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
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(\esh \dashv \flat \dashv \sharp )$
dR-shape modality$\dashv$ dR-flat modality
$\esh_{dR} \dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \Im \dashv \&)$
fermionic modality$\dashv$ bosonic modality $\dashv$ rheonomy modality
$(\rightrightarrows \dashv \rightsquigarrow \dashv Rh)$
Models
Models for Smooth Infinitesimal Analysis
smooth algebra ($C^\infty$-ring)
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
Given a submersion $p\colon E\to B$, one may ask: which differential forms on $E$ are pullbacks of differential forms on $B$?
If the fibers of $p$ are connected (otherwise the characterization given below is valid only locally in $E$), the answer is provided by the notion of a basic form: a differential form $\omega$ is basic if the following two conditions are met:
The contraction of $\omega$ with any $p$-vertical vector field is zero.
The Lie derivative of $\omega$ with respect to any $p$-vertical vector field is zero.
Using Cartan's magic formula, in the presence of the first condition, the second condition can be replaced by the following one:
Given a Lie group $G$ and a smooth $G$-principal bundle $P \overset{p}{\longrightarrow} X$ over a base smooth manifold $X$, a differential form $\omega \in \Omega^\bullet\big( P \big)$ on the total space $P$ is called basic if it is the pullback of differential forms along the bundle projection $p$ of a differential form $\beta \in \Omega^\bullet\big( X \big)$ of the base manifold
Equivalently, if $\mathfrak{g} \simeq T_e G$ denotes the Lie algebra of $G$, and for $v \in \mathfrak{g}$ we write
for the vector field on $P$ which is the derivative of the $G$-action $G \times P \overset{\rho}{\to} P$ along $v$, then differential form $\omega$ is basic precisely if
it is annihilated by the contraction with $\hat v$
it is annihilated by the Lie derivative along $\hat v$:
(where the first equality holds generally by Cartan's magic formula, we are displaying it just for emphasis)
for all $v \in \mathfrak{g}$.
In this form the definition of basic forms makes sense more generally whenever a Cartan calculus is given, not necessarily exhibited by smooth vector fields on actual manifolds. This more general concept of basic differential forms appears notably in the construction of the Weil model for equivariant de Rham cohomology.
Last revised on April 2, 2024 at 12:51:53. See the history of this page for a list of all contributions to it.