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\section*{quaternion-Kähler 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{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{as_part_of_the_berger_classification}{As part of the Berger classification}\dotfill \pageref*{as_part_of_the_berger_classification} \linebreak
\noindent\hyperlink{as_riemannian_manifolds}{As $\mathbb{H}$-Riemannian manifolds}\dotfill \pageref*{as_riemannian_manifolds} \linebreak
\noindent\hyperlink{as_quaternionic_manifolds}{As quaternionic manifolds}\dotfill \pageref*{as_quaternionic_manifolds} \linebreak
\noindent\hyperlink{AsEinsteinManifolds}{As Einstein manifolds}\dotfill \pageref*{AsEinsteinManifolds} \linebreak
\noindent\hyperlink{characteristic_classes}{Characteristic classes}\dotfill \pageref*{characteristic_classes} \linebreak
\noindent\hyperlink{reduction_to_hyperkhler_structure}{Reduction to hyper-Kähler structure}\dotfill \pageref*{reduction_to_hyperkhler_structure} \linebreak
\noindent\hyperlink{PositiveQuaternionKaehlerManifolds}{Positive quaternion-Kähler manifolds}\dotfill \pageref*{PositiveQuaternionKaehlerManifolds} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \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}

A [[Riemannian manifold]] $(X,g)$ of [[dimension]] $4n$ for $n \geq 2$ is called a \emph{quaternion-Kähler manifold} if its [[holonomy group]] is a [[subgroup]] of [[Sp(n).Sp(1)]] (where [[Sp(n)]] is the $n$th [[quaternionic unitary group]], and in particular $Sp(1) \simeq SU(2) \simeq$ [[Spin(3)]], and the [[central product]] is the [[quotient group]] of the [[direct product group]] by the [[diagonal]] [[center]] $\mathbb{Z}/2$).

If the [[holonomy]] group is in fact a [[subgroup]] of just the $Sp(n)$-factor, one speaks of a \emph{[[hyperkähler manifold]]}.

Quaternion-Kähler manifolds are necessarily [[Einstein manifolds]] (see \hyperlink{AsEinsteinManifolds}{below}). In particular their [[scalar curvature]] $R$ is [[constant function|constant]], and hence a [[real number]] $R \in \mathbb{R}$. If the scalar curvature is [[positive number|positive]], then one speaks of a \emph{positive quaternion-Kähler manifold}.

$\backslash$linebreak

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

\hypertarget{as_part_of_the_berger_classification}{}\subsubsection*{{As part of the Berger classification}}\label{as_part_of_the_berger_classification}

[[!include special holonomy table]]

\hypertarget{as_riemannian_manifolds}{}\subsubsection*{{As $\mathbb{H}$-Riemannian manifolds}}\label{as_riemannian_manifolds}

[[!include normed division algebra Riemannian geometry -- table]]

\hypertarget{as_quaternionic_manifolds}{}\subsubsection*{{As quaternionic manifolds}}\label{as_quaternionic_manifolds}

\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 of a G-structure|torsion]]-free. Thus a quaternion-Kähler manifold is automatically a [[quaternionic manifold]].

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{AsEinsteinManifolds}{}\subsubsection*{{As Einstein manifolds}}\label{AsEinsteinManifolds}

[[quaternion-Kähler manifolds]] are [[Einstein manifolds]] (e.g. \hyperlink{Cortes05}{Cortés 05, slide 22})

\hypertarget{characteristic_classes}{}\subsubsection*{{Characteristic classes}}\label{characteristic_classes}

\begin{prop}
\label{CharacteristicClassesForSpin5Spin3Structure}\hypertarget{CharacteristicClassesForSpin5Spin3Structure}{}
Let $X$ be a [[closed manifold|closed]] [[smooth manifold]] of [[dimension]] 8 with [[Spin structure]]. If the [[frame bundle]] moreover admits [[G-structure]] for

$\;\;G =$ [[Sp(n).Sp(1)|Sp(2).Sp(1)]] $\hookrightarrow$ [[SO(8)]]

then the [[Euler class]] $\chi$, the [[second Pontryagin class]] $p_2$ and the [[cup product]]-square $(p_1)^2$ of the [[first Pontryagin class]] of the [[frame bundle]]/[[tangent bundle]] are related by

\begin{equation}
8 \chi
  \;=\;
  4 p_2 - (p_1)^2
  \,.
\label{EulerClassInTermsOfPontryagin}\end{equation}
\end{prop}
(\hyperlink{CadekVanzura98}{Čadek-Vanžura 98, Theorem 8.1 with Remark 8.2})

\begin{remark}
\label{}\hypertarget{}{}
The same conclusion \eqref{EulerClassInTermsOfPontryagin} also holds for $Spin(7)$-structure, see \href{Spin7-manifold#CharacteristicClassesForSpinStructure}{there}

\end{remark}
See also at \emph{[[C-field tadpole cancellation]]}.

\hypertarget{reduction_to_hyperkhler_structure}{}\subsubsection*{{Reduction to hyper-Kähler structure}}\label{reduction_to_hyperkhler_structure}

A quaternion-Kähler manifold $(X,g)$ is a [[hyper-Kähler manifold]], hence has $Sp(n) \hookrightarrow Sp(n)\cdot Sp(1)$-structure, precisely if its [[scalar curvature]], which is a [[constant function|constant]] by $(X,g)$ being an [[Einstein manifold]], vanishes: $R(g) = 0$.

(e.g. \hyperlink{Amann09}{Amann 09, below Def. 1.5})

\hypertarget{PositiveQuaternionKaehlerManifolds}{}\subsubsection*{{Positive quaternion-Kähler manifolds}}\label{PositiveQuaternionKaehlerManifolds}

\begin{defn}
\label{PositiveQuaternionKaehlerManifold}\hypertarget{PositiveQuaternionKaehlerManifold}{}
A quaternion-Kähler manifold $(X,g)$ is called \emph{positive} if

\begin{enumerate}%
\item it is a [[geodesically complete]]


\item its [[scalar curvature]], which is a [[constant function|constant]] by $(X,g)$ being an [[Einstein manifold]], is a [[positive number]], $R(g) \gt 0$.



\end{enumerate}
\end{defn}
(\hyperlink{Salamon82}{Salamon 82, Section 6}, see e.g. \hyperlink{Amann09}{Amann 09, Def. 1.5})

\begin{prop}
\label{}\hypertarget{}{}
A [[connected topological space|connected]] [[positive quaternion-Kähler manifold]] (Def. \ref{PositiveQuaternionKaehlerManifold}) is necessarily [[compact topological space|compact]].

\end{prop}
(\hyperlink{Salamon82}{Salamon 82, p. 158 (16 of 29)})

\begin{prop}
\label{}\hypertarget{}{}
A [[connected topological space|connected]] [[positive quaternion-Kähler manifold]] (Def. \ref{PositiveQuaternionKaehlerManifold}) is necessarily [[simply connected topological space|simply connected]].

\end{prop}
(\hyperlink{Salamon82}{Salamon 82, Theorem 6.6})

\begin{prop}
\label{}\hypertarget{}{}
For each [[dimension]] $dim(X)$ there is a [[finite number]] of [[isometry]] [[isomorphism class|classes]] of [[positive quaternion-Kähler manifolds]] (Def. \ref{PositiveQuaternionKaehlerManifold}).

\end{prop}
(\hyperlink{LeBrunSalamon94}{LeBrun-Salamon 94, Theorem 0.1})

\begin{prop}
\label{WoldSpacesArePositiveQuaternionKaehler}\hypertarget{WoldSpacesArePositiveQuaternionKaehler}{}
(\textbf{[[Wolf spaces]] are [[positive quaternion-Kähler manifolds]])}

Every [[Wolf space]] is a [[positive quaternion-Kähler manifold]].

\end{prop}
In fact the [[Wolf spaces]] are the only known examples of [[positive quaternion-Kähler manifold]] (which is not hyper-Kähler ?!), as of today (e.g. \hyperlink{Salamon82}{Salamon 82, Section 5}).

This leads to the \textbf{conjecture} that in every dimension, the [[Wolf spaces]] are the only [[positive quaternion-Kähler manifolds]].

The conjecture has been proven for the following [[dimensions]]

\begin{itemize}%
\item $d = 4$ (Hitchin)


\item $d = 8$ (\hyperlink{PoonSalamon91}{Poon-Salamon 91}, \hyperlink{LeBrunSalamon94}{LeBrun-Salamon 94})



\end{itemize}
$\backslash$linebreak

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

The archetypical example is

\begin{itemize}%
\item [[quaternionic projective space]] $\mathbb{H}P^2$

\end{itemize}
This is the first of the list of examples of spaces that are both [[quaternion-Kähler manifolds]] as well as a [[symmetric spaces]], called \emph{[[Wolf spaces]]}.

See around Prop. \ref{WoldSpacesArePositiveQuaternionKaehler} above.

$\backslash$linebreak

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

\begin{itemize}%
\item [[quaternionic manifold]], [[Kähler manifold]], [[hyper-Kähler manifold]]

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

Original articles:

\begin{itemize}%
\item [[Simon Salamon]], \emph{Quaternionic Kähler manifolds}, Invent Math (1982) 67: 143. (\href{https://doi.org/10.1007/BF01393378}{doi:10.1007/BF01393378})


\item Y. S. Poon, [[Simon Salamon]], \emph{Quaternionic Kähler 8-manifolds with positive scalar curvature}, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 (\href{https://projecteuclid.org/euclid.jdg/1214446322}{euclid:1214446322})


\item Claude LeBrun, [[Simon Salamon]], \emph{Strong rigidity of positive quaternion Kähler manifolds}, Inventiones Mathematicae 118, 1994, 109–132 (\href{https://eudml.org/doc/144231}{dml:144231}, \href{https://doi.org/10.1007/BF01231528}{doi:10.1007/BF01231528})



\end{itemize}
Exposition

\begin{itemize}%
\item [[Vicente Cortés]], \emph{Quaternionic Kähler manifolds}, 2005 (\href{https://www2.math.hu-berlin.de/gradkoll/Cortes_vorlesung1_handout.pdf}{pdf})

\end{itemize}
Textbook 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}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Quaternion-Kaehler_manifold}{Quaternion-K\"a{}hler manifold}}

\end{itemize}
Articles discussing quaternion-Kähler [[holonomy]], [[connection]], and relation to other [[hypercomplex]] structures:

\begin{itemize}%
\item Andrei Moroianu, [[Uwe Semmelmann]], ``Killing Forms on Quaternion-Kähler Manifolds'', Annals of Global Analysis and Geometry, November 2005, Volume 28, Issue 4, pp 319–335.


\item Pedersen, Poon, and Swann. ``Hypercomplex structures associated to quaternionic manifolds'', Differential Geometry and its Applications (1998) 273-293 North-Holland.


\item [[Misha Verbitsky]], ``Hyperkähler manifolds with torsion, supersymmetry and Hodge theory'', Asian J. Math, V. 6 No. 4, pp. 679-712, Dec. 2002.


\item [[Simon Salamon]], \emph{Differential Geometry of Quaternionic Manifolds}, Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55 (\href{http://www.numdam.org/item/ASENS_1986_4_19_1_31_0/}{numdam:ASENS\_1986\_4\_19\_1\_31\_0})



\end{itemize}
See also

\begin{itemize}%
\item Claude LeBrun, \emph{On complete quaternionic-Kähler manifolds}, Duke Math. J. Volume 63, Number 3 (1991), 723-743 (\href{https://projecteuclid.org/euclid.dmj/1077296077}{euclid:1077296077})


\item Simon G. Chiossi, Óscar Maciá, \emph{SO(3)-Structures on 8-manifolds}, Ann. Glob. Anal. Geom. 43 (1) (2013), 1--18 (\href{https://arxiv.org/abs/1105.1746}{arXiv:1105.1746})



\end{itemize}
On [[positive quaternion-Kähler manifolds]]

\begin{itemize}%
\item Amann, \emph{Positive Quaternion Kähler Manifolds}, 2009 (\href{https://d-nb.info/996176438/34}{pdf})


\item Amann, \emph{Partial Classification Results for Positive Quaternion Kaehler Manifolds} (\href{https://arxiv.org/abs/0911.4587}{arXiv:0911.4587})



\end{itemize}
Discussion of [[characteristic classes]]:

\begin{itemize}%
\item [[Martin Čadek]], [[Jiří Vanžura]], \emph{Almost quaternionic structures on eight-manifolds}, Osaka J. Math. Volume 35, Number 1 (1998), 165-190 (\href{https://projecteuclid.org/euclid.ojm/1200787905}{euclid:1200787905})

\end{itemize}
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[[!redirects quaternion Kähler structure]] [[!redirects quaternion Kähler structures]]

[[!redirects quaternion Kaehler structure]] [[!redirects quaternion Kaehler structures]]

[[!redirects Sp(2).Sp(1)-structure]] [[!redirects Sp(2).Sp(1)-structures]]

[[!redirects Sp(2).Sp(1) structure]] [[!redirects Sp(2).Sp(1) structures]]

[[!redirects positive quaternion-Kähler manifolds]]

[[!redirects positive quaternion Kähler manifold]] [[!redirects positive quaternion Kähler manifols]]

[[!redirects positive quaternion-Kähler manifold]] [[!redirects positive quaternion-Kähler manifols]]

[[!redirects positive quaternion-Kaehler manifold]] [[!redirects positive quaternion-Kaehler manifolds]]

[[!redirects positive quaternion Kaehler manifold]] [[!redirects positive quaternion Kaehler manifolds]]

[[!redirects positive quaternionic Kähler manifold]] [[!redirects positive quaternionic Kähler manifolds]]

[[!redirects positive quaternionic-Kähler manifold]] [[!redirects positive quaternionic-Kahler manifold]] [[!redirects positive quaternionic-Kähler manifolds]] [[!redirects positive quaternionic-Kahler manifolds]]

[[!redirects positive quaternion-Kähler structure]] [[!redirects positive quaternion-Kähler structures]]

[[!redirects quaternion-Kaehler structure]] [[!redirects quaternion-Kaehler structures]]

[[!redirects quaternion Kähler structure]] [[!redirects quaternion Kähler structures]]

[[!redirects quaternion Kaehler structure]] [[!redirects quaternion Kaehler structures]]



\end{document}