Cartan's equivalence method

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

$\phi :M\rightarrow N$

such that

$\phi ^{*}h=g$?

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. (For details see for example Cartan-Karlhede algorithm.)

Cartan successfully applied his equivalence method to many such structures, including projective structures, CR structures, and complex structures, as well as ostensibly non-geometrical structures such as the equivalence of Lagrangians and ordinary differential equations. (His techniques were later developed more fully by many others, such as D. C. Spencer and Chern.)

The equivalence method is an algorithmic procedure for determining when two geometric structures are identical. For Cartan, the primary geometrical information was expressed in a coframe or collection of coframes on a differentiable manifold. See method of moving frames.

Overview of Cartan's method

Specifically, suppose that M and N are a pair of manifolds each carrying a G-structure for a structure group G. This amounts to giving a special class of coframes on M and N. Cartan's method addresses the question of whether there exists a local diffeomorphism φ:M→N under which the G-structure on N pulls back to the given G-structure on M. An equivalence problem has been "solved" if one can give a complete set of structural invariants for the G-structure: meaning that such a diffeomorphism exists if and only if all of the structural invariants agree in a suitably defined sense.

Explicitly, a local system of one-forms θⁱ and γⁱ are given on M and N, respectively, which span the cotangent bundle (a coframe). The question is whether there is a local diffeomorphism φ:M→N such that the pullback satisfies

$\phi ^{*}\gamma ^{i}(y)=g_{j}^{i}(x)\theta ^{i}(x),\ (g_{j}^{i})\in G$ (1)

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 question of whether or not two Riemannian manifolds are isometric is then a question of whether there exists a diffeomorphism φ satisfying (1).

The first step in the Cartan method is to express the pullback relation (1) in as invariant a way as possible through the use of a "prolongation". 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. In particular, later on this article uses a different approach. But for the purposes of an overview, it is convenient to stick with the principal bundle viewpoint.

The second step is to use the diffeomorphism invariance of the exterior derivative to try to isolate any other higher-order invariants of the G-structure. Basically one obtains a connection in the principal bundle PM, with some torsion. The components of the connection and of the torsion are regarded as invariants of the problem.

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.