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
Euler-Lagrange equation, de Donder-Weyl formalism?,
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
For
a differentiable function between differentiable manifolds (e.g. a smooth function between smooth manifolds) a point $q \in f(X) \subset Y$ in the image of $f$ is called a regular value of $f$ if at all points $p \in f^{-1}(\{q\})$ in its preimage, the differential
is a surjective function between the corresponding tangent spaces.
A function all whose values are regular values is called a submersion.
(e.g. Kosinski 93, II (2.4))
(relation to transversality)
That $q \in Y$ is a regular value (Def. ) of $f \colon X \to Y$ means equivalently that $f$ is a transverse map to the submanifold-inclusion $\ast \overset{q}{\hookrightarrow} Y$.
In this sense transversality generalizes the concept of regular values.
The inverse function theorem implies that:
The inverse image $f^{-1}(q) \subset X$ of a smooth function $f \colon X \to Y$ at a regular value $q \in Y$ is a smooth manifold of $X$.
Together with the Thom's transversality theorem, this is the key to the proof of the Pontryagin-Thom isomorphism.
Last revised on March 9, 2019 at 17:32:19. See the history of this page for a list of all contributions to it.