---
title: Almost Quaternionic Skew-Hermitian Manifolds
url: https://www.emergentmind.com/topics/almost-quaternionic-skew-hermitian-manifolds
type: topic
---

# Almost Quaternionic Skew-Hermitian Manifolds

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 $\operatorname{Sym}^4(T^*M)$ [2109.15253], [2601.03094].

## 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 [2601.03094], [2404.05463].
- 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 [1003.2051].

## 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) [2109.15253], [2601.03094].

## 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 $g_{\sim}=\operatorname{Ric}$ [2404.05463].
- 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 [1003.2051].

## 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) [2601.03094], [2109.15253].

## 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 [2404.05463].

## 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 [1003.2051].
- 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 [2109.15253], [2601.03094], [2404.05463].

## 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 [2109.15253], [2601.03094].

*This encyclopedia entry reflects the state-of-the-art as given in [1003.2051], [2109.15253], [2601.03094], and [2404.05463].*

Source: https://www.emergentmind.com/topics/almost-quaternionic-skew-hermitian-manifolds