G-structure - Imported Knowlpedia Content

Definition

Let $M$ be a smooth $n$ -manifold and let $G\subseteq\mathrm{GL}(n,\mathbb R)$ be a Lie subgroup. A $G$ -structure on $M$ is a reduction of structure group of the frame bundle $\mathrm{Fr}(TM)$ from $\mathrm{GL}(n,\mathbb R)$ to $G$ . Concretely, it is a principal $G$ -subbundle $Q\subseteq\mathrm{Fr}(TM)$ whose fiber $Q_x$ consists of the frames at $x$ declared compatible with the structure. The fixed inclusion $G\hookrightarrow\mathrm{GL}(n,\mathbb R)$ is part of the ambient data and determines how $G$ changes those frames.

Geometric examples

An orientation is a $\mathrm{GL}^+(n,\mathbb R)$ -structure, and a Riemannian metric is equivalently an $\mathrm O(n)$ -structure. On a $2m$ -manifold, an almost complex structure gives a $\mathrm{GL}(m,\mathbb C)$ -structure, while a fiberwise nondegenerate alternating form gives an $\mathrm{Sp}(2m,\mathbb R)$ -structure. These correspondences are standard instances of the reduction framework in Kobayashi and Nomizu, Volume I, Chapter I.

Integrability and connections

A $G$ -structure is pointwise reduction data; it need not be locally equivalent to the flat model. Integrability imposes additional differential conditions that depend on $G$ . Likewise, a connection on $\mathrm{Fr}(TM)$ need not preserve $Q$ ; preservation is the separate condition of being a connection compatible with the reduction.

For example, an $\mathrm{Sp}(2m,\mathbb R)$ -structure supplies a nondegenerate two-form, but it defines a symplectic manifold only when that form is also closed.

Conventions and scope

References

Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter I, frame bundles, reductions, and $G$ -structures.
Shlomo Sternberg, Lectures on Differential Geometry, 2nd ed., Chelsea, 1983. AMS record. Relevant: Chapter VII, $G$ -structures and equivalence problems.