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On the existence of various generalizations of semisymmetric and pseudosymmetric type manifolds

Absos Ali Shaikh

TL;DR

The paper surveys a broad spectrum of generalized semisymmetric and pseudosymmetric curvature conditions in semi-Riemannian geometry, emphasizing the existence and properness of these structures across diverse spacetimes. It systematically develops the tensor framework (including $R$, $S$, $C$, $P$, and Tachibana-type constructions) and introduces hierarchies such as RT$_n$, GRT$_n$, Ein$(k)$, and generalized quasi-Einstein models, illustrating them with explicit spacetime examples. A central theme is distinguishing genuine generalizations from consequences of simpler conditions, with numerous explicit spacetimes (e.g., Schwarzschild, pp-waves, Vaidya, LTB, Lifshitz, Kantowski–Sachs, Melvin) serving as evidence for proper existence and for the interplay between Weyl, Ricci, and projective curvature tensors. The work thus clarifies the landscape of curvature-restricted geometries relevant to general relativity and cosmology, providing a structured reference for researchers investigating curvature symmetries and their physical implications.

Abstract

The objective of the paper is to investigate a sequential study of different generalizations of semisymmetric and pseudosymmetric manifolds with their proper existence by several spacetimes. In the literature of differential geometry, there are many generalizations of such notions in various directions by involving different curvature tensors. In this paper, we have systematically and consecutively reviewed various generalized notions of semisymmetry, such as, Ricci semisymmetry, conformal semisymmetry, pseudosymmetry, Ricci pseudosymmetry, Ricci generalized pseudosymmetry, conformal pseudosymmetry, Ricci generalized Weyl pseudosymmetry and also many other semisymmetry type conditions. Most importantly, we have exhibited a plenty of suitable examples to examine the proper existence of such geometric structures, and they are physically significant as several spacetimes admit such geometric structures. By considering an immersion of Schwarzschild black hole metric, we have also provided a new example of Ricci pseudosymmetric manifolds. Finally, the interesting character of Weyl projective curvature tensors of type $(1,3)$ and $(0,4)$ are exhibited with their interesting examples.

On the existence of various generalizations of semisymmetric and pseudosymmetric type manifolds

TL;DR

The paper surveys a broad spectrum of generalized semisymmetric and pseudosymmetric curvature conditions in semi-Riemannian geometry, emphasizing the existence and properness of these structures across diverse spacetimes. It systematically develops the tensor framework (including , , , , and Tachibana-type constructions) and introduces hierarchies such as RT, GRT, Ein, and generalized quasi-Einstein models, illustrating them with explicit spacetime examples. A central theme is distinguishing genuine generalizations from consequences of simpler conditions, with numerous explicit spacetimes (e.g., Schwarzschild, pp-waves, Vaidya, LTB, Lifshitz, Kantowski–Sachs, Melvin) serving as evidence for proper existence and for the interplay between Weyl, Ricci, and projective curvature tensors. The work thus clarifies the landscape of curvature-restricted geometries relevant to general relativity and cosmology, providing a structured reference for researchers investigating curvature symmetries and their physical implications.

Abstract

The objective of the paper is to investigate a sequential study of different generalizations of semisymmetric and pseudosymmetric manifolds with their proper existence by several spacetimes. In the literature of differential geometry, there are many generalizations of such notions in various directions by involving different curvature tensors. In this paper, we have systematically and consecutively reviewed various generalized notions of semisymmetry, such as, Ricci semisymmetry, conformal semisymmetry, pseudosymmetry, Ricci pseudosymmetry, Ricci generalized pseudosymmetry, conformal pseudosymmetry, Ricci generalized Weyl pseudosymmetry and also many other semisymmetry type conditions. Most importantly, we have exhibited a plenty of suitable examples to examine the proper existence of such geometric structures, and they are physically significant as several spacetimes admit such geometric structures. By considering an immersion of Schwarzschild black hole metric, we have also provided a new example of Ricci pseudosymmetric manifolds. Finally, the interesting character of Weyl projective curvature tensors of type and are exhibited with their interesting examples.
Paper Structure (24 sections, 121 equations, 3 figures)

This paper contains 24 sections, 121 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proper existence of the classes of manifolds presented in Figure \ref{['Fig3']}
  4. Flat, Ricci flat and conformally flat manifold
  5. Roter type and generalized Roter type manifold
  6. Einstein, quasi-Einstein and generalized quasi-Einstein manifold
  7. Einstein manifold of level 2, 3 and 4
  8. Existence of various semisymmetric and pseudosymmetric type manifolds
  9. Semisymmetric manifold
  10. Ricci semisymmetric manifold
  11. Conformally semisymmetric manifold
  12. Pseudosymmetric manifold
  13. Ricci pseudosymmetric manifold
  14. Conformally pseudosymmetric manifold
  15. Ricci generalized pseudosymmetric manifold
  16. ...and 9 more sections

Figures (3)

  • Figure 1: Some generalizations of semisymmetric manifold.
  • Figure 2: Some generalizations of semisymmetric manifold due to Weyl conformal curvature.
  • Figure 3: Generalizations of flat manifold in different direction.

Theorems & Definitions (59)

  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4
  • Example 3.5
  • Example 3.6
  • Example 3.7
  • Example 3.8
  • Example 3.9
  • Example 3.10
  • ...and 49 more