation> to be an eigenvector of the Ricci tensor, and the Ricci tensor decomposes into a Weyl-contracted term plus a quasi-Einstein term. Finally, we prove that pseudo quasi-projectively symmetric space-times are of Petrov type I, D, or O, with an irrotational, acceleration-free covector.

Quasi-Projective Curvature Tensor in Space-Time and f(R)-Gravity

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发表日期：

2026年6月16日

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Uday Chand De,
Krishnendu De,
Ayman Elsharkawy,
Füsun Özen Zengin,
Sezgin Altay Demirbag

摘要

We investigate the quasi-projective curvature tensor, a unified object that generalizes the Weyl projective, conharmonic, and M-projective tensors. Our analysis yields several key physical results. Quasi-projectively flat space-times are necessarily Einstein and, under generic conditions, have constant curvature, making them conformally flat and of Petrov type O (de Sitter, anti-de Sitter, or Minkowski). For perfect fluids, this flatness forces the dark energy equation of state $p = -\sigma $ $p = - σ$ , i.e., a cosmological constant. A divergence-free quasi-projective tensor implies the Ricci tensor is of Codazzi type, producing a Yang pure space in Gray’s subspaces $\textbf{B}$ $B$ and $\textbf{B}'$ $B^{'}$ . Also, we provide two examples to validate our results. In f(R)-gravity vacuum solutions, the Codazzi condition forces $\nabla _j R$ $\nabla_{j} R$ to be an eigenvector of the Ricci tensor, and the Ricci tensor decomposes into a Weyl-contracted term plus a quasi-Einstein term. Finally, we prove that pseudo quasi-projectively symmetric space-times are of Petrov type I, D, or O, with an irrotational, acceleration-free covector.