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)
A Killing vector on a (pseudo-)Riemannian manifold is equivalently
a covariantly constant vector field : a vector field $v \in \Gamma(T C)$ that is annihilated by (the symmetrization of) the covariant derivative of the corresponding Levi-Civita connection;
Similarly a Killing spinor is a covariantly constant spinor.
For $(X,g)$ a Riemannian manifold (or pseudo-Riemannian manifold) a vector field $v \in \Gamma(T X)$ is called a Killing vector field if it generates isometries of the metric $g$. More precisely, if, equivalently
the Lie derivative of $g$ along $v$ vanishes: $\mathcal{L}_v g = 0$;
the flow $\exp(v) : X \times \mathbb{R} \to X$ is a flow by isometries.
The flows of Killing vectors are isometries of the Riemannian manifold onto itself.