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The Equivalence principle allows Riemann normal coordinates in which the first derivatives of the metric tensor vanish at a chosen point. A purported local gravitational stress-energy tensor built from those first derivatives would vanish there in that chart and therefore, if tensorial, in all charts. Such an expression cannot encode the gravitational energy assigned by a coordinate-dependent field description. This rules out an ordinary universal local covariant gravitational energy density, not the construction of local curvature tensors or boundary and averaged notions of gravitational energy. The Riemann curvature tensor still describes invariant tidal gravity.

A curvature singularity is an incomplete spacetime boundary approached by curves along which a scalar curvature invariant or a component of the Riemann curvature tensor in a parallel-propagated frame becomes unbounded. Such divergence cannot be removed by changing coordinates.

With the Riemann curvature tensor convention $R^{\mu }{}_{\nu \rho \sigma }={\partial }_{\rho }{\Gamma }^{\mu }{}_{\nu \sigma }-{\partial }_{\sigma }{\Gamma }^{\mu }{}_{\nu \rho }+\cdots$, harmonic coordinates remove the mixed second derivatives from the Ricci tensor. The remainder $Q_{ab}$ contains at most first derivatives of the metric tensor. The vacuum Einstein field equations become a system of quasilinear partial differential equations with a Lorentzian wave equation principal part. Harmonic coordinate constraints must be maintained when using the reduced system.

A circular cylinder of radius $\rho$ has induced metric locally equal to $d{u}^2+d{z}^2$ with $u=\rho \phi$. Its Riemann curvature tensor therefore vanishes, despite its nonzero extrinsic curvature. With outward unit normal and convention $K_{ij}=e_i^a{e}_j^b{\nabla }_a{n}_b$, its trace is $1/\rho$ and its principal curvatures are $1/\rho$ and zero. Its averaged mean curvature is $1/(2\rho )$. The angular identification changes global topology, not local flatness.

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.

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.