Almost Quaternionic Skew-Hermitian Manifolds

• Almost quaternionic skew-Hermitian manifolds are 4n-dimensional spaces equipped with a rank-3 quaternionic structure and a Q-Hermitian 2-form that maintains symmetry under three almost complex structures.
• They feature a unique minimal G-connection that preserves both the quaternionic structure and the symplectic-like form, with torsion capturing key integrability and curvature properties.
• Intrinsic torsion decompositions and explicit curvature-holonomy analyses classify these manifolds, informing symmetric-space models and applications in geometric analysis and mathematical physics.

An almost quaternionic skew-Hermitian manifold (often abbreviated AQSH, and sometimes referred to as a “neutral HKT-manifold” in special cases) is a $4n$-dimensional smooth manifold $M$ equipped with a rank-3 subbundle $Q\subset\mathrm{End}(TM)$ locally generated by an admissible triple $(J_1, J_2, J_3)$ of almost complex structures satisfying the quaternionic relations, and a nondegenerate 2-form $\omega\in\Omega^2(M)$ (the “Q-Hermitian” form) such that $\omega(J_aX, J_aY) = \omega(X, Y)$ for $a=1,2,3$ and all vector fields $X, Y$. These geometries arise as the symplectic analogue of almost quaternionic Hermitian structures, with structure group reduced to $\mathrm{SO}^*(2n)\,\mathrm{Sp}(1)$, and admit a canonical minimal $G$-connection whose torsion encodes deep integrability and curvature properties.

1. Foundational Principles and Equivalent Definitions

An almost quaternionic skew-Hermitian structure on $M^{4n}$ ($n \geq 2$) comprises:

• An almost quaternionic structure $Q$ locally spanned by $H = (I, J, K)$ with $I^2 = J^2 = K^2 = IJK = -\mathrm{Id}$.
• A nondegenerate 2-form $\omega$ that is $Q$-Hermitian ($\omega(J_aX, J_aY)=\omega(X, Y)$).

Key equivalent formulations include:

• Scalar 2-form and local symplectic forms: $\omega_I(X, Y) = \omega(X, IY)$, etc., with compatibility conditions dictated by quaternionic identities.
• Endomorphism-valued skew-Hermitian form: $$h(X, Y) = \omega(X, Y)\,\mathrm{Id} + \omega_I(X, Y)I + \omega_J(X, Y)J + \omega_K(X, Y)K$$; its stabilizer is precisely $\mathrm{SO}^*(2n)\,\mathrm{Sp}(1)$.
• Symmetric 4-tensor: $$S = \omega_I \odot \omega_I + \omega_J \odot \omega_J + \omega_K \odot \omega_K$$, giving a bijective correspondence with $(Q, \omega)$ data and classifying the geometry through its orbit in $\operatorname{Sym}^4(T^*M)$ (Chrysikos et al., 2021, Chrysikos et al., 6 Jan 2026).

2. Canonical Connections, Torsion, and Integrability

Every AQSH manifold admits a unique minimal $G$-connection $\nabla^{Q, \omega}$ preserving both the quaternionic structure $Q$ and $\omega$, constructed explicitly as follows:

• Unimodular Oproiu connection $\nabla^{Q, \mathrm{vol}}$ preserves $Q$ and the volume form $\omega^{2n}$;
• The adapted connection $\nabla^{Q, \omega} = \nabla^{Q, \mathrm{vol}} + A$ where $A\in\Omega^1 \otimes (E \otimes S^2 E^*)$ corrects for $\nabla^{Q, \mathrm{vol}}\omega$ via the relation $$\omega(A(X, Y), Z) = \frac{1}{2}(\nabla^{Q, \mathrm{vol}}_X\omega)(Y, Z)$$.

Integrability is characterized as follows:

• The torsion $T^{Q, \omega}$ vanishes if and only if $Q$ is quaternionic (i.e. $T^Q=0$) and $\nabla^{Q, \mathrm{vol}} \omega = 0$; equivalently, $\omega$ is symplectic ($d\omega=0$), and $(M, Q, \omega)$ is a quaternionic skew-Hermitian manifold (Chrysikos et al., 6 Jan 2026, Chrysikos et al., 2024).
• In the metric setting, for structures $(I, J, K, g)$ with neutral signature $g$, a canonical metric connection $D$ with totally skew-symmetric torsion $T$ exists precisely when $(M, I, g)$ is nearly Kähler and $(M, J, g), (M, K, g)$ are quasi-Kähler of Norden type. The torsion $3$-form is given by $T(X, Y, Z) = F_1(X, Y, IZ)$, and $D$ preserves the full hypercomplex structure and metric (Manev et al., 2010).

3. Intrinsic Torsion, Classification, and Algebraic Types

Intrinsic torsion is analyzed using Salamon’s $EH$-formalism, which realizes $T_pM \simeq E \otimes H$ (standard $\mathrm{SO}^*(2n)$- and $\mathrm{Sp}(1)$-modules). The Spencer differential governs the space of possible torsion components:

• For $n > 3$, the intrinsic torsion splits into at least five irreducible $\mathrm{Sp}(1)$-invariant components $X_1, \ldots, X_5$, corresponding to distinct geometric phenomena:
  • $X_1$ ($K \otimes S^3 H$): Nijenhuis-type obstruction.
  • $X_2$ ($E \otimes S^3 H$): Variation of $\omega$ along $Q$.
  • $X_3$, $X_6$, $X_7$: various higher-order torsion and vector-torsion types.
• Integrability and curvature conditions are encoded by the vanishing of specific projections of the torsion (e.g., $X_6=0$ implies absence of "3-form" part) (Chrysikos et al., 2021, Chrysikos et al., 6 Jan 2026).

4. Curvature Structure and Holonomy

On torsion-free AQSH manifolds:

• The curvature $$R \in \Omega^2(M, \mathfrak{so}^*(2n) \oplus \mathfrak{sp}(1))$$ always realizes $\mathfrak{g}$ as a Berger algebra; the curvature module $K(\mathfrak{g}) \simeq \mathfrak{g}$ and is classified through explicit equivariant splittings.
• The Ricci tensor admits the explicit formula:

$$\operatorname{Ric}_A(y, z) = (2n+1)\kappa\,\omega(Ay, z) + \frac{1}{2n}\kappa \sum_{a=1}^3 \left[g_a(y, z)\,\operatorname{Tr}(J_aA) - \omega(J_aJ_ay, z)\right].$$

• If $A\in\mathfrak{so}^*(2n)$, then $\operatorname{Ric}$ is $Q$-Hermitian and $M$ is pseudo-quaternionic-Kähler and locally symmetric; the uniquely determined connection coincides with the Levi-Civita connection of the Ricci-related metric $g_{\sim}=\operatorname{Ric}$ (Chrysikos et al., 2024).
• For connections with parallel totally skew-symmetric torsion $T$, as in the weak/strong dichotomy, $D$ is strong if $dT=0$ (equivalent to flatness), otherwise $D$ is weak. The curvature of $D$ is of Kähler type, with corrections proportional to the torsion (Manev et al., 2010).

5. Submanifolds and Homogeneous Models

Submanifold theory in the AQSH context largely mirrors that of quaternionic Kähler geometry, but reflects the symplectic structure:

• Almost symplectic submanifolds: Defined via nondegenerate pullbacks $\hat{\omega}$, and closed if ambient torsion vanishes or is of type $X_{15}$.
• Almost complex/pseudo-Hermitian submanifolds: Structure and integrability determined by the existence of $J$-invariant subbundles and torsion projections; Gray–Hervella classes can be read off from the decomposition of $T^{Q, \omega}$ and explicit 1-form obstructions.
• Almost quaternionic submanifolds: $Q$-invariant tangent bundle, with induced AQSH structure and adapted minimal connection inherited from the ambient manifold in the torsion-free integrable case. Homogeneous examples are realized inside symmetric spaces:
  • $$M_1 = \mathrm{SL}(n+1,\mathbb{H})/(\mathrm{GL}(1,\mathbb{H}) \cdot \mathrm{SL}(n, \mathbb{H}))$$ (paracomplex quaternionic series).
  • $$M_2 = \mathrm{SU}(2+p,q)/(\mathrm{SU}(2)\cdot\mathrm{SU}(p,q)\cdot \mathrm{U}(1))$$ (pseudo-Wolf spaces).
  • $$M_3 = \mathrm{SO}^*(2n+2)/[\mathrm{SO}^*(2n)\cdot\mathrm{U}(1)]$$ (quaternionic real form series) (Chrysikos et al., 6 Jan 2026, Chrysikos et al., 2021).

6. Bundle Constructions and Swann Bundle Geometry

The Swann bundle construction generalizes hypercomplex geometry in the AQSH setting:

• For any torsion-free AQSH manifold $(M, Q, \omega)$, the Swann bundle $$\hat{M}=(S\times S^0)/(\mathrm{SO}(3)\times \mathbb{R}^+)$$ admits a canonical triple of almost complex structures $(I_1, I_2, I_3)$, which are always 1-integrable by virtue of the symmetric Ricci tensor of the underlying connection, yielding genuine hypercomplex geometry on $\hat{M}$.
• On $\hat{M}$, one constructs almost symplectic forms $\tilde{\Omega} = \hat{\omega} + \beta$ mixing horizontal and vertical components. Integrability of these structures (closure, vanishing torsion) imposes base curvature constraints; only in the flat base case does $\tilde{\Omega}$ become closed and torsion-free, recovering hyper-Kähler cones (Chrysikos et al., 2024).

7. Geometric, Classification, and Physical Consequences

• The intrinsic torsion decomposition (seven irreducibles $X_1$…$X_7$) governs the landscape of AQSH manifolds: pure/quaternionic/symplectic types, half-flat, and more refined subclasses are explicitly controlled by the vanishing or mix of these components.
• The curvature–holonomy structure identifies AQSH geometries as non-symmetric Berger spaces, with consequences for Ricci tensor properties and local symmetry.
• AQSH manifolds and their weak/strong dichotomy generalize the HKT structures known in supersymmetric sigma-model and string-theory backgrounds to pseudo-Riemannian settings (Manev et al., 2010).
• The explicit construction and analysis of submanifold and symmetric space models provide a platform for further study in differential and Riemannian geometry, representation theory, and mathematical physics (Chrysikos et al., 2021, Chrysikos et al., 6 Jan 2026, Chrysikos et al., 2024).

Table: Classification of Symmetric-space Models (torsion-free AQSH structures)

Model Type	Symmetric Space	Dim.
Quaternionic real	$$\mathrm{SO}^*(2n+2)/[\mathrm{SO}^*(2n)\cdot\mathrm{U}(1)]$$	$4n$
Wolf (noncompact)	$$\mathrm{SU}(2+p,q)/[\mathrm{SU}(2)\cdot\mathrm{SU}(p,q)\cdot \mathrm{U}(1)]$$	$4(p+q)$
Paracomplex	$$\mathrm{SL}(n+1, \mathbb{H})/[\mathrm{GL}(1, \mathbb{H})\cdot\mathrm{SL}(n, \mathbb{H})]$$	$8n$

All such symmetric spaces admit a $K$-invariant torsion-free AQSH structure, and the natural symmetric-space connection coincides with the minimal adapted connection (Chrysikos et al., 2021, Chrysikos et al., 6 Jan 2026).

This encyclopedia entry reflects the state-of-the-art as given in (Manev et al., 2010, Chrysikos et al., 2021, Chrysikos et al., 6 Jan 2026), and (Chrysikos et al., 2024).