Trans-Sasakian Manifolds: A Unified Framework

• Trans-Sasakian manifolds are (2n+1)-dimensional spaces with almost contact metric structures that generalize both contact and cosymplectic geometries.
• They are defined by a pair of smooth functions (α, β) that determine the interpolation between Sasakian, Kenmotsu, and cosymplectic conditions.
• Their rich transverse, curvature, and harmonic properties drive applications in geometric flows, soliton solutions, and complex structure constructions.

A trans-Sasakian manifold is a $(2n+1)$-dimensional smooth manifold equipped with an almost contact metric structure that generalizes both Sasakian and Kenmotsu geometries. These manifolds interpolate between contact and cosymplectic geometries and possess rich transverse and variational structures, making them central in the study of contact metric, Hermitian, and non-Kähler geometries, as well as in the analysis of geometric flows and harmonic mappings.

1. Structure of Trans-Sasakian Manifolds

Let $(M^{2n+1},\phi,\xi,\eta,g)$ be an almost contact metric manifold, where $\phi$ is a $(1,1)$ tensor field, $\xi$ is a global vector field (the Reeb field), $\eta$ is a 1-form, and $g$ is a Riemannian metric. These satisfy

$$\eta(\xi)=1,\qquad \phi^2 = -\mathrm{Id} + \eta\otimes\xi,\qquad g(\phi X, \phi Y) = g(X,Y) - \eta(X)\eta(Y)$$

for all vector fields $X,Y$ on $M$. The fundamental $2$-form is $\Phi(X,Y) = g(X, \phi Y)$.

A trans-Sasakian structure of type $(\alpha,\beta)$, for smooth functions $\alpha, \beta$ on $M$, is defined by the normality condition (vanishing Nijenhuis torsion) and the covariant derivative condition

$$(\nabla_X\phi)Y = \alpha\bigl(g(X,Y)\xi - \eta(Y)X\bigr) + \beta\bigl(g(\phi X, Y)\xi - \eta(Y)\phi X\bigr).$$

Equivalently,

$$d\eta = \alpha\,\Phi,\qquad d\Phi = 2\beta\,\eta\wedge\Phi.$$

Specialized classes within trans-Sasakian manifolds are given by fixed values of $(\alpha,\beta)$:

• $(1,0)$: Sasakian
• $(0,1)$: Kenmotsu
• $(0,0)$: Cosymplectic

2. Geometric and Curvature Properties

Trans-Sasakian manifolds possess a distinguished distribution $\mathcal{D} = \ker\eta$ invariant under $\phi$ and parallel with respect to a transverse Levi-Civita connection. Basic curvature formulas (for $X,Y\in\Gamma(TM)$) include: $$\nabla_X \xi = -\alpha\,\phi X - \beta\,\phi^2 X, \qquad (\nabla_X \eta)(Y) = \alpha\,g(X,\phi Y) + \beta\,g(\phi X,\phi Y).$$ The structure is normal; thus, the Nijenhuis torsion $N_\phi$ vanishes identically, and the distribution $\mathcal{D}$ inherits a natural almost-Kählerian structure when $\beta \equiv 0$.

The curvature operator admits an explicit expression in terms of $\alpha$, $\beta$, $\phi$, and the transverse curvature. For $U,V,W\in\Gamma(\mathcal{D})$,

$$R(U,V)W = R^T(U,V)W + \alpha^2[\Phi(V,W)\phi U - 2\Phi(U,V)\phi W - \Phi(U,W)\phi V] + \cdots,$$

with the full formula given in (Yadav et al., 13 Nov 2025).

3. Canonical Examples and Classification

The local geometry of trans-Sasakian manifolds is determined by the pair $(\alpha,\beta)$:

• Sasakian: $\alpha=1$, $\beta=0$. Classical examples include $S^{2n+1}$ with the standard contact structure.
• Kenmotsu: $\alpha=0$, $\beta=1$; recovers the canonical warped metric example.
• Cosymplectic: $\alpha=\beta=0$; $M$ is locally a product of a symplectic manifold with $\mathbb R$.

In dimensions $\geq 5$, any trans-Sasakian manifold is either Sasakian, Kenmotsu, or cosymplectic (Yadav et al., 13 Nov 2025).

The table below summarizes the defining equations for primary subclasses:

Type	$(\alpha,\beta)$	$d\eta$	$d\Phi$	Contact condition
Sasakian	$(1,0)$	$\Phi$	$0$	$\eta\wedge(d\eta)^n\ne 0$
Kenmotsu	$(0,1)$	$0$	$2\eta\wedge\Phi$	$0$
Cosymplectic	$(0,0)$	$0$	$0$	$0$
Trans-Sasakian	$(\alpha, \beta)$	$\alpha\Phi$	$2\beta\eta\wedge\Phi$	Contact/Noncontact by $\alpha$

(Alegre et al., 2016, Yadav et al., 13 Nov 2025)

Compactness and Ricci conditions yield rigidity results, especially in dimension 3:

• If the Ricci operator $Q$ satisfies $Q(\xi) = \lambda \xi$, $\lambda \ne 0$, then $M$ is homothetic to a Sasakian manifold (Deshmukh et al., 2012).
• Cosymplecticity is characterized by $Q(\xi) = 0$ on compact, connected 3-manifolds (Deshmukh et al., 2012).

4. Hermitian and Harmonic Structures on Products

For products $M_1 \times M_2$ of trans-Sasakian manifolds, canonical Hermitian structures arise via the Morimoto–Tsukada–Watson construction (Yadav et al., 13 Nov 2025). Given $(M_i, \phi_i, \xi_i, \eta_i, g_i)$ of type $(\alpha_i,\beta_i)$, an almost complex structure $J$ and metric $g_{a,b}$ are defined, yielding an integrable Hermitian structure if both factors are normal.

The Dirichlet energy functional for (orthogonal) almost complex structures,

$E(J) = \int_M \|\nabla J\|^2\,\mathrm{vol}_g,$

admits critical points, called harmonic almost complex structures, characterized by the vanishing of the tension field: $[J,\nabla^*\nabla J]=0$. For trans-Sasakian products, harmonicity of $J$ reduces to checking vanishing of certain endomorphisms involving $\alpha_i,\beta_i$ (Yadav et al., 13 Nov 2025). Notably:

• All products of two Sasakian, Kenmotsu, or cosymplectic manifolds carry harmonic Morimoto–Tsukada–Watson complex structures.
• The Calabi–Eckmann manifolds $S^{2p+1}\times S^{2q+1}$ (Sasakian on both factors) admit harmonic Hermitian structures of this type.
• Products of cosymplectic manifolds are astheno-Kähler and hence yield harmonic complex structures.

5. Differential Equations and Soliton Geometries

Trans-Sasakian manifolds serve as settings for various soliton and curvature flow equations.

η-Einstein Solitons: In the 3-dimensional case, a trans-Sasakian manifold $(M^3, \phi, \xi, \eta, g)$ can support an η-Einstein soliton

$$\frac{1}{2} \mathcal{L}_V g + \mathrm{Ric} + (\lambda - r)g + \nu\,\eta\otimes\eta = 0,$$

where $V$ is a vector field, possibly collinear or torse-forming with respect to $\xi$. For $V=\xi$, the Ricci tensor is necessarily η-Einstein and the scalar curvature is constant (Ganguly et al., 2021).

η-Yamabe Solitons: Similarly, an η-Yamabe soliton on a trans-Sasakian 3-manifold is governed by

$L_V g = (r-\lambda)g + \mu\,\eta\otimes\eta.$

Such metrics force the scalar curvature to be constant and the manifold to be η-Einstein (Roy et al., 2020). Ricci symmetry and recurrent structures further constrain $(\alpha,\beta)$. Examples of trans-Sasakian 3-manifolds admitting such solitons include Kenmotsu structures on open domains in $\mathbb R^3$.

6. Submanifold Theory and Transverse Maps

Trans-Sasakian geometry is compatible with submanifold and map-theoretic constructions:

• Invariant Submanifolds: In the Lorentzian setting, a submanifold $N\subset M$ is invariant if $\xi\in TN$ and $\phi(TN)\subset TN$. Such submanifolds inherit strong restrictions, particularly via Tachibana operator techniques—most often, imposing vanishing conditions forces $N$ to be totally geodesic or places differential constraints on $(\alpha,\beta)$ (Atçeken et al., 2024).
• Clairaut Anti-invariant Riemannian Maps: Anti-invariant Riemannian maps from a Riemannian manifold $(N, h)$ to a trans-Sasakian manifold $(M, \varphi, \xi, \eta, g)$, satisfying $$\varphi(\mathrm{Ran}\,\pi_*)\subset (\mathrm{Ran}\,\pi_*)^\perp$$, are Clairaut if a specific geometric conservation law holds for geodesics. Harmonicity of such maps is equivalent to the minimality of the vertical distribution (Zaidi et al., 2023).

7. Almost Contact Curves and Dynamics

In the 3-dimensional case, the behavior of almost contact (Legendre) curves is controlled by the trans-Sasakian structure:

• Any Frenet curve $y:I\to M$ with tangent $T$ in the contact distribution ($\eta(T)=0$) inherits explicit curvature and torsion formulas depending on $\alpha$ and $\beta$: $$\kappa(s) = \sqrt{\vartheta(s)^2 + \beta(s)^2},\qquad \tau(s) = \left| \alpha(s) + \frac{\beta(s)\vartheta'(s) - \beta'(s)\vartheta(s)}{\vartheta(s)^2 + \beta(s)^2} \right|$$ with $\vartheta(s) = g(\nabla_T T, \phi T)$ (Srivastava, 2013). These invariants distinguish dynamical and sub-Riemannian properties, with explicit curvature-torsion behavior acting as signatures of the underlying ambient trans-Sasakian geometry.

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Trans-Sasakian geometry forms a bridge between contact metric structures and almost complex geometry, providing a broad categorical framework that unifies Sasakian, Kenmotsu, and cosymplectic geometries. Its roles in harmonicity, submanifold theory, soliton solutions, and the construction of canonical Hermitian and astheno-Kähler metrics contribute substantially to geometric analysis, with continued developments illuminating connections with mathematical physics and geometric flows (Yadav et al., 13 Nov 2025, Ganguly et al., 2021, Roy et al., 2020, Srivastava, 2013, Zaidi et al., 2023, Atçeken et al., 2024, Alegre et al., 2016, Deshmukh et al., 2012).