Tensor and Curvature Identities

These cards collect the geometric identities most likely to interrupt a gravity calculation when a sign, trace, dimension, or projection is uncertain. Each card states the hypotheses that make its formulas valid, gives a compact independent check, and links to the longer treatment where the result is derived.

Helpful background. Tensor Fields and Natural Operations supplies index operations and weighted antisymmetrization; Affine Connections and Covariant Derivatives supplies covariant differentiation and curvature.

Choose the identity by operation

If you need to…	Use this record	Check before substitution
reduce or differentiate Riemann components	Riemann Symmetries and Bianchi Identities	Levi–Civita hypotheses, curvature sign, index order, and dimension
interchange two covariant derivatives	Covariant-Derivative Commutators	tensor type, torsion, density weight, spin or gauge representation
project ambient curvature onto a hypersurface	Gauss–Codazzi–Mainardi Relations	normal norm, extrinsic-curvature sign, acceleration, and null exclusion
separate tidal curvature from Ricci traces	Weyl–Ricci Decomposition	dimension, trace normalization, and curvature convention
dualize forms or integrate by parts	Hodge-Star and Differential-Form Identities	degree, dimension, metric index, orientation, and boundary term

What controls the visible formula

The Riemann and Weyl cards assume the Levi–Civita connection. Their familiar pair symmetries and trace decomposition do not transfer unchanged to a general connection with torsion or nonmetricity. The commutator card therefore keeps the torsionful affine identity separate and declares the representation before adding spin or gauge curvature.

Hypersurface identities require another sign package: the squared norm of the chosen unit normal, the definition of extrinsic curvature, and the placement of the normal indices. Hodge identities require a different package: form degree, dimension, metric index, and orientation. Recording only the final sign while omitting the package is not enough to translate between sources safely.

Follow the result to its derivation

For curvature definitions, symmetries, and contractions, continue to Curvature, Holonomy, and Geometric Invariants. For embedded geometry, use Hypersurfaces, Extrinsic Curvature, and Embeddings. For duality and boundary integration, use The Hodge Dual in Lorentzian Signature and The Generalized Stokes Theorem. Those pages supply the proofs, geometric interpretation, and limitations that a lookup card intentionally compresses.