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\begin{document}

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\section*{HR manifold}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{differential_geometry}{}\paragraph*{{Differential geometry}}\label{differential_geometry}

[[!include synthetic differential geometry - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{basic_definition}{Basic definition}\dotfill \pageref*{basic_definition} \linebreak
\noindent\hyperlink{hypmanifold}{As $G$-structure}\dotfill \pageref*{hypmanifold} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{other_properties}{Other Properties}\dotfill \pageref*{other_properties} \linebreak
\noindent\hyperlink{other_cliffordtype_hypersurfaces}{Other Clifford-type Hypersurfaces}\dotfill \pageref*{other_cliffordtype_hypersurfaces} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\hypertarget{basic_definition}{}\subsubsection*{{Basic definition}}\label{basic_definition}

We state the definition below in Def. \ref{HRManifold}. First we need the following preliminaries:

Denote by $\mathbf{D}$ the duplex (sometimes called \emph{paracomplex} or \emph{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 differntial equation|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 \emph{paraholomorphic}.

As with [[CR geometry]], one can study real hypersurfaces of manifolds carrying such hyperbolic structure (discussed \hyperlink{hypmanifold}{below}):

\begin{defn}
\label{HRManifold}\hypertarget{HRManifold}{}
\textbf{(HR manifold)}

An \emph{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. $\mathbf{k} \mapsto - \mathbf{k}$.

\end{defn}
\hypertarget{hypmanifold}{}\subsubsection*{{As $G$-structure}}\label{hypmanifold}

[[G-structures]] of this type only exist on even-dimensional differentiable manifolds, and have been known since the classical contributions of Libermann. Explicitly, an \emph{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:

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

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

(\ldots{})

\hypertarget{other_properties}{}\subsection*{{Other Properties}}\label{other_properties}

(\ldots{})

\hypertarget{other_cliffordtype_hypersurfaces}{}\subsection*{{Other Clifford-type Hypersurfaces}}\label{other_cliffordtype_hypersurfaces}

\begin{itemize}%
\item [[BR manifold]]


\item [[CR manifold]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The classical articles are:

\begin{itemize}%
\item P. Libermann, \emph{Sur le probleme d’equivalence de certaines structures infinitesimales}, Ann. Mat. Pura Appl., 36 (1954), 27-120.


\item P. Libermann, \emph{Sur les structures presque paracomplexes}, C.R. Acad. Sci. Paris, 234 (1952), 2517-2519.



\end{itemize}
A convenient modern survey appears in::

\begin{itemize}%
\item V. Cruceanu, P. Fortuny and P. M. Gadea, \emph{A Survey on Paracomplex Geometry} , Rocky Mountain J. Math. Volume 26, Number 1 (1996), 83-115.

\end{itemize}
And a more recent article done in the style of [[generalized complex geometry]] is:

\begin{itemize}%
\item Aïssa Wade, \emph{Dirac structures and paracomplex manifolds}, C. R. Acad. Sci. Paris, Ser. I 338 (2004) 889–894.

\end{itemize}
[[!redirects HR manifolds]]

[[!redirects HR-manifold]] [[!redirects HR-manifolds]]

[[!redirects HR geometry]] [[!redirects HR geometries]]



\end{document}