Einstein Metricity Condition

• Einstein metricity condition is a concept in differential geometry that defines constraints for metric compatibility, including vanishing non-metricity and its extensions in various geometric settings.
• It unites classical metric-affine geometry, Einstein’s nonsymmetric gravitational theory, and algebraic frameworks by recovering Einstein metrics from curvature or projective data.
• This framework informs studies in geometric gravitation and hypersurface theory by clarifying the roles of connection, torsion, and curvature identities.

The expression Einstein metricity condition is used in several distinct but related senses in differential geometry and gravitation. In classical Riemannian and metric-affine settings it denotes metric compatibility, $\nabla_\lambda g_{\mu\nu}=0$, equivalently vanishing non-metricity. In Einstein’s nonsymmetric gravitational theory it is a compatibility condition between a nonsymmetric tensor $G=g+F$, a linear connection, and torsion. In algebraic and projective geometry it denotes conditions under which curvature data or a projective class determines a metric that is Einstein in the sense $\mathrm{Ric}=\lambda g$. In hypersurface theory it may instead refer to a curvature identity involving the Weyl and Riemann tensors rather than to compatibility of a connection with a metric (Benisty et al., 2018, Rovenski et al., 9 Apr 2026, Fodor, 2020, Cap et al., 2012, Deszcz et al., 2018).

1. Terminological scope and basic distinctions

A central source of ambiguity is that metricity and the Einstein condition are not the same statement. In the standard metric-affine usage, metricity means

$\nabla_\lambda g_{\mu\nu}=0,$

or equivalently

$Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$

If torsion also vanishes, this uniquely selects the Levi–Civita connection. By contrast, the Einstein condition is

$\mathrm{Ric}(g)=\lambda g,$

which is an algebraic condition on the Ricci tensor once a metric is given (Benisty et al., 2018).

In the algebraic-curvature setting, the phrase refers to a different problem: given an abstract curvature-type tensor $R_{abcd}$, determine whether there exists a positive-definite metric $g_{ab}$ such that

$R_{abcd}g^{bd}=\lambda g_{ac}.$

This is not a statement about a connection satisfying $\nabla g=0$; it is a metric-recovery problem from curvature data (Fodor, 2020).

In Einstein’s nonsymmetric theory, the fundamental tensor is

$G=g+F$0

with symmetric part $G=g+F$1 and skew part $G=g+F$2, and the relevant condition is

$G=g+F$3

so the failure of $G=g+F$4 to be parallel is controlled by torsion rather than set to zero (Rovenski et al., 9 Apr 2026).

In projective geometry, metricity is encoded by a first BGG equation whose nondegenerate solutions correspond to metrics whose Levi–Civita connections lie in a given projective class, while normal solutions correspond precisely to Einstein metrics in that class (Cap et al., 2012).

A further terminological divergence appears in the hypersurface literature, where a generalized Einstein metric condition may denote the curvature identity

$G=g+F$5

with $G=g+F$6 the Weyl tensor, $G=g+F$7 the Riemann tensor, $G=g+F$8 the Ricci tensor, and $G=g+F$9 the Tachibana tensor. This is a curvature condition, not a compatibility condition for a connection (Deszcz et al., 2018).

2. Metric compatibility, Levi–Civita geometry, and metric-affine relaxations

In the classical sense, Einstein metricity is the requirement that the covariant derivative of the metric vanish: $\mathrm{Ric}=\lambda g$0 This is equivalent to vanishing non-metricity $\mathrm{Ric}=\lambda g$1. In the torsionless sector, the unique compatible connection is the Levi–Civita connection,

$\mathrm{Ric}=\lambda g$2

so imposing metricity and zero torsion forces the connection to be Christoffelian (Benisty et al., 2018).

Metric-affine geometry separates the metric and connection as independent structures. In that framework the connection admits the standard decomposition

$\mathrm{Ric}=\lambda g$3

where $\mathrm{Ric}=\lambda g$4 is the contorsion built from torsion and $\mathrm{Ric}=\lambda g$5 is the disformation built from non-metricity. Imposing metric compatibility removes the disformation, leaving

$\mathrm{Ric}=\lambda g$6

This makes clear that metricity does not force torsion to vanish; it only eliminates non-metricity (Momeni et al., 29 Jul 2025).

The Palatini formulation sharpens this distinction. For an independent torsionless connection $\mathrm{Ric}=\lambda g$7 and matter Lagrangian $\mathrm{Ric}=\lambda g$8, the Palatini equation implies metricity only for an “exceptional class” of matter for which

$\mathrm{Ric}=\lambda g$9

Then the connection equation reduces to $\nabla_\lambda g_{\mu\nu}=0,$0 and hence $\nabla_\lambda g_{\mu\nu}=0,$1. For generic matter, however, $\nabla_\lambda g_{\mu\nu}=0,$2, and the Palatini equations produce non-metricity sourced by matter hypermomentum rather than metric compatibility (Bąk et al., 2022).

A complementary first-order construction enforces metricity directly by a Lagrange multiplier: $\nabla_\lambda g_{\mu\nu}=0,$3 Variation with respect to $\nabla_\lambda g_{\mu\nu}=0,$4 yields $\nabla_\lambda g_{\mu\nu}=0,$5, while variation with respect to $\nabla_\lambda g_{\mu\nu}=0,$6 determines $\nabla_\lambda g_{\mu\nu}=0,$7 algebraically. In the torsionless sector this reproduces the second-order metric equations, including higher-derivative terms in quadratic-curvature theories. The same formalism also admits a Weyl-type relaxation,

$\nabla_\lambda g_{\mu\nu}=0,$8

which replaces strict metricity by controlled non-metricity (Benisty et al., 2018).

Einstein–Cartan theory provides the standard metric-compatible torsionful alternative to Levi–Civita geometry. In the $\nabla_\lambda g_{\mu\nu}=0,$9-dimensional Einstein–Cartan model with a classical spin-$Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$0 source, metricity is imposed from the start, the connection is decomposed as

$Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$1

and torsion is algebraically tied to spin while non-metricity is set to zero. The resulting geometry is Riemann–Cartan rather than metric-affine in the broader sense (0807.4413).

The same metric-compatible viewpoint is built into generalized Cartan formulations in which all ordinary and generalized connections are taken to be $Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$2-valued. There, metricity is not derived as a field equation but imposed kinematically through

$Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$3

and non-metricity is explicitly excluded (Robinson, 1 May 2025).

Metricity also behaves well under hypersurface reduction. For a general affine connection with torsion and non-metricity, if the ambient connection is metric-compatible, then the induced connection on a hypersurface is metric-compatible as well; similarly, ambient torsionlessness implies vanishing induced torsion. In the $Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$4 split of metric-affine general relativity, imposing metricity and torsionlessness collapses the generalized Gauss–Codazzi–Mainardi system to the standard ADM Hamiltonian and momentum constraints (Ariwahjoedi et al., 2020).

3. Einstein connections for nonsymmetric metrics

In Einstein’s nonsymmetric gravitational theory and its geometric descendants, the basic tensor is

$Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$5

where $Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$6 is a nondegenerate symmetric $Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$7-tensor and $Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$8 is a skew-symmetric $Q_{\lambda\mu\nu}:=-\nabla_\lambda g_{\mu\nu}=0.$9-form. Because $\mathrm{Ric}(g)=\lambda g,$0 is nondegenerate, one defines a skew-adjoint $\mathrm{Ric}(g)=\lambda g,$1-tensor $\mathrm{Ric}(g)=\lambda g,$2 or $\mathrm{Ric}(g)=\lambda g,$3 by

$\mathrm{Ric}(g)=\lambda g,$4

The Einstein metricity condition is then not $\mathrm{Ric}(g)=\lambda g,$5, but rather

$\mathrm{Ric}(g)=\lambda g,$6

with torsion $\mathrm{Ric}(g)=\lambda g,$7. In coordinates this becomes

$\mathrm{Ric}(g)=\lambda g,$8

Separating symmetric and skew parts yields coupled formulas for $\mathrm{Ric}(g)=\lambda g,$9 and $R_{abcd}$0, so the condition simultaneously constrains the symmetric metric sector, the skew sector, and the torsion (Rovenski et al., 9 Apr 2026).

For such a nonsymmetric metric, the corresponding Einstein connection can be written in $R_{abcd}$1-covariant form as

$R_{abcd}$2

where $R_{abcd}$3 is the Levi–Civita connection of $R_{abcd}$4. A “special Einstein connection” is characterized by $R_{abcd}$5, so that the symmetric part of $R_{abcd}$6 in $R_{abcd}$7 coincides with that of $R_{abcd}$8 and the deformation is entirely torsional (Rovenski et al., 9 Apr 2026).

Weak almost contact and weak almost Hermitian structures provide the main existence framework for such connections. Under the $R_{abcd}$9–$g_{ab}$0 condition

$g_{ab}$1

the torsion is explicitly determined by $g_{ab}$2 and $g_{ab}$3, and hence the connection is uniquely fixed once $g_{ab}$4 and the structural constraints are given, provided the relevant endomorphism $g_{ab}$5 is invertible on the corresponding distribution (Rovenski et al., 9 Apr 2026).

A related but more restrictive theory imposes totally skew-symmetric torsion together with the $g_{ab}$6-torsion condition

$g_{ab}$7

or, for totally skew torsion,

$g_{ab}$8

In that setting, existence and uniqueness of the Einstein connection are governed by an explicit relation between the Nijenhuis tensor $g_{ab}$9 and $R_{abcd}g^{bd}=\lambda g_{ac}.$0. When the condition holds, the torsion is

$R_{abcd}g^{bd}=\lambda g_{ac}.$1

and the Einstein connection is

$R_{abcd}g^{bd}=\lambda g_{ac}.$2

A notable consequence is that, under totally skew torsion, the $R_{abcd}g^{bd}=\lambda g_{ac}.$3-torsion condition is equivalent to $R_{abcd}g^{bd}=\lambda g_{ac}.$4. In this metric sector the geometry becomes weak nearly Kähler in the weak almost Hermitian case, and weak nearly cosymplectic with splitting results in the weak almost contact case (Zlatanović et al., 11 Aug 2025).

On almost contact metric manifolds, the condition becomes especially rigid. For a generalized metric $R_{abcd}g^{bd}=\lambda g_{ac}.$5 with $R_{abcd}g^{bd}=\lambda g_{ac}.$6, there exists a connection with totally skew-symmetric torsion satisfying the Einstein metricity condition if and only if the structure is almost-nearly cosymplectic. In that case

$R_{abcd}g^{bd}=\lambda g_{ac}.$7

and

$R_{abcd}g^{bd}=\lambda g_{ac}.$8

In dimension five this becomes equivalent to the existence of a Sasaki–Einstein $R_{abcd}g^{bd}=\lambda g_{ac}.$9-manifold; conversely, every Sasaki–Einstein $\nabla g=0$0-manifold yields a two-parameter family of connections with skew torsion satisfying the Einstein metricity condition (Ivanov et al., 2019).

These nonsymmetric theories make explicit a persistent terminological point: here metricity concerns compatibility with the full nonsymmetric tensor $\nabla g=0$1, not merely with $\nabla g=0$2, and it is generally mediated by torsion rather than expressed by $\nabla g=0$3.

4. Algebraic curvature tensors and recovery of an Einstein metric

A purely algebraic version of the Einstein metricity condition arises for curvature-type tensors on a real vector space $\nabla g=0$4 of dimension $\nabla g=0$5. A $\nabla g=0$6 tensor $\nabla g=0$7 is an algebraic curvature-type tensor if it has the local symmetries of a Riemannian curvature tensor: $\nabla g=0$8 together with the first Bianchi identity

$\nabla g=0$9

Such a tensor is strictly sectionally positive if

$G=g+F$00

for every linearly independent pair $G=g+F$01 (Fodor, 2020).

Given a positive-definite metric $G=g+F$02, one forms the Ricci-type contraction

$G=g+F$03

and scalar curvature

$G=g+F$04

The central theorem states that if $G=g+F$05 is strictly sectionally positive and $G=g+F$06, then there exists a positive-definite metric $G=g+F$07 such that

$G=g+F$08

for some scalar $G=g+F$09, and this metric is unique up to an overall constant factor. Moreover,

$G=g+F$10

Thus a strictly sectionally positive algebraic curvature tensor determines a unique ray of positive-definite metrics for which its Ricci contraction is Einstein (Fodor, 2020).

The proof is variational. On the space

$G=g+F$11

consider the scalar-curvature functional

$G=g+F$12

Its critical points are exactly the metrics whose Ricci contraction is proportional to the metric, because the gradient along $G=g+F$13 is the traceless part of $G=g+F$14. Strict sectional positivity supplies lower bounds and compactness, so the functional attains a minimum. Uniqueness follows from strict positivity of the Hessian at an Einstein point: $G=g+F$15 where the $G=g+F$16 are the eigenvalues of a diagonalized variation $G=g+F$17. For $G=g+F$18, strict sectional positivity forces this quadratic form to be positive definite (Fodor, 2020).

This formulation is explicitly purely algebraic. It does not construct a connection and does not assert $G=g+F$19. When $G=g+F$20 happens to be the actual Riemann tensor of the recovered metric, the condition reduces to the usual Einstein equation $G=g+F$21. But in general the statement is that curvature data alone determine, up to scale, a metric for which the Ricci-type contraction is Einstein (Fodor, 2020).

The same work also gives an operator viewpoint. Defining

$G=g+F$22

the Einstein condition becomes the positive-definite eigenvalue problem

$G=g+F$23

A gradient flow

$G=g+F$24

where $G=g+F$25 is the traceless Ricci part with determinant normalization, then provides a practical reconstruction scheme converging to the unique Einstein metric under strict convexity (Fodor, 2020).

5. Projective geometry and normal metricity equations

Projective geometry studies torsion-free affine connections modulo preservation of unparametrized geodesics. Two torsion-free connections are projectively equivalent if they differ by

$G=g+F$26

A projective structure is such an equivalence class $G=g+F$27 (Gover et al., 2012).

Within a projective class, the metrizability problem asks whether some connection is the Levi–Civita connection of a pseudo-Riemannian metric. This is expressed by the first BGG metricity equation. In one standard contravariant formulation, for

$G=g+F$28

the equation is

$G=g+F$29

A nondegenerate solution determines a metric, up to scale, whose Levi–Civita connection lies in the projective class (Gover et al., 2012).

The Einstein refinement is obtained by normality. In tractor language, the BGG splitting operator sends a solution of the metricity equation to a symmetric tractor $G=g+F$30, and a nondegenerate solution is normal exactly when $G=g+F$31 is parallel for the standard projective tractor connection. The main theorem is that nondegenerate normal solutions are equivalent to pseudo-Riemannian Einstein metrics in the projective class; conversely, any Einstein metric produces such a normal solution (Cap et al., 2012).

A complementary tractor characterization states that parallel sub-metrics on the projective cotractor bundle are in one-to-one correspondence with Einstein metrics whose Levi–Civita connection lies in the projective class. In the non-Ricci-flat case the sub-metric is a genuine tractor metric. This recasts Einstein metrizability as a holonomy reduction problem (Gover et al., 2012).

The curvature obstructions are sharp in generic settings. If $G=g+F$32 is projectively equivalent to the Levi–Civita connection of an Einstein metric, then its projective Cotton and Weyl tensors satisfy the $G=g+F$33-space equation

$G=g+F$34

For a weakly generic connection, one can solve for $G=g+F$35 using a left inverse $G=g+F$36 of the Weyl map and build projectively invariant tensors $G=g+F$37 and $G=g+F$38. Then: $G=g+F$39 is equivalent to projective equivalence to a Ricci-flat affine connection, while

$G=g+F$40

is equivalent to projective equivalence to the Levi–Civita connection of a non-Ricci-flat Einstein metric. In that case $G=g+F$41 is, up to scale, the Einstein metric (Gover et al., 2012).

The theory has dimension-specific consequences. In dimension $G=g+F$42, the projective Weyl tensor vanishes identically, and projective Cotton-flatness is the sharp obstruction. In dimension $G=g+F$43, the projective Weyl tensor must vanish for projective–Einstein structures. More generally, the odd curvature forms $G=g+F$44 are projective invariants obstructing metrizability even before the Einstein condition is imposed (Gover et al., 2012).

This projective usage of “metricity” therefore refers neither to $G=g+F$45 for a fixed connection nor to a nonsymmetric $G=g+F$46. It is a projectively invariant PDE whose normal solutions pick out precisely the Einstein members of a projective class (Cap et al., 2012).

6. Generalized curvature conditions and further extensions

In semi-Riemannian hypersurface geometry, the phrase generalized Einstein metric condition can denote the curvature identity

$G=g+F$47

where $G=g+F$48 is the Weyl tensor, $G=g+F$49 the Riemann tensor, $G=g+F$50 the Ricci tensor, $G=g+F$51 the scalar curvature, and $G=g+F$52 the Tachibana tensor. This identity holds trivially when $G=g+F$53, and the cited results show that it is satisfied by all Einstein manifolds, by many quasi-Einstein manifolds, by Roter type manifolds, and by broad classes of hypersurfaces in space forms (Deszcz et al., 2018).

For hypersurfaces in space forms, the analysis proceeds from the Gauss equation

$G=g+F$54

where $G=g+F$55 is the second fundamental form and $G=g+F$56 is the ambient sectional curvature. A principal result states that if, on the locus where the hypersurface is non-quasi-Einstein, the tensor $G=g+F$57 is a linear combination of $G=g+F$58 and $G=g+F$59, then the generalized Einstein metric condition above must hold. In the quasi-Einstein case, the same conclusion follows under additional hypotheses, including a cubic relation for the shape operator (Deszcz et al., 2018).

This curvature identity is explicitly different from Levi–Civita metricity. The same source distinguishes the classical Einstein condition $G=g+F$60 from the generalized Einstein metric condition, which instead controls the commutator $G=g+F$61 by Ricci and scalar-curvature data (Deszcz et al., 2018).

Recent post-Riemannian cosmological models extend the metricity issue further. In Einstein–Cartan–Myrzakulov gravity with torsion and non-metricity, the action

$G=g+F$62

allows curvature, torsion, and non-metricity simultaneously. Imposing metricity sets $G=g+F$63 and eliminates the disformation, while relaxing it keeps $G=g+F$64 as an active dynamical ingredient even in a Weitzenböck sector with $G=g+F$65. In that framework, metricity is presented as a simplifying branch rather than as a default assumption (Momeni et al., 29 Jul 2025).

A related distinction appears in generalized form formulations of Einstein–Cartan geometry. There, all generalized connections are taken to be metric connections, so non-metricity is excluded from the outset, and Einstein’s equations are encoded through generalized torsion and flat generalized metric connections rather than through a relaxation of $G=g+F$66 (Robinson, 1 May 2025).

Taken together, these developments show that “Einstein metricity condition” has become a family of context-dependent notions. The most classical meaning remains metric compatibility, $G=g+F$67. In nonsymmetric geometry it is a torsion-controlled compatibility condition for $G=g+F$68. In algebraic and projective settings it becomes a criterion for recovering an Einstein metric from curvature or geodesic data. In hypersurface theory it may instead designate a higher-order curvature identity. The common theme is not a single equation but a recurring attempt to determine when geometric data are sufficiently constrained to single out a metric, or a metric together with an Einstein-type property.