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\begin{document}

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\section*{quaternionic manifold}

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

\hypertarget{riemannian_geometry}{}\paragraph*{{Riemannian geometry}}\label{riemannian_geometry}

[[!include Riemannian geometry - contents]]

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

\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{holonomy}{Holonomy}\dotfill \pageref*{holonomy} \linebreak
\noindent\hyperlink{twistor_space_of_a_quaternionic_manifold}{Twistor Space of a Quaternionic Manifold}\dotfill \pageref*{twistor_space_of_a_quaternionic_manifold} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


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

There is some variation in the literature on what one calls a ``[[quaternionic]] [[manifold]]''. The most general definition, however, encompassing all the others, is:

\begin{defn}
\label{QuaternionicManifold}\hypertarget{QuaternionicManifold}{}
\textbf{(Quaternionic manifold)}

For $n \geq 2$, a \textbf{quaternionic manifold} is a real 4n-dimensional manifold M with a $\text{GL} (n,\mathbf{H})\cdot \mathbf{H} ^{\times }$-[[G-structure|structure]] which admits a [[torsion of a G-structure|torsion-free]] [[connection]] $\nabla$ (i.e. is integrable as a [[G-structure]]).

\end{defn}
When $\nabla$ does not exist $M$ is called \emph{almost quaternionic}, and this is just a [[reduction of the structure group]] of $M$ along a [[Lie group]] inclusion $\text{GL}(n, \mathbf{H}) \hookrightarrow \text{GL}(4n, \mathbf{R})$. Note that the multiplicative group of the quaternions $\mathbf{H}^{\times}$ can be normalized, so that the second factor of $\text{GL} (n,\mathbf{H})\cdot \mathbf{H} ^{\times }$ is isomorphic to SU(2), or isomorphically again the first quaternionic unitary group Sp(1). Hence one occasionally finds the structure group reduction written $\text{GL} (n,\mathbf{H})\cdot \text{Sp}(1)$.

\begin{defn}
\label{}\hypertarget{}{}
Other definitions have been given, based on the following useful but more naïve reasoning: consider two [[almost complex structures]] on a smooth $4n$-manifold $M$, say $\langle M, I \rangle$ and $\langle M, J \rangle$, given the relation $\{ I, J \} =0$. Then call the structure $\langle M, I, J \rangle$ ``almost quaternionic'', and a map $\varphi:  \langle M, I, J \rangle \rightarrow \langle N, K, F \rangle$ of almost quaternionic structures ``quaternionic'' if it is separately [[complex analytic]] as a map $\langle M, I \rangle \rightarrow \langle N, K \rangle$ and also as a map $\langle M, J \rangle \rightarrow \langle N, F \rangle$.

When $\mathbf{R}^4$ is given a quaternionic multiplication with anti-commuting imaginary units $i$ and $j$, this is actually equivalent to $\varphi : \mathbf{R}^4 \rightarrow \mathbf{R}^4$ satisfying the [[Cauchy-Feuter]] complex:

\begin{displaymath}
i \frac{\partial}{\partial u_3} = \frac{\partial }{\partial u _4}, j \frac{\partial }{ \partial u_2} = \frac{\partial }{\partial u_4}
\end{displaymath}
\begin{displaymath}
i \frac{\partial}{\partial u_1} = \frac{\partial }{\partial u_2}, j \frac{\partial }{\partial u_1 } = \frac{\partial }{\partial u_3}
\end{displaymath}
known as a starting point for defining a notion of ``quaternionic [[holomorphy]]'', since the Cauchy-Feuter complex consists of two separate [[Cauchy-Riemann]] systems in the imaginary units $i, j$. Such a complex exists on $\mathbf{R}^{4n}$ for any $n$ as an extended Cauchy-Fueter complex consisting of the systems above, repeated for each set of four coordinates. Hence, $\mathbf{R}^{4n}$ as a smooth manifold can always be endowed with an almost quaternionic structure. On this account of quaternionic structure, a \emph{quaternionic $n$-manifold} is then an almost quaternionic structure on a smooth manifold $M$ such that $M$ has an atlas of quaternionic maps, considered with respect to the standard quaternionic structure on $\mathbf{R}^{4n}$ just described. However, such a definition has serious drawbacks, such as the fact that quaternionic projective $n$-space $\mathbf{H}P^n := \text{Sp}(n + 1)/\text{Sp}(n)\text{Sp}(1)$ is not a ``quaternionic manifold'' in this sense for any $n$. It is, however, a quaternionic manifold under the official definition above.

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

\begin{example}
\label{QuaternionKaehlermanifoldsAreQuaternionicManifolds}\hypertarget{QuaternionKaehlermanifoldsAreQuaternionicManifolds}{}
\textbf{([[quaternion-Kähler manifolds]] are [[quaternionic manifolds]])}

By definition, a [[quaternion-Kähler manifold]] $M$ has [[holonomy group]] contained in the [[direct product group]] [[Sp(n)]]$\times$[[Spin(3)|Sp(1)]], admitting an extension of the [[Levi-Civita connection]] $\nabla$ on the holonomy bundle as torsion-free. Thus a quaternion-Kähler manifold is automatically quaternionic.

Such extension $\nabla_\text{quat}$ of $\nabla$ however is not unique, since $\nabla_\text{quat} + \mathcal{S}$ is another Sp(n)Sp(1)-preserving connection, where $\mathcal{S}$ is a (1, 2)-tensor such that for every $p \in M$, $\mathcal{S}(p)$ takes values in the first [[prolongation]] of the [[Lie algebra]] for the [[G-structure]].

\end{example}
\hypertarget{properties}{}\subsection*{{Properties}}\label{properties}

\hypertarget{holonomy}{}\subsubsection*{{Holonomy}}\label{holonomy}

(\ldots{})

\hypertarget{twistor_space_of_a_quaternionic_manifold}{}\subsubsection*{{Twistor Space of a Quaternionic Manifold}}\label{twistor_space_of_a_quaternionic_manifold}

(\ldots{})

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[biquaternionic manifold]], [[hypercomplex manifold]],


\item [[quaternionic-Kähler manifold]]


\item [[Kähler manifold]], [[hyper-Kähler manifold]]



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

Book references include:

\begin{itemize}%
\item [[Arthur Besse]], \emph{Einstein Manifolds}, Springer-Verlag 1987.


\item [[Dominic Joyce]], \emph{Compact Manifolds with Special Holonomy}, Oxford University Press, 2000.



\end{itemize}
Classical references:

\begin{itemize}%
\item [[Edmond Bonan]], ``Sur les G-structures de type quaternionien'', Cahiers de topologie et géométrie différentielle catégoriques, tome 9, no 4 (1967), p. 389-463.


\item S.M. Salamon, ``Differential Geometry of Quaternionic Manifolds'', Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55.


\item [[Dmitry Alekseevsky]] and [[Stefano Marchiafava]], ``Quaternionic-like structures on a manifold - note I. 1-integrability and integrability conditions'', Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Serie 9, Vol. 4 (1993), n.1, p. 43–52.



\end{itemize}
Holonomy, connections, and twistor spaces:

\begin{itemize}%
\item Stefan Ivanov, Ivan Minchev, Simeon Zamkovoy, \href{https://arxiv.org/abs/math/0511525}{Twistors of Almost Quaternionic Manifolds}


\item Gueo Grantcharov and Yat Sun Poon, \href{https://arxiv.org/abs/math/9908015v1}{Geometry of Hyper-Kahler Connections with Torsion}



\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Quaternionic_manifold}{Quaternionic manifold}}

\end{itemize}
[[!redirects quaternionic manifolds]] [[!redirects quaternion manifold]]



\end{document}