G-structure on a manifold

In differential geometry, a G-structure on an n-manifold M, for a given structure group^[1] G, is a principal G-subbundle of the tangent frame bundle FM (or GL(M)) of M.

The notion of G-structures includes various classical structures that can be defined on manifolds, which in some cases are tensor fields. For example, for the orthogonal group, an O(n)-structure defines a Riemannian metric, and for the special linear group an SL(n,R)-structure is the same as a volume form. For the trivial group, an {e}-structure consists of an absolute parallelism of the manifold.

Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal $G$ -bundle over a group $G$ "comes from" a subgroup $H$ of $G$ . This is called reduction of the structure group (to $H$ ).

Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures with an additional integrability condition.

^ Which is a Lie group $G\to GL(n,\mathbf {R} )$ mapping to the general linear group $GL(n,\mathbf {R} )$ . This is often but not always a Lie subgroup; for instance, for a spin structure the map is a covering space onto its image.

[1] Which is a Lie group $G\to GL(n,\mathbf {R} )$ mapping to the general linear group $GL(n,\mathbf {R} )$ . This is often but not always a Lie subgroup; for instance, for a spin structure the map is a covering space onto its image.

[1]