---
title: Homogeneous Nearly Kähler Manifolds
url: https://www.emergentmind.com/topics/homogeneous-nearly-kahler-structure
type: topic
---

# Homogeneous Nearly Kähler Manifolds

A homogeneous nearly Kähler structure is a specific type of geometric structure found on a particular class of Riemannian manifolds equipped with an almost complex structure, where the nearly Kähler condition is preserved by a transitive Lie group action. These manifolds are distinguished by the property that their canonical $\mathrm{SU}(3)$-structure satisfies the strict nearly Kähler equations, and the geometry is uniform under the group action, leading to a fully explicit description in terms of Lie-theoretic and differential-geometric data. In dimension six, homogeneous nearly Kähler manifolds play a central role in special holonomy, Killing spinor, and $G_2$-geometry, and their classification is rigid and complete.

## 1. Definition and Characterization

A Riemannian manifold $(M, g, J)$ of real dimension $6$ is said to carry a strictly nearly Kähler structure if there exists an almost complex structure $J$ (not necessarily integrable) compatible with $g$, such that
\[
(\nabla_X J)X = 0 \quad \forall X \in TM, \qquad \text{and} \quad \nabla J \not\equiv 0,
\]
where $\nabla$ is the Levi-Civita connection of $g$. Equivalently, in terms of differential forms for the associated $\mathrm{SU}(3)$-structure $(\omega, \Psi)$, the strict nearly Kähler equations in dimension $6$ are
\[
d\omega = 3\,\psi^+, \qquad d\psi^- = -2\,\omega \wedge \omega,
\]
where $\omega(X,Y) = g(JX, Y)$, and $\Psi = \psi^+ + i \psi^-$ is a complex $(3,0)$-form with appropriate normalization and algebraic constraints. These equations are equivalent to the existence of a real Killing spinor and the Riemannian cone $(\mathbb{R}^+ \times M, dr^2 + r^2 g)$ having holonomy contained in $G_2$ [1601.04400]. In the homogeneous case, all geometric tensors are invariant under the transitive isometry group.

## 2. Classification of Homogeneous Nearly Kähler 6-Manifolds

Butruille and subsequent authors have established that, up to homothety, the only connected, simply connected, irreducible, strictly homogeneous nearly Kähler six-manifolds are the following four 3-symmetric spaces:
\[
\begin{array}{ll}
S^6 = G_2 / SU(3), & \text{(the 6-sphere)} \\
\mathbb{C}P^3 = Sp(2)/(Sp(1) \times U(1)), & \text{(complex projective 3-space)} \\
F_{1,2}(\mathbb{C}^3) = SU(3)/T^2, & \text{(full flag manifold)} \\
S^3 \times S^3 = (SU(2) \times SU(2) \times SU(2)) / \Delta SU(2), & \text{(product of spheres).}
\end{array}
\]
Each carries a unique (up to scaling and sign of $J$) invariant nearly Kähler structure, arising from the canonical $\mathbb{Z}_3$-grading (3-symmetric space) of the Lie algebra $\mathfrak{g}$ of $G$ [1006.2636, 1601.04400, 2411.05675]. There are no nontrivial homogeneous nearly Kähler free quotients except in the $S^3 \times S^3$ case, whose finite free quotients have been fully classified [1410.6912].

Homogeneous nearly Kähler structures in higher dimensions correspond to naturally reductive, compact, 3-symmetric spaces, with the isotropy $K$ as the holonomy of the canonical Hermitian connection [1006.2636]. In all compact homogeneous cases, the intrinsic torsion is of Gray–Hervella $W_1$-type.

## 3. Explicit Constructions and Structural Data

Each homogeneous nearly Kähler $6$-manifold $(M^{6}, g, J)$ admits an explicit Lie-theoretic construction:

- The tangent bundle $TM$ at the basepoint is identified with the complement $\mathfrak{m}$ in a reductive decomposition $\mathfrak{g} = \mathfrak{h} \oplus \mathfrak{m}$.
- The metric $g$ is induced by the (normalized) negative Killing form $-B$ restricted to $\mathfrak{m}$.
- The canonical almost complex structure $J$ is determined by the automorphism of order $3$ on $\mathfrak{g}$:
  \[
  J = \frac{2 \, \sigma|_{\mathfrak{m}} + \mathrm{Id}}{\sqrt{3}},
  \]
  where $\sigma$ is the 3-symmetry automorphism [1006.2636, 2411.05675].
- The intrinsic torsion $T$ is $T(X, Y) = -[X, Y]_{\mathfrak{m}}$, totally skew-symmetric, capturing the non-integrability of $J$.

In the case of $\mathbb{C}P^3$, the nearly Kähler structure is realized via the quaternionic Hopf fibration $\pi: S^7 \to \mathbb{C}P^3$, using the splitting of $TS^7$ into orthogonal summands (vertical and two horizontal distributions), projecting to two orthogonal subbundles $D^1$ (rank $4$) and $D^2$ (rank $2$) on $\mathbb{C}P^3$. The structure is parameterized by a real parameter $a > 0$ corresponding to the ratio of the metric scaling on these distributions:
- $g_a = g_1$ on $D^1$, $g_a = a \, g_1$ on $D^2$ (where $g_1$ is Fubini–Study).
- $J = -J_0$ on $D^1$, $J=+J_0$ on $D^2$ ($J_0$ is the standard Kähler structure).
- The nearly Kähler structure occurs precisely for $a=2$ [2601.18504].

## 4. Curvature and Isometry Groups

The geometry of homogeneous nearly Kähler structures is reflected in their explicit curvature tensors.

For $(\mathbb{C}P^3, g_a, J)$:
\[
R^a(X, Y)Z = \text{(explicit formula in terms of } a,\,J_0,\,J,\,P\text{ and metrics as given in [2601.18504])}.
\]
The Ricci tensor and scalar curvature are:
\[
\mathrm{Ric}_a(X, Y) = 4(1+1/a^2)\, g_a((Id-P)/2 \, X, Y) + 4(3a-1)/a^2\, g_a((Id+P)/2 \, X, Y),
\]
\[
S_a = 8(a^2 + 6a - 1)/a^2,
\]
and the Einstein condition (Ricci proportional to $g$) holds only for $a=1$ (Kähler–Einstein: Fubini–Study) or $a=2$ (nearly Kähler) [2601.18504].

Isometry groups are fully determined for each manifold and are summarized as follows [2411.05675, 2601.18504]:

| Manifold                  | Isometry Group                                                 |
|---------------------------|---------------------------------------------------------------|
| $S^6$                     | $O(7)$                                                        |
| $S^3 \times S^3$          | $(SU(2)^3/\{\pm 1\}) \rtimes S_3$                             |
| $\mathbb{C}P^3$           | $PSp(2) \rtimes \mathbb{Z}_2$ (for $a=2$)                     |
| $F_{1,2}(\mathbb{C}^3)$   | $PSU(3) \times S_3 \rtimes \mathbb{Z}_2$                      |

For $a\neq 1$, any isometry of $(\mathbb{C}P^3, g_a)$ must preserve the almost-product structure $P$, so the isometry group is contained in $Sp(2)$ (up to a discrete extension) [2601.18504].

## 5. Topology, Rigidity, and Deformation Theory

The topological invariants are as follows: $S^6$ is simply connected with trivial $H^2$; $S^3 \times S^3$ and its locally homogeneous finite quotients have $H^2=0$, and their higher rational Betti numbers are identical; $\mathbb{C}P^3$ and $F_{1,2}$ have $H^2 \cong \mathbb{Z}$ and $H^2 \cong \mathbb{Z}^2$, respectively [1601.04400, 1410.6912]. 

Homogeneous nearly Kähler 6-manifolds are rigid in the sense that they admit no nontrivial deformations as nearly Kähler structures. For $F_{1,2}$, infinitesimal deformations do exist (parametrized by $\xi \in \mathfrak{su}(3)$), but all are obstructed at second order by an explicit cubic invariant, so no genuine smooth families arise [1601.04400]. This contrasts with other $G$-structure geometries such as Sasaki–Einstein.

## 6. Special Features and Geometric Invariants

Homogeneous nearly Kähler structures can be characterized intrinsically as the unique strictly type-$W_1$ $SU(3)$-structures in dimension $6$, with real Killing spinors and Riemannian cones of $G_2$ holonomy [1601.04400]. 

Each structure displays special invariant tensors, such as product structures ($P$), triple decompositions, and multi-moment maps. For example, two-torus symmetric homogeneous nearly Kähler six-manifolds admit multi-moment maps whose critical loci and stabilizer jumps organize the orbit structure into trivalent graphs in the orbit space, with explicit manifestations in each homogeneous model [1911.12420].

Canonical fibrations occur in Types III and IV in the full classification, expressing homogeneous nearly Kähler spaces as Riemannian submersions over symmetric spaces with Hermitian or non-Hermitian fibers [1006.2636].

## 7. Extensions and Structure Theory

The homogeneity and 3-symmetry ensure that all geometric objects (connections, torsion, curvature) are $G$-invariant and calculable from the Lie algebra structure. All strictly homogeneous nearly Kähler manifolds arise via this construction, and the $SU(3)$-torsion always lies in the “W$_1$” component only [1006.2636].

Locally homogeneous nearly Kähler 6-manifolds are limited to quotients of $S^3 \times S^3$ by freely acting finite subgroups, whose classification leads to families of spherical space-form type bundles all still supporting strict nearly Kähler $SU(3)$-structures [1410.6912].

This rigidity and explicit Lie-theoretic construction position homogeneous nearly Kähler 6-manifolds as a cornerstone of special geometric structures in Riemannian and complex geometry.

Source: https://www.emergentmind.com/topics/homogeneous-nearly-kahler-structure