synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a vector space and an element in the exterior product (a -covector), then a differential p-form on a smooth manifold whose tangent spaces look like is called definite on (Bryant 05, section 3.1.1) or stable at each point (Hitchin, p. 3) if at each point the restriction is equal to , up to a general linear transformation.
The existence of a definite form implies a G-structure on for the stabilizer subgroup of .
A class of examples of definite forms are the 3-forms on G₂-manifolds, these are definite on the “associative 3-form” on .
The higher prequantization of a definition form is a definite globalization of a WZW term.
Given a vector space and a stable form (hence a form whose orbit under the general linear group is an open subspace in ), and given a smooth manifold modeled on the vector space , then a differential form is definite on if at each point it is in this open orbit.
See at G₂-manifold – Definite forms
Nigel Hitchin, Special holonomy and beyond, Clay Mathematics Proceedings (pdf)
Robert Bryant, Some remarks on -structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf
Last revised on July 18, 2024 at 11:37:37. See the history of this page for a list of all contributions to it.