---
title: Lorentzian Trans-Sasakian Space Form
url: https://www.emergentmind.com/topics/lorentzian-trans-sasakian-space-form
type: topic
---

# Lorentzian Trans-Sasakian Space Form

Searching arXiv for the cited paper and closely related work.
Lorentzian trans-Sasakian space forms are $(2n+1)$-dimensional smooth manifolds equipped with a Lorentzian trans-Sasakian structure and constant $\phi$-holomorphic sectional curvature $c$. In the formulation studied in "Curvature tensors and hyperbolic solitons on Lorentzian trans-Sasakian space form" [2509.18318], such manifolds carry a Lorentzian metric of index one together with the standard almost contact data $(\phi,\xi,\eta)$ and two smooth structure functions $\alpha,\beta$, with the Levi-Civita connection determining both the trans-Sasakian condition and the resulting curvature identities. The same work derives explicit formulas for the Riemann curvature tensor, Ricci tensor, and scalar curvature under constant $\alpha,\beta,c$, and shows that in the odd-dimensional case, the existence of hyperbolic Ricci solitons or hyperbolic conformal Ricci solitons for the choice $V=\xi$ forces the manifold to be $\eta$-Einstein [2509.18318].

## 1. Defining structure

Let $M$ be a $(2n+1)$-dimensional smooth manifold. A Lorentzian trans-Sasakian structure on $M$ consists of a Lorentzian metric $g$ of index one, a $(1,1)$-tensor $\phi$, a vector field $\xi$ (the Reeb field), and a $1$-form $\eta$, satisfying, for all $X,Y\in\mathfrak{X}(M)$,
\[
\eta(\xi)=1,\qquad \phi^2=-I+\eta\otimes\xi,\qquad \phi(\xi)=0,\qquad \eta\circ\phi=0,
\]
and
\[
g(\phi X,\phi Y)=g(X,Y)+\eta(X)\eta(Y).
\]
Within this structure, $\xi$ is timelike, since $g(\xi,\xi)=-1$, and $\phi$ is skew-symmetric in the sense that
\[
g(\phi X,Y)=-g(X,\phi Y).
\]
These relations specify the ambient almost contact Lorentzian geometry studied in [2509.18318].

The trans-Sasakian condition is expressed through the Levi-Civita connection $\nabla$ of $g$ and the smooth structure functions $\alpha,\beta$:
\[
(\nabla_X\phi)Y
= \alpha\,[\,g(X,Y)\xi+\eta(Y)X\,]
+ \beta\,[\,g(\phi X,Y)\xi+\eta(Y)\phi X\,].
\]
Equivalently,
\[
\nabla_X\xi=-\alpha\,\phi X+\beta\,[\,X-\eta(X)\xi\,].
\]
For $\beta\neq 0$, this guarantees that $\{\alpha,\beta\}$ measure the deviation from both Sasakian $(\beta=0)$ and Kenmotsu $(\alpha=0)$ cases [2509.18318].

A Lorentzian trans-Sasakian space form is one for which the sectional curvature of every $\phi$-holomorphic $2$-plane is a constant $c$. The abstract characterizes this as a special type of space form “in which the nature of even and odd dimensional space form both exist” [2509.18318]. This suggests that the class is being positioned as a Lorentzian contact-geometric setting that mixes features usually treated separately in neighboring frameworks.

## 2. Curvature tensor identities

For a Lorentzian trans-Sasakian space form $(M,g,\phi,\xi,\eta;\alpha,\beta;c)$ with $\alpha,\beta$ and $c$ constant, the Riemann curvature tensor is given by
\[
\begin{aligned}
4\,R(X,Y)Z
&= [3(\alpha^2-\beta^2)-c]\,[\,g(X,Z)Y-g(Y,Z)X\,] \\
&\quad + (\alpha^2-\beta^2+c)\Big\{\eta(Z)[\eta(Y)X-\eta(X)Y] \\
&\qquad + [\eta(Y)g(X,Z)-\eta(X)g(Y,Z)]\,\xi \\
&\qquad + g(X,\phi Z)\phi Y-g(Y,\phi Z)\phi X+2\,g(X,\phi Y)\phi Z\Big\} \\
&\quad + 8\Big\{[\eta(X)g(Y,\phi Z)-\eta(Y)g(X,\phi Z)]\,\xi \\
&\qquad + \eta(Z)[\eta(X)\phi Y-\eta(Y)\phi X]\Big\}.
\end{aligned}
\]
The paper states that this may be obtained by combining the general $\phi$-sectional curvature identity with Bianchi symmetries [2509.18318].

Contracting in the $X,Z$ slots yields the Ricci tensor:
\[
Ric(Y,Z)
= \frac12[\,n\,c-(3n-4)(\alpha^2-\beta^2)\,]\,g(Y,Z)
+ \frac12\,n\,(\alpha^2-\beta^2+c)\,\eta(Y)\eta(Z)
+ 2\,g(Y,\phi Z).
\]
The scalar curvature is
\[
\tau = n^2c-(3n^2-2n-2)(\alpha^2-\beta^2).
\]

These formulas isolate the dependence of the intrinsic curvature on the constants $\alpha,\beta,c$. In particular, the Ricci tensor differs from a pure $\eta$-Einstein form by the skew term $2\,g(\cdot,\phi\cdot)$, a point that becomes decisive in the soliton analysis.

## 3. The $\eta$-Einstein condition

A Lorentzian manifold $(M,g)$ is called $\eta$-Einstein if
\[
Ric=a\,g+b\,\eta\otimes\eta
\]
for some constants $a,b$ [2509.18318]. In the present setting, the explicit Ricci tensor obtained from the curvature computation has the form
\[
Ric = \text{(metric term)}+\text{($\eta\otimes\eta$ term)}+2\,g(\cdot,\phi\cdot).
\]
Accordingly, the paper notes that the general Ricci tensor is of $\eta$-Einstein type plus the extra skew piece $2\,g(\cdot,\phi\cdot)$.

The same source therefore observes that, in order to become $\eta$-Einstein, one must have $g(Y,\phi Z)\equiv 0$ or else arrange parameters so that this skew term disappears [2509.18318]. In the subsequent soliton constructions, the $\eta$-Einstein property is recovered through substitution into the hyperbolic soliton equations with the specific vector field choice $V=\xi$. A plausible implication is that the soliton ansatz imposes an algebraic rigidity strong enough to suppress the obstruction represented by the $\phi$-skew contribution.

## 4. Hyperbolic Ricci solitons

A Lorentzian manifold $(M,g)$ admits a hyperbolic Ricci soliton if there is a vector field $V$ and constants $\mu,\lambda$ such that
\[
L_V(L_V g)+2\,\lambda\,L_V g+2\,Ric=2\,\mu\,g.
\]
The soliton is expanding, steady, or shrinking according as $\lambda>0$, $\lambda=0$, or $\lambda<0$ [2509.18318].

For a Lorentzian trans-Sasakian space form with constant $\alpha,\beta$, taking $V=\xi$ and using
\[
\nabla_X\xi=-\alpha\,\phi X+\beta\,[\,X-\eta(X)\xi\,],
\]
one computes
\[
L_\xi g = 2\beta\,(g-\eta\otimes\eta),\qquad
L_\xi(L_\xi g)=(2\beta)^2\,(g-\eta\otimes\eta).
\]
Substitution into the hyperbolic Ricci soliton equation gives
\[
(2\beta)^2\,(g-\eta\otimes\eta)+2\lambda\,2\beta\,(g-\eta\otimes\eta)+2\,Ric=2\,\mu\,g,
\]
hence
\[
Ric = \bigl[\,\mu-2\beta(\beta+\lambda)\bigr]\,g
+2\beta(\beta+\lambda)\,\eta\otimes\eta.
\]
Thus $M$ is $\eta$-Einstein with
\[
a=\mu-2\beta(\beta+\lambda),\qquad
b=2\beta(\beta+\lambda).
\]

Comparing this expression with the explicit Ricci tensor from the curvature calculation fixes $\lambda$ as
\[
\lambda=\frac{2(n-1)(\alpha^2-\beta^2)+\mu}{4\beta}-\beta.
\]
The classification by sign then becomes
\[
\mu \gtrless 4\beta^2-2(n-1)(\alpha^2-\beta^2),
\]
corresponding to expanding, steady, and shrinking, respectively [2509.18318]. The paper states this result for odd-dimensional Lorentzian trans-Sasakian space forms.

## 5. Hyperbolic conformal Ricci solitons

Fix a time-dependent “conformal pressure” $p$. A Lorentzian manifold $(M,g)$ admits a hyperbolic conformal Ricci soliton if there exist $V$ and constants $\mu,\lambda$ such that
\[
L_V(L_V g)+2\,\lambda\,L_V g+2\,Ric
=
2\Bigl(\mu-\tfrac12\!\bigl(p+\tfrac{2}{2n+1}\bigr)\Bigr)\,g.
\]
As in the hyperbolic Ricci soliton case, the soliton is expanding, steady, or shrinking according to the sign of $\lambda$ [2509.18318].

For the same choice $V=\xi$, the previously computed formulas
\[
L_\xi g = 2\beta\,(g-\eta\otimes\eta),\qquad
L_\xi(L_\xi g)=(2\beta)^2\,(g-\eta\otimes\eta)
\]
lead to
\[
(2\beta)^2(g-\eta\otimes\eta)+2\lambda\,2\beta\,(g-\eta\otimes\eta)+2\,Ric
=
2\Bigl(\mu-\tfrac12\!(p+\tfrac{2}{2n+1})\Bigr)\,g.
\]
Therefore,
\[
Ric=\Bigl[\mu-\tfrac12\!(p+\tfrac{2}{2n+1})-2\beta(\beta+\lambda)\Bigr]\,g
+2\beta(\beta+\lambda)\,\eta\otimes\eta,
\]
so the manifold is again $\eta$-Einstein.

The corresponding value of $\lambda$ is
\[
\lambda=
\frac{2(n-1)(\alpha^2-\beta^2)+\mu-\tfrac12\!(p+\tfrac{2}{2n+1})}{4\beta}-\beta.
\]
The paper states that expanding, steady, and shrinking behavior is again determined by the sign of this $\lambda$ [2509.18318]. The parallelism with the hyperbolic Ricci soliton case is explicit in the presentation and indicates that the conformal pressure modifies the metric component of the equation without altering the basic $\eta$-Einstein conclusion.

## 6. Explicit three-dimensional model

An illustrative example is constructed after Mondal–Basu–Bhattacharyya [2509.18318]. Let
\[
M=\mathbb{R}^3\setminus\{y=0\},
\]
with coordinates $(x,y,z)$, and define
\[
e_1=e^z\partial_x,\qquad e_2=e^z\partial_y,\qquad e_3=\partial_z.
\]
The Lorentzian metric is specified by
\[
g(e_1,e_1)=g(e_2,e_2)=+1,\qquad g(e_3,e_3)=-1,
\]
with all other pairings zero. Set
\[
\xi=e_3,\qquad \eta(X)=-g(X,\xi),
\]
and define $\phi$ by
\[
\phi(e_1)=e_2,\qquad \phi(e_2)=-e_1,\qquad \phi(e_3)=0.
\]

Using the Koszul formula, one checks that
\[
\nabla_X\xi=\phi X+(X-\eta(X)\xi),
\]
so $(\alpha,\beta)=(1,1)$, and the $\phi$-sectional curvature is $c=1$ [2509.18318]. The nonzero Riemann and Ricci components are reported to verify the general formulas derived for the space-form case.

Choosing $V=\xi$, one computes
\[
L_V g=2(g-\eta\otimes\eta),\qquad
L_V(L_V g)=4(g-\eta\otimes\eta),
\]
and the hyperbolic Ricci soliton equation holds with
\[
\lambda = 1+\frac{\mu}{4}.
\]
Accordingly, this manifold is an explicit expanding, steady, or shrinking soliton according to $\mu\gtrless -4$ [2509.18318]. The example serves as a concrete realization of the abstract theory and justifies the general results stated in the paper.

## 7. Position within the studied geometry

The central conclusion of the cited work is that, for odd-dimensional Lorentzian trans-Sasakian space forms with constant $\alpha,\beta$ and the distinguished choice $V=\xi$, both hyperbolic Ricci solitons and hyperbolic conformal Ricci solitons imply the $\eta$-Einstein condition [2509.18318]. This places the soliton equations in direct interaction with the almost contact Lorentzian structure, rather than treating them as purely metric evolution constraints.

A common misconception in related contact-metric settings is that constant $\phi$-holomorphic sectional curvature by itself enforces an Einstein-type Ricci tensor. The formulas recorded here show otherwise: the Ricci tensor contains the additional skew term $2\,g(\cdot,\phi\cdot)$, so the $\eta$-Einstein property does not follow automatically from the space-form assumption alone [2509.18318]. Instead, it emerges in the soliton cases through the specific Lie-derivative identities generated by the Reeb field.

The paper’s final example also clarifies that the theory is not purely formal. The three-dimensional model on $\mathbb{R}^3\setminus\{y=0\}$ exhibits the structure equations, curvature identities, and soliton classification explicitly. This suggests that Lorentzian trans-Sasakian space forms provide a workable setting for studying hyperbolic solitons in indefinite almost contact geometry, with the constants $\alpha,\beta,c,\mu,\lambda$, and in the conformal case $p$, governing both curvature and soliton type in a tightly coupled manner [2509.18318].

Source: https://www.emergentmind.com/topics/lorentzian-trans-sasakian-space-form