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\begin{document}

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{\fcolorbox{red}{blue}{\color{white}
{\large\bf Harmonic-Killing vector fields }}\\
{\fcolorbox{red}{blue}{\color{white}
{\large\bf on K\"{a}hler manifolds }}

\hspace*{1cm}
{\small\bf C.T.J. Dodson}

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\hspace*{1cm}
The Lie algebra action of $\Gamma(TM)$ on connections
on $M,$ ($Con(M)$), is given by $$
Con(M)\times\Gamma(TM)\rightarrow Con(M)\,, \quad
(\nabla,X)\mapsto {{\cal L}}_{_X} \nabla\,. $$ Namely, $$ {\cal
L}_{_X} \nabla=ev\vert_{_{t=0}}\circ\,
\displaystyle{\frac{\partial}{\partial t}}\circ\,
\nabla^{^{\varphi_{{}_t}}}, $$ where $\nabla^{^{\varphi_{{}_t}}}$
is the result of the natural action  of
 $\Gamma(TM)$ on $Con(M)$, that is,
$$
\nabla^{^{\varphi_{{}_t}}}_Z Y = {\varphi^*_{_t}} \circ
(\nabla_{_{Z^{\varphi_{-t}}}} Y^{^{\varphi_{-t}}})\circ
{\varphi^*_{_{-t}}},
$$
and by $W^{\varphi}$ we denote the right action of Diff$(M)$  on
$\Gamma(TM)$, i.e., $ W^{\varphi}= \varphi^* \circ W \circ
\varphi^{-1*}, $ $\varphi\in$Diff$(M)$, $W\in\Gamma(TM)$.
}

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For more details see \\                    %% Link to remote document
{\small
\htmladdnormallink{http://www.ma.umist.ac.uk/kd/PREPRINTS/HKaehler.pdf}
{http://www.ma.umist.ac.uk/kd/PREPRINTS/HKaehler.pdf}}
}
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For more details see
{\small \htmladdnormallink{artex.pdf}{artex.pdf}}    %% Link to artex.pdf
}

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{\bf Theorem }\\
$M$ complete, connected, simply-connected\\
$\exists$ naturally reductive homogeneous structure\\
$\Leftrightarrow T_{XYZ}+T_{YXZ}=0 \hspace{1in}\forall X,Y,Z \in {\cal M }$

{\bf Theorem }\\
$T=D-\nabla$ is naturally reductive\\
$\Leftrightarrow \nabla R=0.$
}





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