nLab
definite form

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& Rh & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

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

The existence of a definite form implies a G-structure on XX for GG the stabilizer subgroup of ϕ\phi.

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

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

Definition

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

Examples

G 2G_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 2G_2-structures, Proceedings of the 12th Gökova Geometry-Topology Conference 2005, pp. 75-109 pdf

Revised on May 25, 2017 16:27:07 by Urs Schreiber (92.218.150.85)