G-structure
In differential geometry, a G-structure on an n-manifold M, for a given structure group G, is a G-subbundle of the tangent frame bundle FM (or GL(M)) of M.
The notion of G-structures includes many other structures on manifolds, some of them being defined by 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.
Several structures on manifolds, such as a complex structure, a symplectic structure, or a Kähler structure, are G-structures with an additional integrability condition.
Read more about G-structure: Principal Bundles and G-structures, Integrability Conditions, Isomorphism of G-structures, Connections On G-structures, Higher Order G-structures