Sasakian and Co-Kähler Structures

• Sasakian and Co-Kähler structures are geometric frameworks defined on odd-dimensional manifolds, combining contact, metric, and complex features analogous to Kähler geometry.
• Recent advances extend these notions to generalized and quasi settings, using Poisson-geometric criteria to distinguish and classify Sasakian, co-Kähler, and quasi-Sasakian cases.
• Examples and classification via basic cohomology and foliation reveal topological variations, with explicit models on S³, torus bundles, and Heisenberg nilmanifolds.

A Sasakian structure is a special type of geometric structure on odd-dimensional manifolds, merging contact, metric, and complex geometric concepts in a manner analogous to Kähler geometry on even dimensions. Co-Kähler structures, also known as K-cosymplectic in some literature, represent the co-symplectic and metric counterpart, capturing the product of Kähler and circle geometries. Recent work extends these notions to “generalized” and “quasi-” settings, providing sharp algebraic criteria that delineate the roles and interrelationships of Sasakian, co-Kähler, and related structures in both classical and Poisson-geometric frameworks (Talvacchia, 2022, Gnandi et al., 24 Dec 2025).

1. Foundational Notions: Almost-Contact Metric Structures

On a $(2n+1)$-dimensional smooth manifold $M$, an almost-contact structure is defined by a triple $(\phi, \xi, \eta)$, where $\phi: TM \rightarrow TM$ is a $(1,1)$-tensor, $\xi$ is a global vector field (the Reeb vector field), and $\eta$ is a $1$-form, subject to

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

Equipping this structure with a Riemannian metric $g$ so that

$M$0

defines an almost-contact metric structure. The associated fundamental $M$1-form is $M$2 with the nondegeneracy condition $M$3.

Normality—the vanishing of the Nijenhuis tensor

$M$4

—ensures that the almost-contact metric structure aligns with integrable geometric conditions. Within this framework, key subclasses are:

• Sasakian: $M$5
• co-Kähler (K-cosymplectic): $M$6
• quasi-Sasakian: $M$7 (no restriction on $M$8 beyond being basic).

This taxonomy organizes the landscape of contact-type geometries and sets the basis for higher-level generalizations (Gnandi et al., 24 Dec 2025).

2. Sasakian and co-Kähler Structures: Classical and Quasi Perspectives

Sasakian manifolds generalize the notion of Kähler geometry to odd dimensions: the Riemannian cone $M$9 is Kähler if and only if $(\phi, \xi, \eta)$0 is Sasakian. The co-Kähler structure arises as the product of a Kähler manifold with a circle, characterized by both $(\phi, \xi, \eta)$1 and $(\phi, \xi, \eta)$2 being closed and the metric satisfying adapted compatibility conditions.

In three dimensions, quasi-Sasakian manifolds are completely classified: any closed orientable $(\phi, \xi, \eta)$3-manifold with a quasi-Sasakian structure is either Sasakian or supports the structure of a Kähler mapping torus (and is thus co-Kähler). The deformation $(\phi, \xi, \eta)$4 with $(\phi, \xi, \eta)$5 constant and $(\phi, \xi, \eta)$6 exact basic, governs the transition:

• If $(\phi, \xi, \eta)$7, the manifold can be deformed to a co-Kähler structure.
• If $(\phi, \xi, \eta)$8, it may be deformed to a Sasakian structure (Gnandi et al., 24 Dec 2025).

Quasi-Sasakian geometry therefore interpolates between Sasakian and co-Kähler, with deformations implicitly classifying all quasi-Sasakian cases.

3. Generalized Structures and the Poisson-Geometric Perspective

Generalized contact and contact metric geometries, formulated in the language of Courant algebroids and Poisson geometry, extend the classical frameworks. A generalized almost-contact structure consists of $(\phi, \xi, \eta)$9 where

• $\phi: TM \rightarrow TM$0 is skew-adjoint and satisfies $\phi: TM \rightarrow TM$1,
• $\phi: TM \rightarrow TM$2 are null with respect to the natural inner product and normalized by $\phi: TM \rightarrow TM$3.

A generalized contact metric manifold is normal if both $\phi: TM \rightarrow TM$4 and $\phi: TM \rightarrow TM$5 (the $\phi: TM \rightarrow TM$6-eigenbundles) are Courant-involutive, and $\phi: TM \rightarrow TM$7 in the Courant bracket. On any such normal manifold there exists a canonically defined Poisson bivector,

$\phi: TM \rightarrow TM$8

where $\phi: TM \rightarrow TM$9 and $(1,1)$0 is the canonical transverse Poisson structure. This Poisson tensor’s properties completely encode the distinction between the Sasakian and co-Kähler worlds (Talvacchia, 2022).

4. Distinguishing Criteria and Algebraic Characterization

The precise algebraic separatrix between generalized Sasakian and generalized co-Kähler structures is an invertibility criterion formulated in terms of the canonical Poisson bivector. On a normal generalized contact metric manifold $(1,1)$1, form $(1,1)$2 and define

$(1,1)$3

requiring this operator to be invertible for all $(1,1)$4. If $(1,1)$5 and this invertibility holds, the structure is generalized Sasakian. Failure of invertibility signals a generalized co-Kähler (or co-symplectic) structure. This criterion elegantly recovers the Sasakian/co-Kähler dichotomy in both the classical and generalized contexts (Talvacchia, 2022).

From a cohomological perspective, co-Kählerity is characterized by the vanishing of the Poisson cohomology class $(1,1)$6, equivalently by the vanishing of the basic torsion $(1,1)$7. For Sasakian structures, this torsion is everywhere nondegenerate and $(1,1)$8 pairs to the symplectic leaf of $(1,1)$9.

5. Foliation, Basic Cohomology, and Topological Classification

The Reeb foliation defined by the vector field $\xi$0 underlies the deeper topological structure of Sasakian and co-Kähler manifolds. In dimension three, any 2-form $\xi$1 is basic with respect to this foliation, since normality enforces $\xi$2. The basic cohomology $\xi$3 is always generated by $\xi$4 under standard hypotheses (Gnandi et al., 24 Dec 2025). This allows expressions of the form

$\xi$5

with $\xi$6 basic. As shown in (Gnandi et al., 24 Dec 2025), the possible diffeomorphism types admitting quasi-Sasakian structures correspond precisely to:

• Sasakian cases: quotients of round $\xi$7, nilmanifolds, or universal covers of $\xi$8.
• co-Kähler cases: $\xi$9, torus bundles, or quotients of $\eta$0.

The first Betti number distinguishes these: $\eta$1 or even for Sasakian, $\eta$2 odd for co-Kähler (mapping torus) cases.

6. Examples and Illustrative Constructions

Key explicit models include:

• Sasakian: The standard Sasakian structure on $\eta$3 or its lens space quotients, the left-invariant structure on the Heisenberg nilmanifold, and similar normal structures on compact quotients of $\eta$4.
• co-Kähler: The $\eta$5-torus $\eta$6 with $\eta$7, $\eta$8; the product $\eta$9 with area form $1$0 and $1$1.
• Generalized structures: On $1$2 with standard contact form, the Poisson bivector is trivial on the transverse leaves, yielding an invertible gauge transformation and thus a Sasakian structure. On products with $1$3 vanishing in directions transverse to $1$4, the invertibility criterion fails, and these are genuinely co-Kähler.

This dichotomy and the explicit Poisson-geometric constructs provide a canonical, algebraically precise mechanism distinguishing Sasakian from co-Kähler geometries in both classical and generalized settings (Talvacchia, 2022, Gnandi et al., 24 Dec 2025).

Table: Characteristic Properties

Structure Type	Normality Condition	$1$5	Basic $1$6-form $1$7	Canonical Example
Sasakian	$1$8	$1$9	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$0	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$1, nilmanifolds
co-Kähler	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$2	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$3	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$4	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$5, $$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$6
quasi-Sasakian	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$7	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$8 basic	$$\phi^2 = -\mathrm{Id} + \eta \otimes \xi, \qquad \eta(\xi) = 1.$$9	mapping tori, both above

The table presents the structural conditions and explicit instances distinguishing each case, directly reflecting the connections and dichotomies articulated in the cited works.