Parallel Skew-Symmetric Torsion in Geometry

• Parallel skew-symmetric torsion is defined as a covariantly constant 3-form in metric connections that influences curvature and preserves geodesics.
• It induces a canonical splitting of the tangent space and enables local classifications in various geometries like nearly Kähler, G2, and naturally reductive spaces.
• This theory informs applications in special holonomy, Lorentzian supergravity, and Einstein equations by generating symmetry operators in spin geometry.

Parallel skew-symmetric torsion is the condition that a metric connection has torsion represented by a differential $3$-form and that this torsion is covariantly constant with respect to the same connection. Two standard conventions occur in the literature: one writes $\nabla^\tau=\nabla^g+\tau$ with $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$, or $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$ for a $3$-form $H$; the parallelism requirement is then $\nabla^\tau\tau=0$ or $\nabla^{g,H}H=0$ (Cleyton et al., 2018, Agricola et al., 2012). Geometries with this property occur on naturally reductive homogeneous spaces, nearly Kähler and nearly parallel $\mathrm G_2$ manifolds, Sasakian and $3$-Sasakian manifolds, twistor spaces, almost contact and almost product manifolds with adapted characteristic connections, and Lorentzian backgrounds arising in supergravity (Cleyton et al., 2018, Manev, 2010, Mekerov et al., 2010, Ernst et al., 2022).

1. Basic formalism and curvature identities

For a metric connection with totally skew-symmetric torsion, the torsion tensor is identified with a $\nabla^\tau=\nabla^g+\tau$0-form, and the connection has the same geodesics as the Levi-Civita connection. In the Riemannian setting, one basic curvature relation is that the curvature of $\nabla^\tau=\nabla^g+\tau$1 is pair symmetric, and in the notation of Einstein manifolds with skew torsion one has

$\nabla^\tau=\nabla^g+\tau$2

while in the almost contact $\nabla^\tau=\nabla^g+\tau$3-metric setting the scalar curvature satisfies

$\nabla^\tau=\nabla^g+\tau$4

These formulas make explicit that torsion enters curvature quadratically and, in general, through first derivatives of the torsion form (Agricola et al., 2012, Manev, 2010).

Several papers isolate the additional condition that the torsion be closed. In the Hermitian and hyper-Hermitian literature this is the “strong” condition, while the 2026 classification paper abbreviates the conjunction of parallel, skew-symmetric, and closed torsion as “PSCT.” For geometries with parallel skew-symmetric torsion, the closedness condition $\nabla^\tau=\nabla^g+\tau$5 is equivalent to $\nabla^\tau=\nabla^g+\tau$6, to the first Bianchi identity for the torsion connection curvature, and to the algebraic condition that the Lie algebra generated by the endomorphisms $\nabla^\tau=\nabla^g+\tau$7 lies in the stabilizer of $\nabla^\tau=\nabla^g+\tau$8 (Houri et al., 2010, Mekerov et al., 2010, Moroianu et al., 13 May 2026).

2. Riemannian structure theory and classification

The modern Riemannian structure theory begins from the observation that a geometry with parallel skew-symmetric torsion admits a canonical splitting mechanism even though a direct de Rham theorem is unavailable. Cleyton, Moroianu, and Semmelmann define a standard decomposition

$\nabla^\tau=\nabla^g+\tau$9

where both summands are $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$0-parallel and orthogonal; the vertical distribution is totally geodesic and integrable, the leaves are locally naturally reductive homogeneous spaces, and locally one obtains a Riemannian submersion onto a lower-dimensional base that again carries a geometry with parallel skew-symmetric torsion (Cleyton et al., 2018). The later submersion theory of admissible splittings and canonical $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$1-splittings extends this viewpoint and shows that geometries with parallel skew torsion can be analyzed iteratively through locally defined Riemannian submersions with totally geodesic fibers, yielding a replacement for de Rham decomposition in the torsion context (Moroianu et al., 2024).

Closed parallel torsion admits a sharper local classification. Any PSCT geometry splits locally as a Riemannian product into PSCT factors that are $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$2-irreducible, and if the manifold is complete and simply connected the product is global. The irreducible nontrivial PSCT factors are locally isomorphic to a simple compact Lie group with bi-invariant metric, a flat Lie algebra regarded as Euclidean space with a Lie bracket and flat metric, an irreducible noncompact symmetric space of the form $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$3, or a $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$4-dimensional Riemannian manifold with torsion $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$5 (Moroianu et al., 13 May 2026).

A complementary holonomy-based rigidity statement is the Berger-type theorem for metric connections with skew-symmetric torsion. If $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$6 is simply connected, complete, and irreducible, and if the orthogonal subgroup generated by the torsion is neither trivial nor all of $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$7, then $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$8 is isometric to a Lie group with a bi-invariant metric or to its symmetric dual. On a simple Lie group with a bi-invariant metric, the only flat metric connections with skew-symmetric torsion are the two flat canonical connections $g(T^\tau(X,Y),Z)=2\tau(X,Y,Z)$9, with torsion $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$0; away from these flat endpoints, the torsion connection holonomy generically coincides with the Riemannian holonomy (Reggiani, 2011).

3. Adapted connections on almost contact, hypercomplex, almost product, and metric $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$1-manifolds

On almost contact manifolds with $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$2-metric, Manev and Ivanova construct a natural connection preserving $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$3 and having totally skew-symmetric torsion, called the $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$4KT-connection. It exists if and only if $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$5 is a Killing vector field and the fundamental tensor $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$6 has vanishing cyclic sum, equivalently on the class $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$7. Its torsion can be written as

$$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$8

or equivalently through $$\nabla^{g,H}_X Y=\nabla^g_XY+\tfrac12 H(X,Y,\cdot)^\sharp$$9. When $3$0, the curvature simplifies to

$3$1

and $3$2 is of $3$3-Kähler type if and only if the torsion $3$4-form is closed (Manev, 2010).

On almost hypercomplex manifolds with Hermitian and anti-Hermitian metrics in the class $3$5, a unique natural pHKT-connection exists and is given by

$3$6

with torsion

$3$7

A primary result in this setting is that the torsion is $3$8-parallel:

$3$9

The same paper proves that the connection is strong if and only if it is flat, and that if it is not flat then it is weak (Manev et al., 2010).

On a Riemannian almost product manifold $H$0, the analogous natural connection with totally skew-symmetric torsion is the RPT-connection. It exists precisely on the Staikova–Gribachev class $H$1, is unique, and its torsion is

$H$2

If the torsion is parallel, then

$H$3

and the curvature of the RPT-connection is a Riemannian $H$4-tensor exactly when the associated $H$5-form $H$6 vanishes (Mekerov et al., 2010).

A broad higher-codimension extension is provided by metric $H$7-manifolds $H$8 with commuting Reeb fields. Such a manifold admits a unique metric connection with skew-torsion preserving the full structure if and only if each $H$9 is Killing and the associated Nijenhuis tensor $\nabla^\tau\tau=0$0 is totally skew-symmetric. The torsion is

$\nabla^\tau\tau=0$1

For contact metric $\nabla^\tau\tau=0$2-manifolds, also called almost $\nabla^\tau\tau=0$3-manifolds, such a connection exists exactly in the normal case; then

$\nabla^\tau\tau=0$4

is $\nabla^\tau\tau=0$5-parallel. For $\nabla^\tau\tau=0$6 this yields new parallel skew-torsion geometries in all dimensions $\nabla^\tau\tau=0$7, with degenerate torsion $\nabla^\tau\tau=0$8-form (Borówka et al., 18 Nov 2025).

4. Exceptional holonomy and special $\nabla^\tau\tau=0$9-structures

A large part of the contemporary theory concerns special holonomy reductions. In dimension seven, the 2025 classification of $\nabla^{g,H}H=0$0-structures with parallel skew-symmetric torsion studies metric connections $\nabla^{g,H}H=0$1 with $\nabla^{g,H}H=0$2 and $\nabla^{g,H}H=0$3, where $\nabla^{g,H}H=0$4 is the defining stable $\nabla^{g,H}H=0$5-form. Up to naturally reductive homogeneous spaces and nearly parallel $\nabla^{g,H}H=0$6-structures, the local models are organized by the dimension $\nabla^{g,H}H=0$7 of the space of $\nabla^{g,H}H=0$8-parallel vector fields. The list includes torsion-free $\nabla^{g,H}H=0$9 structures, splittings $\mathrm G_2$0 with $\mathrm G_2$1 Calabi–Yau or strict nearly Kähler, $\mathrm G_2$2-Sasaki and twistor-space constructions, products such as $\mathrm G_2$3 or $\mathrm G_2$4 with $\mathrm G_2$5 hyperkähler, parallel $\mathrm G_2$6-Sasaki structures, generic $\mathrm G_2$7-$\mathrm G_2$8-Sasaki manifolds, and nearly parallel $\mathrm G_2$9 structures with $3$0 (Moroianu et al., 3 Oct 2025).

The same classification implies an $3$1 statement in dimension six: metric connections with parallel skew-symmetric torsion preserving an $3$2-structure are locally naturally reductive homogeneous, Calabi–Yau, strict nearly Kähler, or of the form $3$3 with $3$4 an $3$5-Sasaki $3$6-manifold fibering over a Kähler–Einstein base (Moroianu et al., 3 Oct 2025).

For integrable $3$7 manifolds of constant type, the characteristic connection furnishes a more restrictive equivalence. The characteristic curvature is symmetric under exchange of the first and second pair and Ricci flat if and only if the torsion $3$8-form is parallel with respect to both the Levi-Civita and characteristic connections simultaneously, and this is equivalent to the characteristic curvature satisfying the Riemannian first Bianchi identity. The same paper shows that the Hull connection is a $3$9-instanton exactly when the torsion is closed, and that any compact integrable $\nabla^\tau=\nabla^g+\tau$00 manifold with closed torsion is a generalized gradient Ricci soliton precisely when a certain vector field is parallel with respect to the characteristic connection (Ivanov et al., 2023).

These results fit the broader submersion picture for parallel skew torsion. In the reducible-holonomy regime, Gray manifolds with complex reducible canonical holonomy are locally either homogeneous or twistor spaces over anti-self-dual Einstein $\nabla^\tau=\nabla^g+\tau$01-manifolds, nearly parallel $\nabla^\tau=\nabla^g+\tau$02-manifolds with reducible holonomy are exactly the $\nabla^\tau=\nabla^g+\tau$03-$\nabla^\tau=\nabla^g+\tau$04-Sasaki manifolds with $\nabla^\tau=\nabla^g+\tau$05, and Sasaki manifolds appear when the stabilizer acts almost irreducibly (Moroianu et al., 2024).

5. Lorentzian geometry, null structures, and supergravity

Lorentzian signature exhibits phenomena absent in the Riemannian case. Ernst and Galaev show that metric Lorentzian connections with parallel skew-symmetric torsion admit a complete description of holonomy algebras, torsion, and curvature up to the corresponding Riemannian objects. In the simply connected indecomposable weakly irreducible case that is not locally symmetric, the manifold admits a parallel isotropic vector field $\nabla^\tau=\nabla^g+\tau$06 and the torsion has the form

$\nabla^\tau=\nabla^g+\tau$07

with $\nabla^\tau=\nabla^g+\tau$08 a parallel $\nabla^\tau=\nabla^g+\tau$09-form on the screen bundle. The same work proves that all simply connected Lorentzian naturally reductive homogeneous spaces of arbitrary dimension can be constructed from Riemannian naturally reductive homogeneous spaces, and this yields a low-dimensional classification of Lorentzian naturally reductive homogeneous spaces (Ernst et al., 2022).

A related Lorentzian classification concerns closed parallel skew-symmetric torsion and twistor-free torsion. Lorentzian manifolds carrying a parallel $\nabla^\tau=\nabla^g+\tau$10-form $\nabla^\tau=\nabla^g+\tau$11 with $\nabla^\tau=\nabla^g+\tau$12 are classified into the $\nabla^\tau=\nabla^g+\tau$13-dimensional orientable case with $\nabla^\tau=\nabla^g+\tau$14 proportional to the volume form, higher-dimensional geometries with a parallel isotropic vector field $\nabla^\tau=\nabla^g+\tau$15 and $\nabla^\tau=\nabla^g+\tau$16 for a parallel $\nabla^\tau=\nabla^g+\tau$17-form on the screen bundle, the trivial case $\nabla^\tau=\nabla^g+\tau$18, and direct products with Riemannian factors of the corresponding Riemannian classification. When the vectorial component of a twistor-free torsion is non-isotropic, completeness fails in all causal senses; isotropic cases lead naturally to Walker and Kundt geometries (Ernst et al., 2023).

In dimension three, the spinorial description becomes particularly explicit. A Lorentzian $\nabla^\tau=\nabla^g+\tau$19-manifold admits a parallel skew-torsion spinor if and only if it admits a null one-form $\nabla^\tau=\nabla^g+\tau$20 satisfying

$\nabla^\tau=\nabla^g+\tau$21

for some smooth function $\nabla^\tau=\nabla^g+\tau$22. Equivalently, $\nabla^\tau=\nabla^g+\tau$23 is parallel for the unique metric connection with torsion $\nabla^\tau=\nabla^g+\tau$24, where $\nabla^\tau=\nabla^g+\tau$25 is the volume form. Such manifolds are necessarily Kundt, and in the compact case geodesic completeness is governed by an explicit differential condition on the relevant one-forms parameterizing null coframes (Shahbazi, 2024).

The four-dimensional Lorentzian theory developed through spinorial polyforms and bundle gerbes interprets a skew-symmetric torsion $\nabla^\tau=\nabla^g+\tau$26-form $\nabla^\tau=\nabla^g+\tau$27 as the curvature of a gerbe curving. The basic supersymmetric system is

$\nabla^\tau=\nabla^g+\tau$28

and torsion-parallel spinors are translated into null coframe data and exterior differential systems. This provides a gauge-theoretic interpretation of torsion as higher-curvature data rather than an independent tensor field (Shahbazi, 8 Jul 2025).

Compact three-dimensional Heterotic solitons with parallel non-trivial torsion are rigid: they are either hyperbolic three-manifolds or compact quotients of the Heisenberg group with a left-invariant metric. In particular, the Heisenberg quotients occur both with completely skew-symmetric torsion and with non-vanishing twistorial component. In the skew-symmetric case, the paper derives the universal bound $\nabla^\tau=\nabla^g+\tau$29 for the scalar curvature of such Heterotic solitons, preventing it from becoming arbitrarily large in the stated normalization (Moroianu et al., 15 Jan 2026).

6. Einstein equations, Dirac symmetries, and hidden symmetry algebras

The Einstein theory for skew torsion is based on the functional

$\nabla^\tau=\nabla^g+\tau$30

whose Euler–Lagrange equation is

$\nabla^\tau=\nabla^g+\tau$31

When $\nabla^\tau=\nabla^g+\tau$32, any Einstein manifold with parallel skew torsion has constant scalar curvature, and if it is complete, connected, and has $\nabla^\tau=\nabla^g+\tau$33, then it is compact with finite fundamental group. The paper constructs large families of examples: bi-invariant Lie groups, nearly Kähler manifolds, almost Hermitian six-manifolds with holonomy in $\nabla^\tau=\nabla^g+\tau$34, Einstein-Sasaki and $\nabla^\tau=\nabla^g+\tau$35-Einstein-Sasaki manifolds after Tanno deformation, nearly parallel $\nabla^\tau=\nabla^g+\tau$36 manifolds, $\nabla^\tau=\nabla^g+\tau$37-dimensional $\nabla^\tau=\nabla^g+\tau$38-Sasakian manifolds, and examples on Aloff–Wallach spaces $\nabla^\tau=\nabla^g+\tau$39 (Agricola et al., 2012).

Parallel skew torsion also controls symmetry operators for spin geometry. For a connection $\nabla^\tau=\nabla^g+\tau$40 with totally skew-symmetric torsion $\nabla^\tau=\nabla^g+\tau$41, the relevant Dirac operator is

$\nabla^\tau=\nabla^g+\tau$42

Generalized conformal Killing–Yano tensors then define symmetry operators for the massless Dirac equation provided an explicit anomaly vanishes. In strong KT and strong HKT manifolds, the torsion is closed and the canonical Kähler forms are parallel generalized conformal Killing–Yano forms; consequently, a strong KT metric admits one operator commuting with $\nabla^\tau=\nabla^g+\tau$43, and a strong HKT metric admits three such commuting operators (Houri et al., 2010).

At the level of hidden symmetry algebras, a graded Lie bracket for torsionful conformal Killing–Yano forms is available under stronger hypotheses. For a closed and parallel skew-symmetric torsion $\nabla^\tau=\nabla^g+\tau$44-form $\nabla^\tau=\nabla^g+\tau$45, and for torsionful CKY forms that are parallel along torsion directions,

$\nabla^\tau=\nabla^g+\tau$46

the HCKY bracket closes on constant curvature manifolds and on Einstein manifolds for normal torsionful CKY forms. The same construction extends to generalized geometry and yields a graded Lie algebra of generalized hidden symmetries (Ertem et al., 7 Aug 2025).