Homogeneous Nearly Kähler Manifolds

• Homogeneous nearly Kähler structures are geometric frameworks on 6-manifolds featuring an almost complex structure and invariant SU(3)-conditions maintained by a transitive Lie group action.
• They are exemplified by four canonical 3-symmetric spaces—S⁶, ℂP³, F₁,₂(ℂ³), and S³×S³—each carrying a unique invariant nearly Kähler structure derived via explicit Lie-theoretic constructions.
• Their rigorous formulation provides explicit descriptions of curvature, torsion, and deformation rigidity, underpinning applications in special holonomy, Killing spinor theory, and differential geometry.

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 $G_2$ (Foscolo, 2016). 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 $G$ (Dávila et al., 2010, Foscolo, 2016, Anarella et al., 2024). 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 (Cortés et al., 2014).

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 (Dávila et al., 2010). 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 (Dávila et al., 2010, Anarella et al., 2024).

• 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 $a=2$ (Liefsoens et al., 26 Jan 2026).

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) (Liefsoens et al., 26 Jan 2026).

Isometry groups are fully determined for each manifold and are summarized as follows (Anarella et al., 2024, Liefsoens et al., 26 Jan 2026):

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) (Liefsoens et al., 26 Jan 2026).

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 (Foscolo, 2016, Cortés et al., 2014).

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 (Foscolo, 2016). 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 (Foscolo, 2016).

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 (Russo, 2019).

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 (Dávila et al., 2010).

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 (Dávila et al., 2010).

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 (Cortés et al., 2014).

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.