In general relativity, the Carminati–McLenaghan invariants or CM scalars are a set of 16 scalar curvature invariants for the Riemann tensor. This set is usually supplemented with at least two additional invariants.
The CM invariants consist of 6 real scalars plus 5 complex scalars, making a total of 16 invariants. They are defined in terms of the Weyl tensor and its right (or left) dual , the Ricci tensor, and the trace-free Ricci tensor
In the following, it may be helpful to note that if we regard as a matrix, then is the square of this matrix, so the trace of the square is , and so forth.
comprise a of invariants for the Riemann tensor. In the case of vacuum solutions, electrovacuum solutions and perfect fluid solutions, the CM scalars comprise a complete set. Additional invariants may be required for more general spacetimes; determining the exact number (and possible syzygies among the various invariants) is an open problem.
See also[]
Curvature invariant, for more about curvature invariants in (semi)-Riemannian geometry in general