In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism.
For example, if M and N are two Riemannian manifolds with metrics g and h, respectively, when is there a diffeomorphism such that Although the answer to this particular question was known in dimension 2 to Gauss and in higher dimensions to Christoffel and perhaps Riemann as well, Élie Cartan and his intellectual heirs developed a technique for answering similar questions for radically different geometric structures.
The equivalence method is an essentially algorithmic procedure for determining when two geometric structures are identical.
The question is whether there is a local diffeomorphism φ:M→N such that the pullback of the coframe on N satisfies where the coefficient g is a function on M taking values in the Lie group G. For example, if M and N are Riemannian manifolds, then G=O(n) is the orthogonal group and θi and γi are orthonormal coframes of M and N respectively.
The most economical way to do this is to use a G-subbundle PM of the principal bundle of linear coframes LM, although this approach can lead to unnecessary complications when performing actual calculations.
The third step is that if the remaining torsion coefficients are not constant in the fibres of the principal bundle PM, it is often possible (although sometimes difficult), to normalize them by setting them equal to a convenient constant value and solving these normalization equations, thereby reducing the effective dimension of the Lie group G. If this occurs, one goes back to step one, now having a Lie group of one lower dimension to work with.
The Cartan test generalizes the Frobenius theorem on the solubility of first-order linear systems of partial differential equations.
If the coframes on M and N (obtained by a thorough application of the first three steps of the algorithm) agree and satisfy the Cartan test, then the two G-structures are equivalent.
Once this is done, one obtains a new coframe on the prolonged space and has to return to the first step of the equivalence method.
Of course, depending on the particular application, a great deal of information can still be obtained with the equivalence method.