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)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR} \dashv \flat_{dR}$
(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)
An ordinary vielbein/orthogonal structure is a reduction of the structure group of the tangent bundle of a smooth manifold from the general linear group $GL_n$ to its maximal compact subgroup, the orthogonal group.
Accordingly, whenever we have a reduction of structure groups along the inclusion $H \hookrightarrow G$ of a maximal compact subgroup, we may speak of a generalized vielbein.
Let $G$ be a Lie group and let $H \hookrightarrow G$ be the inclusion of a maximal compact subgroup. Write
for the induced morphism of smooth moduli stacks of principal bundles.
Notice that
exhibiting the coset $G/H$ as the homotopy fiber of $i$;
under geometric realization $i$ becomes an equivalence
Then for $X$ a smooth manifold or more generally a smooth infinity-groupoid equiped with a map $g : X \to \mathbf{B}G$ an $i$-generalized vielbein is a lift $e$ in
The moduli space of $i$-generalized vielbeing relative $g$ is the twisted cohomology
The ordinary notion of vielbein is obtained for
in the context of generalized complex geometry one considers generalized vielbeins arising from reduction along $O(n)\times O(n) \to O(n,n)$ of the generalized tangent bundle
in the context of exceptional generalized geometry one considers vielbeins arising from reduction along $H_n \to E_n$ for $E_n$ an exceptional Lie group.
Section Fields at
Last revised on January 17, 2015 at 10:14:41. See the history of this page for a list of all contributions to it.