Lorentzian Trans-Sasakian Space Form

• Lorentzian trans-Sasakian space forms are (2n+1)-dimensional manifolds combining a Lorentzian metric, almost contact structure (φ, ξ, η), and constant φ-holomorphic sectional curvature.
• They feature smooth structure functions α and β that dictate deviation from classical Sasakian and Kenmotsu cases, leading to explicit curvature tensor identities.
• The study shows that, for V = ξ, both hyperbolic Ricci and conformal Ricci solitons enforce an η-Einstein condition, clarifying soliton behavior in indefinite almost contact geometry.

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" (Mondal et al., 22 Sep 2025), 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 (Mondal et al., 22 Sep 2025).

1. Defining structure

Let $M$ be a $(2n+1)$-dimensional smooth manifold. A Lorentzian trans-Sasakian structure on $\phi$0 consists of a Lorentzian metric $\phi$1 of index one, a $\phi$2-tensor $\phi$3, a vector field $\phi$4 (the Reeb field), and a $\phi$5-form $\phi$6, satisfying, for all $\phi$7,

$\phi$8

and

$\phi$9

Within this structure, $c$0 is timelike, since $c$1, and $c$2 is skew-symmetric in the sense that

$c$3

These relations specify the ambient almost contact Lorentzian geometry studied in (Mondal et al., 22 Sep 2025).

The trans-Sasakian condition is expressed through the Levi-Civita connection $c$4 of $c$5 and the smooth structure functions $c$6: $c$7 Equivalently,

$c$8

For $c$9, this guarantees that $(\phi,\xi,\eta)$0 measure the deviation from both Sasakian $(\phi,\xi,\eta)$1 and Kenmotsu $(\phi,\xi,\eta)$2 cases (Mondal et al., 22 Sep 2025).

A Lorentzian trans-Sasakian space form is one for which the sectional curvature of every $(\phi,\xi,\eta)$3-holomorphic $(\phi,\xi,\eta)$4-plane is a constant $(\phi,\xi,\eta)$5. The abstract characterizes this as a special type of space form “in which the nature of even and odd dimensional space form both exist” (Mondal et al., 22 Sep 2025). 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 $(\phi,\xi,\eta)$6 with $(\phi,\xi,\eta)$7 and $(\phi,\xi,\eta)$8 constant, the Riemann curvature tensor is given by

$(\phi,\xi,\eta)$9

The paper states that this may be obtained by combining the general $\alpha,\beta$0-sectional curvature identity with Bianchi symmetries (Mondal et al., 22 Sep 2025).

Contracting in the $\alpha,\beta$1 slots yields the Ricci tensor: $\alpha,\beta$2 The scalar curvature is

$\alpha,\beta$3

These formulas isolate the dependence of the intrinsic curvature on the constants $\alpha,\beta$4. In particular, the Ricci tensor differs from a pure $\alpha,\beta$5-Einstein form by the skew term $\alpha,\beta$6, a point that becomes decisive in the soliton analysis.

3. The $\alpha,\beta$7-Einstein condition

A Lorentzian manifold $\alpha,\beta$8 is called $\alpha,\beta$9-Einstein if

$\alpha,\beta,c$0

for some constants $\alpha,\beta,c$1 (Mondal et al., 22 Sep 2025). In the present setting, the explicit Ricci tensor obtained from the curvature computation has the form

$\alpha,\beta,c$2

Accordingly, the paper notes that the general Ricci tensor is of $\alpha,\beta,c$3-Einstein type plus the extra skew piece $\alpha,\beta,c$4.

The same source therefore observes that, in order to become $\alpha,\beta,c$5-Einstein, one must have $\alpha,\beta,c$6 or else arrange parameters so that this skew term disappears (Mondal et al., 22 Sep 2025). In the subsequent soliton constructions, the $\alpha,\beta,c$7-Einstein property is recovered through substitution into the hyperbolic soliton equations with the specific vector field choice $\alpha,\beta,c$8. A plausible implication is that the soliton ansatz imposes an algebraic rigidity strong enough to suppress the obstruction represented by the $\alpha,\beta,c$9-skew contribution.

4. Hyperbolic Ricci solitons

A Lorentzian manifold $V=\xi$0 admits a hyperbolic Ricci soliton if there is a vector field $V=\xi$1 and constants $V=\xi$2 such that

$V=\xi$3

The soliton is expanding, steady, or shrinking according as $V=\xi$4, $V=\xi$5, or $V=\xi$6 (Mondal et al., 22 Sep 2025).

For a Lorentzian trans-Sasakian space form with constant $V=\xi$7, taking $V=\xi$8 and using

$V=\xi$9

one computes

$\eta$0

Substitution into the hyperbolic Ricci soliton equation gives

$\eta$1

hence

$\eta$2

Thus $\eta$3 is $\eta$4-Einstein with

$\eta$5

Comparing this expression with the explicit Ricci tensor from the curvature calculation fixes $\eta$6 as

$\eta$7

The classification by sign then becomes

$\eta$8

corresponding to expanding, steady, and shrinking, respectively (Mondal et al., 22 Sep 2025). The paper states this result for odd-dimensional Lorentzian trans-Sasakian space forms.

5. Hyperbolic conformal Ricci solitons

Fix a time-dependent “conformal pressure” $\eta$9. A Lorentzian manifold $M$0 admits a hyperbolic conformal Ricci soliton if there exist $M$1 and constants $M$2 such that

$M$3

As in the hyperbolic Ricci soliton case, the soliton is expanding, steady, or shrinking according to the sign of $M$4 (Mondal et al., 22 Sep 2025).

For the same choice $M$5, the previously computed formulas

$M$6

lead to

$M$7

Therefore,

$M$8

so the manifold is again $M$9-Einstein.

The corresponding value of $(2n+1)$0 is

$(2n+1)$1

The paper states that expanding, steady, and shrinking behavior is again determined by the sign of this $(2n+1)$2 (Mondal et al., 22 Sep 2025). 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 $(2n+1)$3-Einstein conclusion.

6. Explicit three-dimensional model

An illustrative example is constructed after Mondal–Basu–Bhattacharyya (Mondal et al., 22 Sep 2025). Let

$(2n+1)$4

with coordinates $(2n+1)$5, and define

$(2n+1)$6

The Lorentzian metric is specified by

$(2n+1)$7

with all other pairings zero. Set

$(2n+1)$8

and define $(2n+1)$9 by

$\phi$00

Using the Koszul formula, one checks that

$\phi$01

so $\phi$02, and the $\phi$03-sectional curvature is $\phi$04 (Mondal et al., 22 Sep 2025). The nonzero Riemann and Ricci components are reported to verify the general formulas derived for the space-form case.

Choosing $\phi$05, one computes

$\phi$06

and the hyperbolic Ricci soliton equation holds with

$\phi$07

Accordingly, this manifold is an explicit expanding, steady, or shrinking soliton according to $\phi$08 (Mondal et al., 22 Sep 2025). 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 $\phi$09 and the distinguished choice $\phi$10, both hyperbolic Ricci solitons and hyperbolic conformal Ricci solitons imply the $\phi$11-Einstein condition (Mondal et al., 22 Sep 2025). 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$12-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 $\phi$13, so the $\phi$14-Einstein property does not follow automatically from the space-form assumption alone (Mondal et al., 22 Sep 2025). 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 $\phi$15 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 $\phi$16, and in the conformal case $\phi$17, governing both curvature and soliton type in a tightly coupled manner (Mondal et al., 22 Sep 2025).