nLab HR manifold

Definition

Basic definition

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):

Definition

(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. $\mathbf{k} \mapsto - \mathbf{k}$.

As $G$-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 $2n$-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:

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

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.

Examples

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Other Properties

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Other Clifford-type Hypersurfaces

• BR manifold?

• CR manifold

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.