Hyperbolic solitons on trans-Sasakian space forms and its submanifolds

Abstract: Kong and Liu introduced the concept of hyperbolic Ricci flow in 2007 and used it to study the wave character of metrics. After that, many mathematicians have used this new geometric flow to study the evolution of manifolds and their structures. Ricci solitons and hyperbolic Ricci solitons are self-similar solitons of the Ricci flow and hyperbolic Ricci flow respectively. In this paper, we introduce the concept of hyperbolic $*-$Ricci solitons and hyperbolic Ricci-Yamabe solitons on a trans-Sasakian space forms and characterized the nature of some hyperbolic solitons. Additionally, we deduce the Ricci tensors of submanifolds of trans-Sasakian space forms and conformal trans-Sasakian space form and found the nature of solitons on submanifolds. Finally, we have included an example which will justify our result.

• The paper introduces hyperbolic *-Ricci and Ricci-Yamabe solitons and classifies their shrinking, steady, or expanding behavior through curvature thresholds involving the trans-Sasakian functions α, β, and the soliton constants.
• It proves that the studied three-dimensional trans-Sasakian space forms are η-Einstein and shows that conformal changes modify soliton parameters through the conformal factor, its Laplacian, and gradient norm rather than curvature alone.
• For totally umbilical two-dimensional submanifolds containing the Reeb field, the paper derives mean-curvature rigidity by fixing the squared mean curvature as ||H||² = μ − 2(α² − β²), and verifies the theory with an explicit β-Kenmotsu example.

Overview

This paper studies self-similar solutions of hyperbolic geometric flows on three-dimensional trans-Sasakian space forms and their submanifolds. The hyperbolic geometric flow, introduced by Kong and Liu to capture the wave character of metrics, evolves a metric by a second-order equation $\frac{\partial^2 g}{\partial t^2} + 2S = 0$, in contrast to the parabolic Ricci flow of Hamilton. Building on the hyperbolic Ricci soliton of Faraji–Azami–Fasihi-Ramandi, the authors introduce two new soliton notions — the hyperbolic $*$-Ricci soliton and the hyperbolic Ricci-Yamabe soliton — and classify their behavior (shrinking, steady, expanding) on trans-Sasakian space forms, conformal trans-Sasakian space forms, and totally umbilical submanifolds thereof.

Background structures

The ambient setting is an almost contact metric manifold $(M,\varphi,\xi,\eta,g)$ that is trans-Sasakian, i.e., satisfies

$$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$

with $\alpha$, $\beta$ smooth functions; by Marrero's classification, trans-Sasakian manifolds of dimension $\geq 5$ are locally cosymplectic, $\alpha$-Sasakian, or $\beta$-Kenmotsu. A trans-Sasakian space form carries constant $\varphi$-sectional curvature $*$0, with Ricci tensor $*$1 and scalar curvature $*$2. Throughout, the authors assume $*$3 and take the soliton vector field to be the Reeb field $*$4.

Two computational identities drive all results: since $*$5, one obtains

$*$6

Thus both Lie derivatives are pointwise determined by $*$7 alone, which is why every classification below reduces to algebraic conditions on $*$8, $*$9, $(M,\varphi,\xi,\eta,g)$0.

Hyperbolic Ricci-Yamabe flow and soliton

The paper defines the hyperbolic Ricci-Yamabe flow

$(M,\varphi,\xi,\eta,g)$1

where $(M,\varphi,\xi,\eta,g)$2, interpolating between the hyperbolic Ricci flow ($(M,\varphi,\xi,\eta,g)$3) and a hyperbolic analogue of the Yamabe flow ($(M,\varphi,\xi,\eta,g)$4). Via a self-similar ansatz $(M,\varphi,\xi,\eta,g)$5 with $(M,\varphi,\xi,\eta,g)$6, $(M,\varphi,\xi,\eta,g)$7, stationary solutions satisfy

$(M,\varphi,\xi,\eta,g)$8

Theorem (Ricci-Yamabe case). A three-dimensional trans-Sasakian space form with $(M,\varphi,\xi,\eta,g)$9 admitting such a soliton along $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$0 is $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$1-Einstein, with

$$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$2

Consequently the soliton is shrinking, steady, or expanding according as $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$3, equality, or $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$4. The classification therefore depends on the $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$5-sectional curvature relative to a threshold set by the structure functions and the flow parameters.

Hyperbolic $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$6-Ricci solitons

The authors first compute the $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$7-Ricci tensor of a trans-Sasakian space form:

$$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$8

which in dimension three specializes to $$(\nabla_X\varphi)Y = \alpha\,[g(X,Y)\xi - \eta(Y)X] + \beta\,[g(\varphi X,Y)\xi - \eta(Y)\varphi X],$$9; hence every three-dimensional trans-Sasakian space form is automatically $\alpha$0-$\alpha$1-Einstein. This structural fact underlies the following classifications.

For the classical $\alpha$2-Ricci soliton $\alpha$3 with $\alpha$4, contraction yields $\alpha$5: the soliton shrinks when $\alpha$6, is steady at $\alpha$7, and expands when $\alpha$8. Notably, this condition is independent of $\alpha$9 and $\beta$0.

For the newly introduced hyperbolic $\beta$1-Ricci soliton,

$\beta$2

the authors obtain $\beta$3, so the soliton expands, is steady, or shrinks according as $\beta$4, equality, or $\beta$5. Here the threshold involves only $\beta$6 and $\beta$7, not $\beta$8.

Hyperbolic Ricci, conformal Ricci, and Yamabe solitons

Three further classifications follow from the same mechanism:

Soliton type	Key relations	Shrinking condition
Hyperbolic Ricci	$\beta$9, $\geq 5$0	$\geq 5$1
Hyperbolic conformal Ricci	$\geq 5$2, $\geq 5$3	$\geq 5$4
Hyperbolic Yamabe	$\geq 5$5, $\geq 5$6	$\geq 5$7

In each case the underlying manifold is shown to be $\geq 5$8-Einstein. The hyperbolic Yamabe result is particularly rigid: $\geq 5$9 forces the soliton type to be dictated entirely by the sign of the $\alpha$0-function of the trans-Sasakian structure, independent of curvature.

Conformal trans-Sasakian space forms

For a conformal change $\alpha$1 preserving the trans-Sasakian type, the connection transforms via the Lee vector field $\alpha$2, and the Ricci tensors relate by

$\alpha$3

where $\alpha$4. Substituting into the hyperbolic Ricci soliton equation gives

$\alpha$5

The soliton parameter thus depends explicitly on the conformal factor through its Laplacian and gradient norm, so no fixed shrinking/steady/expanding dichotomy in terms of $\alpha$6 alone survives under conformal deformation — a qualitative departure from the unwarped case.

Submanifolds

For a two-dimensional totally umbilical submanifold $\alpha$7 of a three-dimensional trans-Sasakian space form containing $\alpha$8, the Gauss equation together with umbilicity ($\alpha$9) yields

$\beta$0

Imposing the hyperbolic Ricci soliton equation then forces

$\beta$1

The first relation is a rigidity statement: existence of the soliton pins down the squared mean curvature of the submanifold in terms of the ambient structure functions and the soliton constant $\beta$2.

Example

The authors construct an explicit model on $\beta$3 with frame $\beta$4, $\beta$5, $\beta$6. This is a trans-Sasakian manifold of type $\beta$7 (hence $\beta$8-Kenmotsu) with $\beta$9-sectional curvature $\varphi$0 and Ricci tensor $\varphi$1. Taking $\varphi$2, direct computation of the Lie derivatives gives $\varphi$3 from the soliton equation, while the general formulas give $\varphi$4, $\varphi$5 — consistent with the derived relation. The example confirms the classification theorems in a concrete $\varphi$6-Kenmotsu space form.

Limitations and open questions

Several restrictions qualify the results. All classifications assume the soliton vector field is the Reeb field $\varphi$7 rather than an arbitrary vector field or a gradient potential, so the analysis does not address non-Reeb or gradient hyperbolic solitons on these spaces. The hypothesis $\varphi$8 excludes cosymplectic and $\varphi$9-Sasakian cases, where the identities for $*$00 degenerate and separate treatment would be required. The submanifold theorem is restricted to dimension two and total umbilicity; whether analogous mean-curvature rigidity holds for higher-dimensional or non-umbilical submanifolds remains open. Finally, the conformal case produces $*$01 in terms of $*$02, $*$03, and $*$04 without yielding a clean sign-based classification, leaving open the question of which conformal factors admit shrinking or expanding hyperbolic solitons.

Conclusion

The paper extends the theory of hyperbolic geometric flow solitons to contact metric geometry by introducing hyperbolic $*$05-Ricci and hyperbolic Ricci-Yamabe solitons and classifying them, together with hyperbolic Ricci, conformal Ricci, and Yamabe solitons, on three-dimensional trans-Sasakian space forms. The unifying mechanism — explicit formulas for the first and second Lie derivatives of the metric along $*$06 — reduces each classification to threshold inequalities on the $*$07-sectional curvature $*$08 involving $*$09, $*$10, and the soliton constants. The conformal and submanifold results show how these thresholds deform under conformal change and how soliton existence imposes mean-curvature rigidity, and the explicit $*$11-Kenmotsu example verifies the framework computationally.