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)
We state the definition below in Def. . First we need the following preliminaries:
Denote by $\mathbf{D}$ the duplex (sometimes called paracomplex or hyperbolic) numbers, which is the associative algebra over the real numbers $\mathbf{R}$ generated by the elements $1, \mathbf{k}$ s.t. $\mathbf{k}^2 = 1$, in other words the real Clifford algebra $C\ell _{1, 0} (\mathbf{R})$. A PDE theory analogous to complex holomorphy may be developed based on this algebra; for a function $\psi = (\psi_1 , \psi_2) : \mathbf{D} \rightarrow \mathbf{D}$ (under an identification of $\mathbf{D}$ with $\mathbf{R}^2$), paracomplex linearity of $d \psi$ means the real components of $\psi$ must satisfy the equations $\partial_1 \psi_2 = \partial_2 \psi_1$ and $\partial_1 \psi_1 = \partial_2 \psi_2$. These are the hyperbolic analogue of the Cauchy-Riemann equations, although clearly not defining an elliptic system? since the components of $\psi$ therefore satisfy the wave equations $\Box \psi_i =0$. In the context of differential geometry over $\mathbf{D}$, such functions are sometimes called paraholomorphic.
As with CR geometry, one can study real hypersurfaces of manifolds carrying such hyperbolic structure (discussed below):
(HR manifold)
An HR manifold (for “hyperbolic-real”) is a differentiable manifold $M$ together with a sub-bundle $H$ of the hyperbolified tangent bundle, $H \subset TM \otimes_\mathbf{R} \mathbf{D}$ such that $[H, H ] \subset H$ and $H \cap H^{\dagger} =\{ 0 \}$, where $\dagger$ is the bundle involution s.t. $\mathbf{k} \mapsto - \mathbf{k}$.
G-structures of this type only exist on even-dimensional differentiable manifolds, and have been known since the classical contributions of Libermann. Explicitly, an almost-hyperbolic structure on a real $2n$-manifold $M$ is determined by a reduction of the structure group $\text{GL}(n, \mathbf{D}) \hookrightarrow \text{GL}(2n, \mathbf{R})$, defining a bundle automorphism $K \in \text{End}(TM)$ s.t. $K^2 = \text{id}_{TM}$. Locally this means that $K$, when integrable, is of the form:
on fibers, so that the transition functions of $M$ satisfy the wave equations just discussed. One can also give various integrability conditions of $K$, although as a Dirac structure the simplest to state is the vanishing of the Nijenhuis tensor $N_K (X, Y) = [KX, KY] + [X, Y] - K ([KX, Y] + [X, KY])$, a sign away from its complex analogue.
(…)
(…)
BR manifold?
The classical articles are:
P. Libermann, Sur le probleme d’equivalence de certaines structures infinitesimales, Ann. Mat. Pura Appl., 36 (1954), 27-120.
P. Libermann, Sur les structures presque paracomplexes, C.R. Acad. Sci. Paris, 234 (1952), 2517-2519.
A convenient modern survey appears in::
And a more recent article done in the style of generalized complex geometry is: