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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For $(X,g)$ a Riemannian manifold or pseudo-Riemannian manifold its isometry group
is that subgroup of the group of all diffeomorphisms $\phi : X \to X$ that are isometries: which preserve the metric $g$ in that
The Lie algebra of $Iso(X,g)$ is spanned by the Killing vectors of $(X,g)$.
The isometry group of Minkowski spacetime is the Poincaré group.
The isometry group of anti de Sitter spacetime is the anti de Sitter group.
The isometry group of de Sitter spacetime is the de Sitter group.
Special subgroups of isometry groups:
Last revised on November 13, 2018 at 05:44:25. See the history of this page for a list of all contributions to it.