Frame Bundle - G-structures

G-structures

If a smooth manifold M comes with additional structure it is often natural to consider a subbundle of the full frame bundle of M which is adapted to the given structure. For example, if M is a Riemannian manifold we saw above that it is natural to consider the orthonormal frame bundle of M. The orthonormal frame bundle is just a reduction of the structure group of F_GL(M) to the orthogonal group O(n).

In general, if M is a smooth n-manifold and G is a Lie subgroup of GL_n(R) we define a G-structure on M to be a reduction of the structure group of F_GL(M) to G. Explicitly, this is a principal G-bundle F_G(M) over M together with a G-equivariant bundle map

over M.

In this language, a Riemannian metric on M gives rise to an O(n)-structure on M. The following are some other examples.

• Every oriented manifold has an oriented frame bundle which is just a GL_n+(R)-structure on M.
• A volume form on M determines a SL_n(R)-structure on M.
• A 2n-dimensional symplectic manifold has a natural Sp_2n(R)-structure.
• A 2n-dimensional complex or almost complex manifold has a natural GL_n(C)-structure.

In many of these instances, a G-structure on M uniquely determines the corresponding structure on M. For example, a SL_n(R)-structure on M determines a volume form on M. However, in some cases, such as for symplectic and complex manifolds, an added integrability condition is needed. A Sp_2n(R)-structure on M uniquely determines a nondegenerate 2-form on M, but for M to be symplectic, this 2-form must also be closed.