Certain invariants formed from these curvature tensors play an important role in classifying spacetimes.
The second two names are somewhat anachronistic, but since the integrals of the last two are related to the instanton number and Euler characteristic respectively, they have some justification.
, not necessarily geodesic or hypersurface orthogonal, consists of three pieces Because these are all transverse (i.e. projected to the spatial hyperplane elements orthogonal to our timelike unit vector field), they can be represented as linear operators on three-dimensional vectors, or as three by three real matrices.
If we write these operators as E, B, L respectively, the principal invariants of the Riemann tensor are obtained as follows: In terms of the Weyl scalars in the Newman–Penrose formalism, the principal invariants of the Weyl tensor may be obtained by taking the real and imaginary parts of the expression (But note the minus sign!)
, may be obtained as a more complicated expression involving the Ricci scalars (see the paper by Cherubini et al. cited below).
Examples of such are fully general Petrov type I spacetimes with no Killing vectors, see Coley et al. below.