Riemann curvature tensor

The Riemann tensor measures the failure of covariant derivatives to commute.

In dimension three the Riemann curvature tensor is $R_{ijpq}=g_{ip}{S}_{jq}-g_{iq}{S}_{jp}-g_{jp}{S}_{iq}+g_{jq}{S}_{ip}$, where $S$ is the Schouten tensor. Consequently every three-dimensional Ricci-flat Riemannian manifold is a flat Riemannian manifold. The implication fails in higher dimensions.

The Kretschmann scalar is the scalar tensor contraction of two copies of the Riemann curvature tensor. In Schwarzschild spacetime, with metric coefficient $1-2M/r$ and $G=c=1$, it is $48M^2/r^6$. Divergence of this scalar along a curve approaching a proposed regular point proves that there is no twice continuously differentiable, nondegenerate metric extension through that point: such an extension would have continuous finite curvature components and hence a finite scalar contraction. This gives a coordinate-independent test of a curvature singularity. Boundedness of this single scalar does not prove regularity.

For a rank-two covariant tensor, the derivative commutator contributes one curvature action on each index:

For a type-$(2,0)$ tensor, the Ricci identity acts once on each contravariant index:

For a family of affinely parametrized geodesics with tangent $U$ and connecting field $J$, the Jacobi equation in the convention used here isIt describes the relative acceleration produced by spacetime curvature.

In constant sectional curvature $\kappa$, a deviation vector orthogonal to a unit timelike geodesic obeys $D^{2}S=\kappa S$, where $D={\nabla }_T$. Therefore $K=g(DS,DS)-\kappa g(S,S)$ is constant, since its derivative is $2g(DS,D^{2}S)-2\kappa g(S,DS)=0$. Its squared length $f=g(S,S)$ satisfies $\ddot{f}=2K+4\kappa f$. In four spacetime dimensions, $\kappa =R/12$.

In a parallel-propagated orthonormal frame, orthogonal geodesic deviation in constant sectional curvature reduces to three scalar equations ${\ddot{S}}^i=\kappa S^i$. For $\kappa >0$, $S=A{e}^{\sqrt{\kappa }\tau }+B{e}^{-\sqrt{\kappa }\tau }$: generic initial data grow exponentially, but the pure decaying mode is a counterexample to universal growth. Initially comoving neighbors have $S=S(0)\cosh (\sqrt{\kappa }\tau )$. For $\kappa <0$, $S=A\cos (\sqrt{-\kappa }\tau )+B\sin (\sqrt{-\kappa }\tau )$, so the deviation is bounded and oscillatory. Simultaneous focusing requires compatible vector initial data and is not automatic for every oscillatory solution.

For a geodesic variation of unit timelike geodesics, proper time normalization holds across the entire variation: $g(T,T)=-1$. Metric compatibility and ${\nabla }_{T}S={\nabla }_{S}T$ then give $dg(S,T)/d\tau =\frac{1}{2}S(g(T,T))=0$. Consequently an initially orthogonal deviation vector stays orthogonal. The common normalization is essential: an arbitrary family of affinely parametrized curves with different tangent norms does not imply this conclusion.

A Jacobi field is the infinitesimal connecting vector between neighboring members of a one-parameter family of geodesics. It satisfies the geodesic-deviation equation along the reference geodesic.

For $\gamma (t)={\exp }_p(tV)$, every Jacobi field with $J(0)=0$ is uniquely of the formDifferentiate the geodesic variation ${\exp }_p(t(V+sB))$ in $s$ to obtain the formula. Uniqueness of the Jacobi equation with prescribed initial value and derivative shows that all such fields are obtained. A nontrivial field vanishing at $t=L>0$ therefore corresponds to a nonzero vector in the kernel of $(d{\exp }_p{)}_{LV}$, giving the conjugate point criterion.

For the Riemann curvature tensor convention $R(X,Y)Z={\nabla }_X{\nabla }_{Y}Z-{\nabla }_Y{\nabla }_{X}Z-{\nabla }_{[X,Y]}Z$, define ${\mathcal{R}}_X(v)=R(v,X)X$. Curvature pair interchange makes this operator self-adjoint. If $v\perp X$, then $⟨{\mathcal{R}}_{X}v,v⟩=K(v,X)|v{|}^2|X{|}^2$. The Jacobi field equation is $D_t^{2}J+{\mathcal{R}}_{\dot{\gamma }}J=0$. The operator $v\mapsto R(X,v)X$ is $-{\mathcal{R}}_X$ and is likewise self-adjoint; this sign distinction matters when testing curvature inequalities.

For a Lie group with a bi-invariant Riemannian metric, vary $\gamma (t)=e^{tA}$ by conjugation: $F(s,t)=e^{sX}{e}^{tA}{e}^{-sX}$. Its Jacobi field is $J_X(t)=X{e}^{tA}-e^{tA}X$. If $e^A$ is central, all these fields vanish at both endpoints. Their space has dimension $\dim \mathfrak{g}-\dim \mathfrak{z}(A)$, since the kernel is the centralizer of an element of a Lie algebra.

In dimension $n\ge 2$, along a nonconstant geodesic, such a Jacobi field is determined injectively by $D_{t}J(0)$, which lies in the $(n-1)$-dimensional normal space. On the round unit sphere the geodesic from a point to its antipode, parametrized with speed $\pi$ on $[0,1]$, has fields $J(t)=\sin (\pi t)E(t)$ for all parallel normal fields $E$, attaining the bound. For a constant geodesic the endpoint-vanishing Jacobi field space is zero.

For a torsion-free connection, $U_{\mu }={\partial }_{\mu }\varphi$ satisfies ${\nabla }_{\alpha }{U}_{\beta }={\nabla }_{\beta }{U}_{\alpha }$.

For a torsion-free connection, $R^{\mu }{}_{\alpha \beta \gamma }+R^{\mu }{}_{\beta \gamma \alpha }+R^{\mu }{}_{\gamma \alpha \beta }=0$.

For the Levi-Civita connection, the cyclic covariant derivative of curvature vanishes: ${\nabla }_{[e}{R}^a{}_{|b|cd]}=0$. Contracting this identity gives the contracted Bianchi identity.

The Weyl tensor is the completely trace-free part of the Riemann curvature tensor. In four dimensions,

In four spacetime dimensions, relative to a timelike unit normal $n$, define $B_{ab}=\frac{1}{2}{ϵ}_{acde}{C}^{de}{}_{bf}{n}^c{n}^f$, with the chosen spacetime metric volume tensor. This is the dual tidal part of the Weyl tensor. Normal contractions vanish by curvature and volume-tensor antisymmetry. Its sign depends on the specified orientation and dualization convention.

Relative to a timelike unit normal $n$, the electric part is $E_{ab}=C_{acbd}{n}^c{n}^d$. It describes the tidal component of the Weyl tensor measured by that observer. Antisymmetry in each curvature pair makes it spatial in both slots; pair symmetry and vanishing traces make it symmetric and trace-free.

Every contraction of the Weyl tensor vanishes. Contracting its first and third indices directly cancels the Ricci and scalar-curvature terms; antisymmetry in each pair and symmetry under exchanging the two pairs reduce all other contractions to this one or make them vanish immediately.

Every conformally flat metric in dimension at least four has identically zero Weyl tensor. The Weyl tensor is the obstruction to local conformal flatness in dimensions at least four.

Applying the curvature commutator to the parallel metric gives $R_{\mu \alpha \beta \gamma }=-R_{\alpha \mu \beta \gamma }$.

The Ricci tensor is the contraction $R_{\alpha \beta }=R^{\mu }{}_{\alpha \mu \beta }$ and is symmetric for the Levi-Civita connection.

For dimension $n\ge 3$, the Schouten tensor packages the Ricci tensor and scalar curvature in the trace part of the Riemann curvature tensor. In dimension three it is $S_{ij}=R_{ij}-R{g}_{ij}/4$, and the full Riemann curvature tensor is reconstructed from it.

The algebraic symmetries of the Riemann curvature tensor leave only one independent curvature component in two dimensions and forceConsequently the Einstein tensor vanishes identically.

With no cosmological term, the two-dimensional Einstein field equations reduce identically on the geometric side to $G_{\alpha \beta }=0$. They impose $T_{\alpha \beta }=0$ and supply no local metric dynamics; the Einstein-Hilbert action is topological.

The Ricci tensor and scalar satisfy ${\nabla }^{\alpha }{R}_{\alpha \beta }=\frac{1}{2}{\nabla }_{\beta }R$.

The Einstein tensor isThe contracted Bianchi identity and metric compatibility give
${\nabla }^{\nu }{G}_{\mu \nu }=0$, ensuring compatibility of the Einstein equations with covariant stress-energy conservation.

For $d{s}^2=-d{t}^2+a(t{)}^{2}d{\mathbf{x}}^2$, the nonzero connection coefficients areWith $G_t^t=-3H^2$ and
$G_i^j=-(2\ddot{a}/a+H^2){\delta }_i^j$, direct substitution into the mixed-tensor covariant derivative gives zero.

The vacuum equation with positive cosmological constant gives
$H^2=\Lambda /3$. Its expanding flat-slicing solution is

A pseudo-Riemannian manifold has constant sectional curvature $K$ when $R_{abcd}=K(g_{ac}{g}_{bd}-g_{ad}{g}_{bc})$. In dimension $n$, its Ricci tensor and scalar are $R_{ab}=(n-1)K{g}_{ab}$ and $R=n(n-1)K$.

In dimension greater than two, if sectional curvature is pointwise independent of the two-plane, the contracted Bianchi identity forces that curvature to be constant on each connected component.

In a Ricci-flat spacetime, the Penrose wave equation is a nonlinear covariant wave equation for the Riemann tensor. Its principal part is ${\nabla }^e{\nabla }_e{R}_{abcd}$ and its lower-order terms are quadratic contractions of curvature.