Cartan Torsion: Geometry and Applications

• Cartan torsion is a geometric property defined in two settings: as the antisymmetric part of an affine connection in Cartan geometry and as a cubic measure in Finsler spaces.
• In Einstein–Cartan theory, torsion is algebraically sourced by matter’s intrinsic spin, leading to non-propagating modifications of classical gravity.
• In Finsler geometry, the vanishing of Cartan torsion (per Deicke’s theorem) marks a Riemannian metric, while its boundedness is key for isometric immersions.

In the cited literature, Cartan torsion denotes two distinct objects. In Cartan and Riemann–Cartan geometry it is the torsion associated with a metric-compatible affine or spin connection, encoded by the torsion $2$-form

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$

or, in components, by the antisymmetric part of the connection $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$ up to normalization conventions; it measures the failure of infinitesimal parallelograms to close and, in Einstein–Cartan theory, is sourced algebraically by intrinsic spin (Scholz, 2018, Kapoor, 2020). In Finsler geometry, by contrast, Cartan torsion is the cubic tensor

$$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$

whose vanishing is equivalent to the Riemannian case by Deicke’s theorem (Gangopadhyay, 5 Aug 2025).

1. Historical emergence and terminology

Élie Cartan’s 1922–24 work introduced a generalized notion of curvature in which an infinitesimal closed loop determines not only a rotation, as in Riemannian geometry, but also a net translation. He called this translational curvature torsion, and the historical motivation was explicitly dual: a reformulation of Einstein’s gravitation in moving-frame language and an analogy with Cosserat elasticity, where surface elements transmit both forces and couples (Scholz, 2018).

In Cartan’s formulation, a connection on an $n$-dimensional manifold is split into a translational part $\theta^a$ and a rotational part $\omega^a{}_b$. The corresponding curvatures are the torsion $2$-form $T^a$ and the rotational curvature $R^a{}_b$. Cartan further used Grassmann dualization to reinterpret translational curvature as a torquelike object, which explains why the translational sector received the name “torsion” rather than a purely translational label (Scholz, 2018).

Modern Einstein–Cartan theory realizes Cartan’s original physical intuition in a precise way: variation with respect to the coframe sources curvature by energy–momentum, while variation with respect to the spin connection sources torsion by spin current. This establishes the now-standard statement that spacetime torsion is the geometric response to matter’s internal angular momentum (Scholz, 2018).

2. Differential-geometric definitions in Riemann–Cartan geometry

In first-order formalisms one takes an orthonormal coframe $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$0 and a metric-compatible spin connection $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$1. The torsion $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$2-form is defined by

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$3

while the curvature $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$4-form is

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$5

These are Cartan’s first and second structure equations, and they retain the same formal appearance in torsionful geometry as in the torsionless case (Montesinos et al., 2020, Cardoso, 24 Jan 2025).

In coordinate language, the full affine connection $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$6 is not required to be symmetric. The torsion tensor is its antisymmetric part, written in the cited literature either as

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$7

or as

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$8

reflecting different normalization conventions across papers (Kapoor, 2020, Cabral et al., 2020). The same convention dependence appears in the decomposition of the full connection into a Levi–Civita part plus contortion. Representative forms in the cited literature are

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$9

and

$T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$0

with corresponding sign changes in the algebraic relation between torsion and contortion (Cardoso, 24 Jan 2025, Kapoor, 2020).

Metric compatibility is a standing assumption in several of the cited treatments, expressed as $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$1 or $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$2; equivalently, nonmetricity is set to zero (Kapoor, 2020, Ivanov et al., 2016). Within this metric-compatible setting, torsion is geometrically interpreted as the non-closure of infinitesimal parallelograms under parallel transport, whereas curvature measures the failure of frame orientation to return to itself after infinitesimal transport (Cardoso, 24 Jan 2025, Scholz, 2018).

The first Bianchi identity couples the two sectors: $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$3 Thus torsion and curvature are complementary rather than interchangeable: the first encodes translational curvature, the second rotational curvature (Scholz, 2018).

3. Einstein–Cartan dynamics and the algebraic sourcing of torsion

Einstein–Cartan theory extends general relativity by allowing torsion while retaining metric compatibility. In one reviewed formulation, the Einstein–Hilbert action on a Riemann–Cartan manifold can be written as

$T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$4

or, up to a surface term, as

$T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$5

where $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$6 is a scalar potential for the torsion trace and $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$7 is the traceless part of the contortion (Kapoor, 2020).

Variation with respect to $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$8, $T^\rho{}_{\mu\nu}=2\Gamma^\rho{}_{[\mu\nu]}$9, and $$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$0 yields vacuum equations in which $$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$1, $$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$2, and

$$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$3

The review emphasizes that torsion does not propagate in vacuum and that in empty regions the equations reduce to the usual torsionless general-relativistic ones (Kapoor, 2020).

With matter present, minimal coupling replaces $$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$4 and couples the matter Lagrangian to the vielbein and spin connection. Variation of the total action with respect to the metric or vielbein gives an Einstein equation of the form

$$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$5

while variation with respect to torsion or contortion gives the algebraic Cartan equation

$$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$6

The direct consequence is that intrinsic spin sources torsion algebraically and that torsion vanishes outside spinning matter (Kapoor, 2020).

The same structure appears in the two-component spinor transcription of Einstein–Cartan theory, where the world-tensor equations are written as

$$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$7

The spinor formulation makes explicit that Einstein–Cartan theory extends general relativity by a purely algebraic torsion sector and collapses smoothly to Einstein’s theory when torsion vanishes (Cardoso, 24 Jan 2025).

A recurrent misconception is that adding torsion automatically introduces new long-range propagating gravitational modes. The cited Einstein–Cartan reviews state the opposite for the standard algebraic theory: torsion is non-propagating and instead generates contact or short-distance corrections controlled by spin density (Kapoor, 2020, Cabral et al., 2020).

4. Gauge, spinor, and non-Riemannian extensions

In $$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$8-dimensional $$C_{ijk}(x,y)=\tfrac14\,\frac{\partial^3}{\partial y^i\partial y^j\partial y^k}\bigl[F^2(x,y)\bigr] =\tfrac12\,\frac{\partial g_{ij}}{\partial y^k}(x,y),$$9 gravity with torsion, formulated in Cartan variables $n$0, torsion is again defined by $n$1, but the connection equation no longer enforces $n$2. Instead, the cited derivation gives

$n$3

showing that nonlinearity in $n$4 dynamically sources torsion even in vacuum (Montesinos et al., 2020).

That same analysis identifies a new internal gauge symmetry $n$5, obtained from the converse of Noether’s second theorem. Together with local Lorentz transformations, this symmetry reproduces infinitesimal diffeomorphisms on shell, so diffeomorphisms become a derived symmetry in the first-order description. The resulting gauge algebra is open, closing only up to equations of motion, which is presented as characteristic of gravity in first-order form and directly tied to the presence of torsion in the gauge algebra (Montesinos et al., 2020).

The two-component spinor formalism for torsionful spacetimes introduces soldering forms $n$6 and defines a spinor covariant derivative compatible with both $n$7 and the skew spin-metrics. In the presence of torsion, the curvature decomposes into two independent pieces,

$n$8

where $n$9 generalizes the Weyl spinor and $\theta^a$0 carries the torsional contributions to the Riemann tensor. The usual index-pair symmetry of $\theta^a$1 is lost unless torsion vanishes (Cardoso, 24 Jan 2025).

Other gauge-geometric extensions retain the same Cartan definition of torsion but alter its kinematical rôle. In Newton–Cartan geometry, the temporal and spatial torsions are

$\theta^a$2

The cited comparison shows that gauging the Bargmann algebra allows arbitrary torsion off shell, whereas null reduction of the Einstein equations imposes zero torsion on shell (Bergshoeff et al., 2017). In three-dimensional $\theta^a$3 Newton–Cartan supergravity, torsion is similarly encoded by the curvatures $\theta^a$4 and $\theta^a$5, and the spatial components of the dilatation gauge field $\theta^a$6 are precisely what obstruct a torsionless reduction without further unconventional constraints (Bergshoeff et al., 2015).

5. Matter couplings, string backgrounds, and phenomenology

One reviewed application interprets the NS–NS Kalb–Ramond field as totally antisymmetric torsion on a $\theta^a$7-brane. In type II supergravity, the NS–NS $\theta^a$8-form $\theta^a$9 has field strength $\omega^a{}_b$0, and on the brane worldvolume the $\omega^a{}_b$1-flux is reinterpreted as a totally antisymmetric torsion with

$\omega^a{}_b$2

The resulting Dirac–Born–Infeld action is

$\omega^a{}_b$3

so the gauge-invariant combination entering the DBI determinant is extended by the torsional contortion term $\omega^a{}_b$4. The derivation assumes metric compatibility, retains only the totally antisymmetric part of torsion, and sets the dilaton, R–R fields, and fermions to zero while treating $\omega^a{}_b$5 as a weak, slowly varying background (Kapoor, 2020).

In Einstein–Cartan–Dirac–Maxwell theory, the Dirac sector couples directly to the totally antisymmetric axial part of torsion and yields the non-linear Hehl–Datta equation

$\omega^a{}_b$6

The minimally coupled Maxwell tensor becomes

$\omega^a{}_b$7

and the modified Maxwell equation contains a torsion current $\omega^a{}_b$8 together with effective mass terms of the form $\omega^a{}_b$9 (Cabral et al., 2020).

More recent Einstein–Cartan model building decomposes torsion into Lorentz-irreducible vector, axial, and tensor pieces,

$2$0

together with $2$1, and shows that operators built from these components up to dimension four can induce a scalar degree of freedom, the scalaron. After solving the torsion equations and Weyl-rescaling, one obtains Einstein-frame interactions of the schematic form

$2$2

as well as axion-like couplings

$2$3

The cited applications include reheating, spontaneous baryogenesis, and the connection to the QCD $2$4 term (He et al., 16 Jul 2025).

Cosmological applications use symmetry restrictions to reduce torsion to a small number of time-dependent functions. In the FLRW implementation of Einstein–Cartan–Dirac–Maxwell gravity, homogeneity and isotropy allow only

$2$5

and the resulting Friedmann-like equations contain effective torsion density and pressure terms

$2$6

The same analysis states that $2$7 at high density, replacing the Big Bang by a non-singular bounce and allowing early accelerated expansion without an inflaton (Cabral et al., 2020).

A separate Einstein–Cartan cosmology compatible with the cosmological principle writes torsion as

$2$8

with $2$9 a “torsion scalar” obeying

$T^a$0

In that framework the torsion term renders the matter–dark-energy ratio $T^a$1 dynamical even when the phenomenological interaction $T^a$2 is absent, and shifts the holographic-dark-energy equation of state toward negative values (Yun et al., 27 May 2026).

Other phenomenological studies emphasize vacuum torsion. In a quadratic Poincaré gauge model, nonzero vacuum torsion can induce a de Sitter background and an additional force on spinning bodies. The cited nonrelativistic circular-orbit equation is

$T^a$3

and the paper argues that the additional term is negligible in the solar system but can flatten galactic rotation curves and affect cluster dynamics (Minkevich, 2017). A conceptually distinct proposal treats torsion as an external background field and identifies its effective stress tensor with a cosmological-constant form,

$T^a$4

so that torsion acts as a vacuum-energy density (Ivanov et al., 2016).

Direct experimental searches have also been proposed. In the nonrelativistic limit of Einstein–Cartan theory for a spin-$T^a$5 beam crossing a spin-polarized slab, the reflected and transmitted beams acquire spin-deflection angles $T^a$6 and $T^a$7 proportional to $T^a$8; these effects vanish identically in torsion-free general relativity. The estimates given for ultra-cold neutrons and a polarized target are extremely small, with present neutron-spin-rotation sensitivity still many orders of magnitude away from the required scale (Costa et al., 2023).

6. Cartan torsion in Finsler geometry

In Finsler geometry, Cartan torsion is not the antisymmetric part of an affine connection. It is defined from the Finsler metric $T^a$9 and its fundamental tensor

$R^a{}_b$0

by

$R^a{}_b$1

Its mean is

$R^a{}_b$2

Deicke’s theorem states that $R^a{}_b$3 is Riemannian if and only if $R^a{}_b$4, equivalently $R^a{}_b$5 (Gangopadhyay, 5 Aug 2025).

The norm of the Cartan torsion plays a central rôle in immersion theory. One cited result states that if a Finsler manifold admits an isometric immersion into any Minkowski space, then its Cartan torsion must be uniformly bounded; unbounded $R^a{}_b$6 is therefore an obstruction to such an embedding (Tayebi et al., 2013). The same paper exhibits two subclasses of $R^a{}_b$7-metrics with globally bounded Cartan torsion: $R^a{}_b$8 and

$R^a{}_b$9

under the respective coefficient restrictions stated in the source (Tayebi et al., 2013).

For Minkowskian products of Finsler manifolds, the cited analysis considers

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$00

with $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$01 and $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$02, and derives explicit formulas for both Cartan torsion and mean Cartan torsion. The full torsion has mixed components unless $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$03 is affine-linear in $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$04 and $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$05. The precise splitting criterion is

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$06

Thus Cartan torsion decomposes additively if and only if the Minkowskian product is Euclidean (Gangopadhyay, 5 Aug 2025).

The same equivalence holds for the mean Cartan torsion: $$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$07 In the Euclidean case one moreover has

$$T^a=D\theta^a\equiv d\theta^a+\omega^a{}_b\wedge\theta^b$$08

so boundedness of the mean Cartan torsion on the product is equivalent to boundedness on each factor. The cited paper notes, in particular, that the Euclidean Minkowskian product of two Randers metrics has bounded mean Cartan torsion (Gangopadhyay, 5 Aug 2025).

Taken together, these Finsler results show that the phrase “Cartan torsion” also names a vertical, cubic measure of non-Riemannianity of a Finsler metric. This is a different tensorial object from the torsion of a Cartan or Einstein–Cartan connection, even though both usages descend from Cartan’s broader geometric legacy.