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)
Around every point of a Riemannian manifold there is a coordinate system such that the Levi-Civita connection of the metric pulled back to these coordinates vanishes at that point. (Notice that the Riemann curvature will not in general vanish even at that point).
In the context of general relativity this reflects aspects of the equivalence princile (physics)?.
In the sense of integrability of G-structures, Riemann normal coordinates exhibit the first-order integrability of orthogonal structure, see at integrability of G-structures – Examples – Orthogonal structure.
Last revised on January 15, 2015 at 14:18:58. See the history of this page for a list of all contributions to it.