Reduction of the structure group

In differential geometry, a G-structure on an n-manifold M, for a given structure groupG, 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.

In mathematics, in particular the theory of principal bundles, one can ask if a principal ${\displaystyle G}$-bundle over a group ${\displaystyle G}$ "comes from" a subgroup ${\displaystyle H}$ of ${\displaystyle G}$. This is called reduction of the structure group (to ${\displaystyle H}$).

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