On Lorentzian torse-forming structure manifolds
Abstract
In this paper we introduce a class of non-flat Lorentzian manifolds, called Lorentzian torse-forming structure manifolds (briefly, LTFSMs). This class contains Lorentzian concircular structure manifolds (briefly, LCSMs), LSP-Sasakian manifolds, LP-Sasakian manifolds with coefficient α, LP-α-Sasakian manifolds, LP-Sasakian manifolds, and LP-α-Kenmotsu manifolds. The interrelations among these various classes of manifolds are illustrated in Figures 1 and 2. We investigate several geometric properties of LTFSMs. In particular, we show that the expansion scalar, given by the divergence of ξ, is nonzero and nonconstant, and that the flow of ξ is irrotational. In an LCSM, the Ricci operator commutes with the structure tensor ϕ [12]. However, it is shown that, in an LTFSM, the Ricci operator does not generally commute with ϕ. If the gradient of the torse-forming scalar α is collinear with the torse-forming vector field ξ, then the Ricci operator commutes with ϕ, and the manifold reduces to an LCSM. It is known that a conformally flat LCSM is quasi-Einstein and hence is a manifold of quasi-constant curvature. We show that a conformally flat LTFSM is a generalized quasi-Einstein manifold in the sense of Chaki [1] and is also a manifold of generalized quasi-constant curvature. Several curvature-related properties of LTFSMs are established, and their sectional curvature is determined. The existence of LTFSMs is demonstrated through several proper nontrivial examples. We also present examples that are simultaneously LCSMs and LTFSMs, examples of LTFMs characterizing twisted cubic spacetimes, and generalized Robertson-Walker (GRW) spacetimes characterizing LCSMs. Finally, we provide examples of LTFSMs in which the vector field ξ is a unit timelike torse-forming vector field but is not concircular; see Examples 5.6--5.8.
Related
- Paired Statistical Solitons and Torse-Forming Rigidity on Statistical Manifolds
- Area-constrained Willmore surfaces in Lorentzian manifolds: curvature inequalities and the Hawking quasi-local energy
- On the structure of $2$-step nilpotent Lorentzian naturally reductive Lie groups
- On the Structure of Sub-Riemannian Geodesic Orbit Manifolds
- On the Structure of the Cut Locus in Certain Classes of Warped Product Manifolds
BibTeX
@misc{shaikh2026lorentzian,
title = {{On Lorentzian torse-forming structure manifolds}},
author = {Absos Ali Shaikh},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28651},
eprint = {2610.28651},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28651},
note = {CC BY-SA 4.0},
}
RIS
TY - UNPB
TI - On Lorentzian torse-forming structure manifolds
AU - Absos Ali Shaikh
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2610.28651
UR - https://arxiv.org/abs/2610.28651
AB - In this paper we introduce a class of non-flat Lorentzian manifolds, called Lorentzian torse-forming structure manifolds (briefly, LTFSMs). This class contains Lorentzian concircular structure manifolds (briefly, LCSMs), LSP-Sasakian manifolds, LP-Sasakian manifolds with coefficient α, LP-α-Sasakian manifolds, LP-Sasakian manifolds, and LP-α-Kenmotsu manifolds. The interrelations among these various classes of manifolds are illustrated in Figures 1 and 2. We investigate several geometric properties of LTFSMs. In particular, we show that the expansion scalar, given by the divergence of ξ, is nonzero and nonconstant, and that the flow of ξ is irrotational. In an LCSM, the Ricci operator commutes with the structure tensor ϕ [12]. However, it is shown that, in an LTFSM, the Ricci operator does not generally commute with ϕ. If the gradient of the torse-forming scalar α is collinear with the torse-forming vector field ξ, then the Ricci operator commutes with ϕ, and the manifold reduces to an LCSM. It is known that a conformally flat LCSM is quasi-Einstein and hence is a manifold of quasi-constant curvature. We show that a conformally flat LTFSM is a generalized quasi-Einstein manifold in the sense of Chaki [1] and is also a manifold of generalized quasi-constant curvature. Several curvature-related properties of LTFSMs are established, and their sectional curvature is determined. The existence of LTFSMs is demonstrated through several proper nontrivial examples. We also present examples that are simultaneously LCSMs and LTFSMs, examples of LTFMs characterizing twisted cubic spacetimes, and generalized Robertson-Walker (GRW) spacetimes characterizing LCSMs. Finally, we provide examples of LTFSMs in which the vector field ξ is a unit timelike torse-forming vector field but is not concircular; see Examples 5.6--5.8.
ER -