--- title: Einstein Metricity Condition url: https://www.emergentmind.com/topics/einstein-metricity-condition type: topic --- # Einstein Metricity Condition 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 [1805.09667] [2604.08486] [2006.01611] [1207.0128] [1810.01402]. ## 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 [1805.09667]. 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 [2006.01611]. In Einstein’s nonsymmetric theory, the fundamental tensor is \[ G=g+F, \] with symmetric part $g$ and skew part $F$, and the relevant condition is \[ (\nabla_X G)(Y,Z)=-G(T(X,Y),Z), \] so the failure of $G$ to be parallel is controlled by torsion rather than set to zero [2604.08486]. 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 [1207.0128]. A further terminological divergence appears in the hypersurface literature, where a **generalized Einstein metric condition** may denote the curvature identity \[ C\cdot R-R\cdot C=Q(S,C)-\frac{k}{n-1}Q(g,C), \] with $C$ the Weyl tensor, $R$ the Riemann tensor, $S$ the Ricci tensor, and $Q$ the Tachibana tensor. This is a curvature condition, not a compatibility condition for a connection [1810.01402]. ## 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: \[ \nabla_\lambda g_{\mu\nu}=0. \] This is equivalent to vanishing non-metricity $Q_{\lambda\mu\nu}=0$. In the torsionless sector, the unique compatible connection is the Levi–Civita connection, \[ \Gamma^\rho_{\mu\nu}=\{^\rho_{\mu\nu}\}(g) =\frac12 g^{\rho\lambda} (\partial_\mu g_{\lambda\nu}+\partial_\nu g_{\lambda\mu}-\partial_\lambda g_{\mu\nu}), \] so imposing metricity and zero torsion forces the connection to be Christoffelian [1805.09667]. Metric-affine geometry separates the metric and connection as independent structures. In that framework the connection admits the standard decomposition \[ \Gamma^\rho{}_{\mu\nu} =\{^\rho{}_{\mu\nu}\}+K^\rho{}_{\mu\nu}+L^\rho{}_{\mu\nu}, \] where $K^\rho{}_{\mu\nu}$ is the contorsion built from torsion and $L^\rho{}_{\mu\nu}$ is the disformation built from non-metricity. Imposing metric compatibility removes the disformation, leaving \[ \Gamma^\rho{}_{\mu\nu}=\{^\rho{}_{\mu\nu}\}+K^\rho{}_{\mu\nu}. \] This makes clear that metricity does not force torsion to vanish; it only eliminates non-metricity [2508.00026]. The Palatini formulation sharpens this distinction. For an independent torsionless connection $\Gamma$ and matter Lagrangian $L_{\mathrm{matt}}$, the Palatini equation implies metricity only for an “exceptional class” of matter for which \[ P^\lambda{}_{\mu\nu}:=\frac{\partial L_{\mathrm{matt}}}{\partial \Gamma^\lambda{}_{\mu\nu}}=0. \] Then the connection equation reduces to $\nabla g=0$ and hence $\Gamma=\{\}$. For generic matter, however, $P^\lambda{}_{\mu\nu}\neq 0$, and the Palatini equations produce non-metricity sourced by matter hypermomentum rather than metric compatibility [2209.05936]. A complementary first-order construction enforces metricity directly by a Lagrange multiplier: \[ S[g,\Gamma,k] =\int d^4x\,\sqrt{-g}\,\big[L(g,\Gamma)+k^{\alpha\beta\gamma}\nabla_\gamma g_{\alpha\beta}\big]. \] Variation with respect to $k^{\alpha\beta\gamma}$ yields $\nabla_\gamma g_{\alpha\beta}=0$, while variation with respect to $\Gamma$ determines $k$ 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, \[ \nabla_\lambda g_{\mu\nu}=eA_\lambda g_{\mu\nu}, \] which replaces strict metricity by controlled non-metricity [1805.09667]. Einstein–Cartan theory provides the standard metric-compatible torsionful alternative to Levi–Civita geometry. In the $2+1$-dimensional Einstein–Cartan model with a classical spin-$1/2$ source, metricity is imposed from the start, the connection is decomposed as \[ \Gamma^\rho{}_{\mu\nu}=\{^\rho{}_{\mu\nu}\}+K^\rho{}_{\mu\nu}, \] 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 $\mathfrak{so}(1,3)$-valued. There, metricity is not derived as a field equation but imposed kinematically through \[ Dg=0, \qquad \omega_{ab}=-\omega_{ba}, \] and non-metricity is explicitly excluded [2505.00617]. 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 $3+1$ 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 [2012.12171]. ## 3. Einstein connections for nonsymmetric metrics In Einstein’s nonsymmetric gravitational theory and its geometric descendants, the basic tensor is \[ G=g+F, \] where $g$ is a nondegenerate symmetric $(0,2)$-tensor and $F$ is a skew-symmetric $2$-form. Because $g$ is nondegenerate, one defines a skew-adjoint $(1,1)$-tensor $f$ or $A$ by \[ g(X,fY)=F(X,Y) \quad\text{or}\quad g(AX,Y)=F(X,Y). \] The Einstein metricity condition is then not $\nabla G=0$, but rather \[ (\nabla_X G)(Y,Z)=-G(T(X,Y),Z), \] with torsion $T$. In coordinates this becomes \[ \nabla_\lambda G_{\mu\nu}=-G_{\rho\nu}T^\rho{}_{\lambda\mu}. \] Separating symmetric and skew parts yields coupled formulas for $\nabla g$ and $\nabla F$, so the condition simultaneously constrains the symmetric metric sector, the skew sector, and the torsion [2604.08486]. For such a nonsymmetric metric, the corresponding Einstein connection can be written in $g$-covariant form as \[ g(\nabla_XY,Z) = g(\nabla^g_XY,Z) +\frac12\{T(X,Y,Z)-T(Z,X,fY)+T(Y,Z,fX)\}, \] where $\nabla^g$ is the Levi–Civita connection of $g$. A “special Einstein connection” is characterized by $K_XY=-K_YX$, so that the symmetric part of $\nabla$ in $(X,Y)$ coincides with that of $\nabla^g$ and the deformation is entirely torsional [2604.08486]. Weak almost contact and weak almost Hermitian structures provide the main existence framework for such connections. Under the $Q$–$T$ condition \[ T(QX,Y,Z)=T(X,QY,Z)=T(X,Y,QZ), \] the torsion is explicitly determined by $\nabla^gF$ and $dF$, and hence the connection is uniquely fixed once $(g,f)$ and the structural constraints are given, provided the relevant endomorphism $P=I-f^2$ is invertible on the corresponding distribution [2604.08486]. A related but more restrictive theory imposes **totally skew-symmetric torsion** together with the $A$-torsion condition \[ T(AX,Y)=T(X,AY), \] or, for totally skew torsion, \[ T(AX,Y,Z)=T(X,AY,Z)=T(X,Y,AZ). \] In that setting, existence and uniqueness of the Einstein connection are governed by an explicit relation between the Nijenhuis tensor $N_A$ and $dF$. When the condition holds, the torsion is \[ T(X,Y,Z)=-\frac13\,dF(X,Y,Z), \] and the Einstein connection is \[ g(\nabla_XY,Z) = g(\nabla^g_XY,Z) +\frac16\big[dF(AX,Y,Z)-dF(X,Y,Z)-dF(X,AY,Z)\big]. \] A notable consequence is that, under totally skew torsion, the $A$-torsion condition is equivalent to $\nabla g=0$. 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 [2508.08021]. On almost contact metric manifolds, the condition becomes especially rigid. For a generalized metric $G=g+F$ with $F(X,Y)=g(\phi X,Y)$, 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 \[ T(X,Y,Z)=-3\,dF(X,Y,Z), \] and \[ g(\nabla^{\mathrm{ngt}}_X Y,Z) = g(\nabla^g_X Y,Z)-dF(X,Y,Z)+\eta(X)d\eta(Y,Z)+\eta(Y)d\eta(X,Z). \] In dimension five this becomes equivalent to the existence of a Sasaki–Einstein $5$-manifold; conversely, every Sasaki–Einstein $5$-manifold yields a two-parameter family of connections with skew torsion satisfying the Einstein metricity condition [1905.04013]. These nonsymmetric theories make explicit a persistent terminological point: here **metricity** concerns compatibility with the full nonsymmetric tensor $G=g+F$, not merely with $g$, and it is generally mediated by torsion rather than expressed by $\nabla G=0$. ## 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 $V$ of dimension $n\ge 3$. A $(4,0)$ tensor $R_{abcd}$ is an algebraic curvature-type tensor if it has the local symmetries of a Riemannian curvature tensor: \[ R_{abcd}=-R_{bacd},\qquad R_{abcd}=-R_{abdc},\qquad R_{abcd}=R_{cdab}, \] together with the first Bianchi identity \[ R_{abcd}+R_{acdb}+R_{adbc}=0. \] Such a tensor is **strictly sectionally positive** if \[ R(x,y,y,x)>0 \] for every linearly independent pair $(x,y)$ [2006.01611]. Given a positive-definite metric $g_{ab}$, one forms the Ricci-type contraction \[ \mathrm{Ric}_{ac}(g)=R_{abcd}g^{bd} \] and scalar curvature \[ \mathrm{Scal}(g)=R_{abcd}g^{ac}g^{bd}. \] The central theorem states that if $R_{abcd}$ is strictly sectionally positive and $n\ge 3$, then there exists a positive-definite metric $g_{ab}$ such that \[ R_{abcd}g^{bd}=\lambda g_{ac} \] for some scalar $\lambda$, and this metric is unique up to an overall constant factor. Moreover, \[ \lambda=\frac1n\,\mathrm{Scal}(g)>0. \] Thus a strictly sectionally positive algebraic curvature tensor determines a unique ray of positive-definite metrics for which its Ricci contraction is Einstein [2006.01611]. The proof is variational. On the space \[ S=\{g_{ab}>0:\det(g)=1\}, \] consider the scalar-curvature functional \[ R(g)=R_{abcd}g^{ac}g^{bd}=\mathrm{Scal}(g). \] Its critical points are exactly the metrics whose Ricci contraction is proportional to the metric, because the gradient along $S$ is the traceless part of $\mathrm{Ric}(g)$. 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: \[ D^2R_g[h,h] = 2\sum_{1\le k