Curvature invariant (general relativity)

In general relativity, curvature invariants are a set of scalars formed from the Riemann, Weyl and Ricci tensors - which represent curvature, hence the name, - and possibly operations on them such as contraction, covariant differentiation and dualisation.

Certain invariants formed from these curvature tensors play an important role in classifying spacetimes. Invariants are actually less powerful for distinguishing locally non-isometric Lorentzian manifolds than they are for distinguishing Riemannian manifolds. This means that they are more limited in their applications than for manifolds endowed with a positive definite metric tensor.

The principal invariants of the Riemann and Weyl tensors are certain quadratic polynomial invariants (i.e., sums of squares of components).

The principal invariants of the Riemann tensor of a four-dimensional Lorentzian manifold are

These are quadratic polynomial invariants (sums of squares of components). (Some authors define the Chern-Pontryagin scalar using the right dual instead of the left dual.)

The first of these was introduced by Erich Kretschmann. 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.

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