nLab definite form

Idea

Given a vector space $V$ and an element $\phi$ in the exterior product $\wedge^p V^\ast$ (a $p$-covector), then a differential p-form $\omega$ on a smooth manifold $X$ whose tangent spaces look like $V$ is called definite on $\phi$ (Bryant 05, section 3.1.1) or stable at each point (Hitchin, p. 3) if at each point $x \in X$ the restriction $\omega|_x \in \wedge^p T^\ast X \simeq \wedge^p V^\ast$ is equal to $\phi$, up to a general linear transformation.

The existence of a definite form implies a G-structure on $X$ for $G$ the stabilizer subgroup of $\phi$.

A class of examples of definite forms are the 3-forms on G₂-manifolds, these are definite on the “associative 3-form” on $\mathbb{R}^7$.

The higher prequantization of a definite form is a definite globalization of a WZW term.

Definition

Given a vector space $V$ and a stable form $\phi \in \wedge^p V^\ast$ (hence a form whose orbit under the general linear group $GL(V)$ is an open subspace in $\wedge^p V$), and given a smooth manifold modeled on the vector space $V$, then a differential form $\omega \in \Omega^p(X)$ is definite on $\phi$ if at each point it is in this open orbit.

Examples

$G_2$-manifolds

See at G₂-manifold – Definite forms

References

• Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)

• Robert Bryant, Some remarks on $G_2$-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf