Category:Differential manifolds with reduced structure group

The smaller the structure group, the stronger the structure on the differential manifold. In other words, a differential manifold with a given structure group automatically has any bigger group containing that, as structure group. The important subgroups of the general linear group occurring here are:

• ${\displaystyle U(n,\mathbb {C} )}$ which corresponds to almost Hermitian manifold
• ${\displaystyle GL(n,\mathbb {C} )}$ which contains ${\displaystyle U(n,\mathbb {C} )}$ and corresponds to almost complex manifold. Thus almost Hermitian manifold is a stronger structure than almost complex manifold.
• ${\displaystyle Sp(2n,\mathbb {R} )}$ which contains ${\displaystyle U(n,\mathbb {C} )}$ and corresponds to almost symplectic manifold. Thus almost Hermitian manifold is a stronger structure than almost symplectic manifold.
• ${\displaystyle O(n,\mathbb {R} )}$ which corresponds to Riemannian manifold. When ${\displaystyle n=2m}$, this contains ${\displaystyle U(n,\mathbb {C} )}$. Thus, almost Hermitian manifold is a stronger structure than Riemannian manifold
• ${\displaystyle GL^{+}(n,\mathbb {R} )}$ which contains ${\displaystyle Sp(2m,\mathbb {R} )}$ when ${\displaystyle n=2m}$, and corresponds to oriented manifold. Thus oriented manifold is a stronger structure than almost symplectic manifold.

Lorentzian and pseudo-Riemannian manifolds correspond to pseudo-orthogonal groups ${\displaystyle O(p,q)}$ and ${\displaystyle O(p,1)}$.