Curvature as the Error Term: The Ladder of Tensor Invariants in Curved Space

An earlier article built a ladder of tensor invariants in flat space. The derivative tensor of a vector field splits into its trace, antisymmetric and deviatoric slices, each with an integral theorem; second derivatives add null, Green and Piola identities. This article carries the ladder onto a Riemannian manifold and states, slice by slice, what curvature adds. The antisymmetric column survives unchanged; it needs at most the volume form. Every other divergence-type second-floor composition of a vector field acquires a Ricci term and nothing more, while uncontracted compositions see the full Riemann tensor. A slice-weighted combination of the 3 divergences equals 2Ric(C). On 2-forms the Weyl tensor enters, and in dimension 4 the self-dual and anti-self-dual halves of the jad rung see the 2 halves of the Weyl tensor. The Saint-Venant chain closes with a second-order operator exactly at constant curvature. A closed curvature formula gives the divergence of every Piola tensor. For maps between manifolds of equal dimension the cofactor identity holds without such a term. The energy identity of the slices acquires minus the integrated Ricci energy. On the round sphere a rotation field therefore breaks the flat Korn inequality. The determinant of the Hessian is no longer a null Lagrangian. The tensor-valued integral theorems survive when each slice is paired with the kernel of its dual operator. Symmetric fields pair with Killing fields, trace-free ones with conformal Killing fields, antisymmetric ones with closed fields. In Lorentz signature the pointwise identities hold unchanged, and the pairings give the conserved and conformal currents. One floor up, the Bianchi identity reads as the curvature's jad rung, and the Lovelock tensors as transcripts of Piola tensors. The identities are checked exactly on rational Taylor jets and by quadrature on a curved 3-torus, the round 2-sphere and a band of it.