G-structure on a manifold

In differential geometry, a G-structure on an $n$-manifold $M$, for a given structure group $G$, is a principal $G$-subbundle of the tangent frame bundle ${\text{F}}M$ (or $\operatorname {GL} (M)$) 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 $\operatorname {O} (n)$-structure defines a Riemannian metric, and for the special linear group an $\operatorname {SL} (n,\mathbb {R} )$-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.