The Cartan–Karlhede algorithm is a procedure for completely classifying and comparing Riemannian manifolds.
[1] Élie Cartan, using his exterior calculus with his method of moving frames, showed that it is always possible to compare the manifolds.
Carl Brans developed the method further,[2] and the first practical implementation was presented by Anders Karlhede [sv] in 1980.
[4][5] For most problems considered, far fewer derivatives than the maximum are actually required, and the algorithm is more manageable on modern computers.
In 4 dimensions, Karlhede's improvement to Cartan's program reduces the maximal number of covariant derivatives of the Riemann tensor needed to compare metrics to 7.