In mathematics, the Maurer–Cartan form for a Lie group G is a distinguished differential one-form on G that carries the basic infinitesimal information about the structure of G. It was much used by Élie Cartan as a basic ingredient of his method of moving frames, and bears his name together with that of Ludwig Maurer.
The Maurer–Cartan form ω is thus a one-form defined globally on G, that is, a linear mapping of the tangent space TgG at each g ∈ G into TeG.
The geometries of interest were homogeneous spaces G/H, but usually without a fixed choice of origin corresponding to the coset eH.
A left-invariant vector field is a section X of TG such that [2] The Maurer–Cartan form ω is a g-valued one-form on G defined on vectors v ∈ TgG by the formula If G is embedded in GL(n) by a matrix valued mapping g =(gij), then one can write ω explicitly as In this sense, the Maurer–Cartan form is always the left logarithmic derivative of the identity map of G. If we regard the Lie group G as a principal bundle over a manifold consisting of a single point then the Maurer–Cartan form can also be characterized abstractly as the unique principal connection on the principal bundle G. Indeed, it is the unique g = TeG valued 1-form on G satisfying where Rh* is the pullback of forms along the right-translation in the group and Ad(h) is the adjoint action on the Lie algebra.
In this context, one may view the Maurer–Cartan form as a 1-form defined on the tautological principal bundle associated with a homogeneous space.
The Maurer–Cartan form on the Lie group G yields a flat Cartan connection for this principal bundle.
The Maurer–Cartan equation implies that Moreover, if sU and sV are a pair of local sections defined, respectively, over open sets U and V, then they are related by an element of H in each fibre of the bundle: The differential of h gives a compatibility condition relating the two sections on the overlap region: where ωH is the Maurer–Cartan form on the group H. A system of non-degenerate g-valued 1-forms θU defined on open sets in a manifold M, satisfying the Maurer–Cartan structural equations and the compatibility conditions endows the manifold M locally with the structure of the homogeneous space G/H.
In other words, there is locally a diffeomorphism of M into the homogeneous space, such that θU is the pullback of the Maurer–Cartan form along some section of the tautological bundle.