nLab
HR manifold

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential 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}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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):

Definition

(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.

Examples

(…)

Other Properties

(…)

Other Clifford-type Hypersurfaces

References

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::

  • V. Cruceanu, P. Fortuny and P. M. Gadea, A Survey on Paracomplex Geometry , Rocky Mountain J. Math. Volume 26, Number 1 (1996), 83-115.

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

  • Aïssa Wade, Dirac structures and paracomplex manifolds, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 889–894.

Last revised on March 14, 2019 at 04:09:31. See the history of this page for a list of all contributions to it.