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)
For $(X,g)$ a (pseudo-)Riemannian manifold a Killing tensor is a section of a symmetric power of the tangent bundle
which is covariantly constant in that
For $k = 1$ this reduces to the notion of Killing vector.
For every Killing tensor $K$ on $(X,g)$ the dynamics of the relativistic particle on $X$ has a further conserved quantity. In the canonical case $K = g$ this quantity is the Hamiltonian of the particle (in the case of a relativistic particle its four-velocity normalization).
The analog of this for spinning particles and superparticles are Killing-Yano tensors.