---
title: Sasakian and Co-Kähler Structures
url: https://www.emergentmind.com/topics/sasakian-and-co-kahler-structures
type: topic
---

# Sasakian and Co-Kähler Structures

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 [2205.12067][2512.21378].

## 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
\[
g(\phi X, \phi Y) = g(X, Y) - \eta(X)\eta(Y), \quad \forall X, Y
\]
defines an almost-contact metric structure. The associated fundamental $2$-form is $\Omega(X, Y) = g(X, \phi Y)$ with the nondegeneracy condition $\eta \wedge \Omega^n \ne 0$.

Normality—the vanishing of the Nijenhuis tensor
\[
N = [\phi, \phi] + 2\, d\eta \otimes \xi
\]
—ensures that the almost-contact metric structure aligns with integrable geometric conditions. Within this framework, key subclasses are:

- **Sasakian**: $N = 0, \ d\eta = \Omega$
- **co-Kähler (K-cosymplectic)**: $N = 0, \ d\eta=0, \ d\Omega=0$
- **quasi-Sasakian**: $N = 0, \ d\Omega=0$ (no restriction on $d\eta$ beyond being basic).

This taxonomy organizes the landscape of contact-type geometries and sets the basis for higher-level generalizations [2512.21378].

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

Sasakian manifolds generalize the notion of Kähler geometry to odd dimensions: the Riemannian cone $(C(M), \bar{g}) = (M \times \mathbb{R}^+, dr^2 + r^2 g)$ is Kähler if and only if $(M, \phi, \xi, \eta, g)$ is Sasakian. The co-Kähler structure arises as the product of a Kähler manifold with a circle, characterized by both $\eta$ and $\Omega$ being closed and the metric satisfying adapted compatibility conditions.

In three dimensions, quasi-Sasakian manifolds are completely classified: any closed orientable $3$-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 $d\eta = r\,\Omega + d\alpha$ with $r$ constant and $d\alpha$ exact basic, governs the transition:
- If $r=0$, the manifold can be deformed to a co-Kähler structure.
- If $r \ne 0$, it may be deformed to a Sasakian structure [2512.21378].

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 $(\Phi, E_+, E_-)$ where
- $\Phi \in \operatorname{End}(TM \oplus T^*M)$ is skew-adjoint and satisfies $\Phi^3 + \Phi = 0$,
- $E_+, E_- \in \Gamma(TM \oplus T^*M)$ are null with respect to the natural inner product and normalized by $2\langle E_+, E_-\rangle = 1$.

A generalized contact metric manifold is normal if both $L_+$ and $L_-$ (the $\pm i$-eigenbundles) are Courant-involutive, and $[E_+, E_-] = 0$ in the Courant bracket. On any such normal manifold there exists a canonically defined Poisson bivector,
\[
\Pi_M = \Pi_0 + e_+ \wedge e_-
\]
where $e_\pm = \operatorname{pr}_{TM}(E_\pm)$ and $\Pi_0$ is the canonical transverse Poisson structure. This Poisson tensor’s properties completely encode the distinction between the Sasakian and co-Kähler worlds [2205.12067].

## 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 $(M,\Phi,E_+,E_-,G)$, form $\eta = \operatorname{pr}_{T^*M}(E_+) + \operatorname{pr}_{T^*M}(E_-)$ and define
\[
I + e^t (d\eta)\Pi_M \in \operatorname{End}(T^*M),
\]
requiring this operator to be invertible for all $t \in \mathbb{R}$. If $d\eta \ne 0$ 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 [2205.12067].

From a cohomological perspective, co-Kählerity is characterized by the vanishing of the Poisson cohomology class $[\Pi_M] \cup [d\eta] \in H^2_\Pi(M)$, equivalently by the vanishing of the basic torsion $T = d\eta|_{\ker\eta} \equiv 0$. For Sasakian structures, this torsion is everywhere nondegenerate and $d\eta$ pairs to the symplectic leaf of $\Pi_M$.

## 5. Foliation, Basic Cohomology, and Topological Classification

The Reeb foliation defined by the vector field $\xi$ underlies the deeper topological structure of Sasakian and co-Kähler manifolds. In dimension three, any 2-form $d\eta$ is basic with respect to this foliation, since normality enforces $\iota_\xi d\eta = 0$. The basic cohomology $H_B^2(\mathcal{F}_\Omega) \cong \mathbb{R}$ is always generated by $[\Omega]$ under standard hypotheses [2512.21378]. This allows expressions of the form
\[
d\eta = r\,\Omega + d\alpha
\]
with $\alpha$ basic. As shown in [2512.21378], the possible diffeomorphism types admitting quasi-Sasakian structures correspond precisely to:
- Sasakian cases: quotients of round $S^3$, nilmanifolds, or universal covers of $\mathrm{SL}(2, \mathbb{R})$.
- co-Kähler cases: $S^2 \times S^1$, torus bundles, or quotients of $(\mathbb{H}^2 \times \mathbb{R})$.

The first Betti number distinguishes these: $b_1=0$ or even for Sasakian, $b_1$ odd for co-Kähler (mapping torus) cases.

## 6. Examples and Illustrative Constructions

Key explicit models include:
- **Sasakian**: The standard Sasakian structure on $S^3$ or its lens space quotients, the left-invariant structure on the Heisenberg nilmanifold, and similar normal structures on compact quotients of $\widetilde{\mathrm{SL}(2, \mathbb{R})}$.
- **co-Kähler**: The $3$-torus $T^3 = T^2 \times S^1$ with $\eta=dt$, $\omega=dx \wedge dy$; the product $S^2 \times S^1$ with area form $\omega$ and $\eta = dt$.
- **Generalized structures**: On $SU(2)$ 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 $d\eta$ vanishing in directions transverse to $\xi$, 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 [2205.12067][2512.21378].

## Table: Characteristic Properties

| Structure Type      | Normality Condition | $d\eta$          | Basic $2$-form $d\Omega$ | Canonical Example         |
|---------------------|--------------------|------------------|--------------------------|--------------------------|
| Sasakian            | $N=0$              | $d\eta=\Omega$   | $d\Omega=0$              | $S^3$, nilmanifolds      |
| co-Kähler           | $N=0$              | $d\eta=0$        | $d\Omega=0$              | $T^3$, $S^2 \times S^1$  |
| quasi-Sasakian      | $N=0$              | $d\eta$ basic    | $d\Omega=0$              | 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.

Source: https://www.emergentmind.com/topics/sasakian-and-co-kahler-structures