In differential geometry, a G-structure on a n-manifold M, for a given structure group G (which is a Lie subgroup of the general linear group GL(n)) is a G-subbundle of the frame bundle on 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)-structure is the same as a volume form . On the other hand a symplectic manifold structure is a stronger concept than a G-structure for the symplectic group; the latter would correspond to specifying a two-form ω on M that is non-degenerate, but the former to the extra condition (an integrability condition) that dω = 0. Subgroups of GL(n) defined as types of block matrix define a range of G-structures that include foliations. Complex matrix groups define almost complex manifolds.

The set of diffeomorphisms of M that preserve a G-structure is called the automorphism group of that structure. For an O(n)-structure they are the group of isometries of the Riemannian metric and for an SL(n)-structure volume preserving maps.

References

• S. Kobayashi, Transformation Groups in Differential Geometry, Springer, 1972. ISBN 0-387-05848-6.