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. Mathematical definition 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. The real CM scalars are: (the trace of the Ricci tensor) The complex CM scalars are: The CM scalars have the following degrees: is linear, are quadratic, are cubic, are quartic, are quintic. They can all be expressed directly in terms of the Ricci spinors and Weyl spinors, using Newman–Penrose formalism; see the link below. Complete sets of invariants In the case of spherically symmetric spacetimes or planar symmetric spacetimes, it is known that comprise a complete set 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 Curvature invariant (general relativity), for other curvature invariants which are useful in general relativity References External links The GRTensor II website includes a manual with definitions and discussions of the CM scalars. Implementation in the Maxima computer algebra system Tensors in general relativity