HR manifold


Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



Basic definition

We state the definition below in Def. . First we need the following preliminaries:

Denote by D\mathbf{D} the duplex (sometimes called paracomplex or hyperbolic) numbers, which is the associative algebra over the real numbers R\mathbf{R} generated by the elements 1,k1, \mathbf{k} s.t. k 2=1\mathbf{k}^2 = 1 , in other words the real Clifford algebra C 1,0(R)C\ell _{1, 0} (\mathbf{R}). A PDE theory analogous to complex holomorphy may be developed based on this algebra; for a function ψ=(ψ 1,ψ 2):DD\psi = (\psi_1 , \psi_2) : \mathbf{D} \rightarrow \mathbf{D} (under an identification of D\mathbf{D} with R 2\mathbf{R}^2), paracomplex linearity of dψd \psi means the real components of ψ\psi must satisfy the equations 1ψ 2= 2ψ 1\partial_1 \psi_2 = \partial_2 \psi_1 and 1ψ 1= 2ψ 2\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 ψ\psi therefore satisfy the wave equations ψ i=0\Box \psi_i =0. In the context of differential geometry over D\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 MM together with a sub-bundle HH of the hyperbolified tangent bundle, HTM RDH \subset TM \otimes_\mathbf{R} \mathbf{D} such that [H,H]H[H, H ] \subset H and HH ={0}H \cap H^{\dagger} =\{ 0 \} , where \dagger is the bundle involution s.t. kk\mathbf{k} \mapsto - \mathbf{k}.

As GG-structure

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 2n2n-manifold MM is determined by a reduction of the structure group GL(n,D)GL(2n,R)\text{GL}(n, \mathbf{D}) \hookrightarrow \text{GL}(2n, \mathbf{R}), defining a bundle automorphism KEnd(TM)K \in \text{End}(TM) s.t. K 2=id TMK^2 = \text{id}_{TM}. Locally this means that KK, when integrable, is of the form:

(0 I n I n 0) \left( \begin{matrix} 0 & I_n \\ I_n & 0 \end{matrix} \right)

on fibers, so that the transition functions of MM satisfy the wave equations just discussed. One can also give various integrability conditions of KK, 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])N_K (X, Y) = [KX, KY] + [X, Y] - K ([KX, Y] + [X, KY]) , a sign away from its complex analogue.



Other Properties


Other Clifford-type Hypersurfaces


The classical articles are:

A convenient modern survey appears in::

And a more recent article done in the style of generalized complex geometry is: