# On Kähler Structures of Taub-NUT and Kerr Spaces

Özgür Kelekçi

(  Department of Basic Sciences & Faculty of Engineering,
University of Turkish Aeronautical Association, Ankara, Turkey
<EMAIL_ADDRESS>)

###### Abstract

In this paper, we study the Kählerian nature of Taub-NUT and Kerr spaces which
are gravitational instanton and black hole solutions in general relativity. We
show that Euclidean Taub-NUT metric is hyper-Kähler with respect to the usual
almost complex structures by employing an alternative explicit coframe, and
Euclidean Kerr metric is globally conformally Kähler. We also show that
conformally scaled Euclidean Kerr space admits a Kähler structure by applying
a conformal scaling factor stemming from the Lee-form of the original metric
or alternatively a factor coming from self-dual part of the Weyl tensor
$(W^{+})$.

Keywords: Kerr space; Taub-NUT space; Kähler manifold; Hyper-Kähler manifold;
locally conformally Kähler; Lee form.

##  1 Introduction

Taub-NUT and Kerr spaces, in addition to their significance in general
relativity, possess rich geometric properties which provide interesting
mathematical research problems. The Taub-NUT metrics were discovered by Taub
[1] and later extended by Newmann-Unti-Tamburino [2]. They are complete Ricci-
flat self-dual metrics on $\mathbb{R}^{4}$ and Hawking showed that they are
non-trivial examples of the gravitational instantons [3]. Taub-NUT space has
attracted a significant interest from both mathematics and physics research
community [4, 5, 6, 7, 8]. Despite the existence of several studies on them
there are few studies which are concentrated on their explicit hyper-Kähler
structure. Gaeta etal. obtained Taub-NUT space from Euclidean eight-
dimensional space $\mathbb{R}^{8}$ through a momentum map construction by
implementing the HKLR theorem (Hitchin, Karlhede, Lindström, Roček) [7, 9].
However, there is no explicit connection of that study to the widely used
Taub-NUT metrics such as given in [4] and [10].

The Kerr metric mathematically describes a type of rotating black hole, and it
generalizes the rotating version of Schwarzschild metric [11, 12]. Recently,
geometric modeling of observed M87* black hole as a Kerr black hole was
investigated [13]. Our focus will be on Euclidean versions of Kerr and Taub-
NUT spaces rather than Lorentzian ones since we are searching for integrable
almost complex structures. Lorentzian metrics can be transformed into
Euclidean metrics by a coordinate transformation of the form $t\rightarrow
i\tau$, known as Wick rotation in physics literature [14]. Gibbons etal.
discovered Kerr-Taub-bolt instanton solution as a rotating generalization of
the Taub-bolt solution in a similar way that the Kerr solution generalizes the
Schwarzschild solution [15]. Dixon showed that Wick-rotated Kerr metric, also
called Riemannian Kerr metric, is ambitoric [16]. Aliev etal studied the
general Kerr-Taub-bolt instanton and obtained a harmonic self-dual 2-form
[17]. However, it is not clear whether the obtained self-dual 2-form is the
genuine Kähler form with respect to the given metric and there is no mention
of its almost complex structure. Babak etal studied some Kähler metrics via
Lorentzian geometry where they have given Kähler structure only on some
specific sections of Kerr space [18]. Despite the existence of these studies
about Kähler structures of Kerr and Taub-NUT spaces, alternative explicit
derivations of their Kähler structures starting directly from the commonly
used metrics will be helpful for future studies.

In this paper, we first introduce an alternative explicit co-frame for the
commonly used form of Euclidean Taub-NUT metric. We explicitly compute self-
dual Kähler forms and corresponding almost complex structures
$(J_{1},J_{2},J_{3})$ by using this alternative co-frame. We also show that
they are integrable and satisfy the quaternionic relations which completes the
hyper-Kähler structure. Second, we focus on Wick-rotated Kerr metric
(Euclidean Kerr) and show that it is globally conformally Kähler by
calculating its Kähler form, Lee-form, and almost complex structure. Then, by
applying a conformal scaling factor we show that there exists a genuine closed
Kähler form and integrable almost complex structure for the conformally scaled
Euclidean Kerr metric. As another verification we computed matrix form of
$W^{+}$ and showed that the conformal factor obtained from exact Lee-form
matches with the conformal factor of Derdziński’s theorem.

##  2 Preliminaries

In this section we shall give some definitions and theorems needed to prove
our results. Note that we use Einstein summation convention throughout this
paper, i.e. summing over repeated indices is assumed.

######  Definition 1.

Let M be an n-dimensional complex manifold and let ${z^{\mu}}$ be local
coordinates on a coordinate neighborhood U. A mixed tensor
${J_{\alpha}}^{\beta}$ can be defined such that

$J=i\ dz^{\mu}\otimes\frac{\partial}{\partial z^{\mu}}-\ i\
d{\bar{z}}^{\mu}\otimes\frac{\partial}{\partial{\bar{z}}^{\mu}}~{}\
,\quad\mu=1,2,...,n$ (2.1)

J is called almost complex structure and has the following properties:

  * (i)

J is a tensor.

  * (ii)

J is real.

  * (iii)

${J_{\alpha}}^{\beta}\ {J_{\beta}}^{\sigma}=-{\delta_{\alpha}}^{\sigma}~{}\
,\quad\alpha\ ,\ \sigma=1,...\ ,2n.$

######  Theorem 1.

An almost complex structure J on a manifold $M$ is integrable if and only if
the Nijenhuis tensor $N(X,Y)$ vanishes for any two vector fields X and Y [19].
Nijenhuis tensor is defined for any vector fields $X,Y\ \in T_{p}M$ as follows

$N\left(X,Y\right)=\left[X,Y\right]+J\left[JX,Y\right]+J\left[X,JY\right]-\left[JX,JY\right]$
(2.2)

######  Theorem 2.

Let $(M,J)$ be an almost complex manifold. If J is integrable, the manifold
$M$ is complex. This is also known as Newlander-Nirenberg theorem [20].

######  Definition 2.

Let $\left(M,J\right)$ be a complex manifold. A Riemannian metric on M is
called Hermitian if it is compatible with the complex structure J of M,

$g(JX,JY)=g(X,Y)\ ~{}\text{for all}\ ~{}X,Y\in T_{p}M\ ~{}\text{and }\ ~{}p\in
M.$ (2.3)

Then the associated differential two-form $\omega$ defined by

$\omega(X,Y)=g(JX,Y)$ (2.4)

is called the Kähler form of the Hermitian metric. If moreover the form
$\omega$ is closed, the metric is called Kähler.

######  Definition 3.

Let $(M,g)$ be a 4k-dimensional Hermitian manifold admitting three almost
complex structures I, J, K. $(M,g)$ is called hyper-Kähler if I, J, K
satisfies the quaternionic relations such as
$I^{2}=J^{2}=K^{2}=-Id,I.J=-J.I=K$. Moreover it can be shown that $(M,g)$ is
hyper-Kähler if and only if $Hol(g)\subset Sp(k)$ [19].

######  Theorem 3.

A manifold with a metric of Lorentz signature cannot admit an almost Hermitian
structure [21].

######  Theorem 4.

The Hermitian manifold $(M^{2n},J,g)$ is locally conformally Kähler (l.c.K) if
and only if there exists a globally defined closed 1-form (called Lee form)
$\xi$ on $M^{2n}$ so that [22]

$d\ \omega=\xi\wedge\omega$ (2.5)

Furthermore, the Lee form $\xi$ is uniquely determined by any of the following
formulae [23].

$\xi=-\frac{1}{n-1}\left(\delta\omega\right)\circ
J\quad\text{or}\quad\xi_{i}=-\frac{2}{m-2}{(\nabla}_{\alpha\
}J_{\beta}^{\alpha}){\ J}_{i}^{\beta}$ (2.6)

where $\delta$ is the co-differential operator and
$\text{dim}\left(M\right)=m=2n$. If moreover the Lee form $\xi$ is exact then
$(M^{2n},J,g)$ is globally conformally Kähler. On a l.c.K. manifold,
differentials of the negative of the logarithm of local conformal factors can
be shown to agree on the overlaps so to form a global 1-form $\xi$.
Conversely, starting with a closed 1-form on $M^{2n}$ satisfying (2.5),
Poincaré lemma guarantees that there is an open cover
$\mathcal{U}=\\{U_{k}\\}_{k\in I}$ of $M^{2n}$ and a family of $C^{\infty}$
functions $f_{k}:U_{k}\rightarrow\mathbb{R}$ so that $\xi=df_{k}$ on $U_{k}$.
Also, by multiplying $\omega$ with $e^{-f_{k}}$ one gets $d(e^{-f_{k}}\
\omega)=0$ and hence $\hat{g}=e^{-f_{k}}\ g$ becomes a Kähler metric on
$U_{k}$.

######  Theorem 5.

Let $(M,g)$ be an oriented 4-dimensional Einstein manifold. Assume that the
self-dual Weyl tensor $W^{+}$ does not vanish. Then there exist two equivalent
properties [24]

i) $W^{+}$ has, at each point, at most two distinct eigenvalues on
$\Lambda_{2}^{+}$

ii) up to a double cover, the metric $|W^{+}|^{2/3}g$ is Kähler (with respect
to some complex structure on $M$ compatible with the original orientation).

where $\Lambda_{2}^{+}$ is bundle of self-dual 2-forms on $M$.

Theorem 5 can be used for an alternative verification of the conformal Kähler
structure. Let us define self-dual ($\Omega_{+}$) and anti-self-dual
($\Omega_{-}$) bases for $\Lambda^{2}$ (bundle of 2-forms on $M$) using
orthonormal basis one-forms $e^{i}$ as,

$\Omega_{\pm}=\left\\{e^{1}\wedge e^{2}\pm e^{3}\wedge e^{4},e^{1}\wedge
e^{3}\pm e^{4}\wedge e^{2},e^{1}\wedge e^{4}\pm e^{2}\wedge e^{3}\right\\}$

The Riemann curvature tensor defines a self-adjoint transformation
$\mathcal{R}:\Lambda^{2}\rightarrow\Lambda^{2}$ and it can be considered as a
$6\times 6$ block matrix with respect to the decomposition
$\Lambda^{2}=\Lambda_{+}^{2}\oplus\Lambda_{-}^{2}$ [25]

$\mathcal{R}\Lambda^{2}=\biggl{(}\begin{array}[]{c;{2pt/2pt}c}W^{+}+\frac{R}{12}I&\mathring{Ric}\\\
\hdashline[2pt/2pt]\overset{}{\mathring{Ric}}&W^- + \frac{R}{12}
I\end{array}\biggr{)}\biggl{(}\begin{array}[]{c}\Omega_{+}\\\
\hdashline[2pt/2pt]\Omega_{-}\end{array}\biggr{)}$ (2.7)

where $W^{\pm}$ are the self-dual and anti-self-dual parts of Weyl tensor in
matrix form, $I$ is the identity map of $\Lambda^{2}$ and $\mathring{Ric}$ is
the trace-free Ricci curvature ($Ric-\frac{R}{4}g$). Upper left block matrix
is also represented by a $3\times 3$ matrix $A$ whose entries can be computed
by the following equation [10]

$\displaystyle A_{ij}$ $\displaystyle=$
$\displaystyle(R_{0i0j}+\frac{1}{2}\epsilon_{jkl}R_{0ikl})+\frac{1}{2}\epsilon_{imn}(R_{mn0j}+\epsilon_{jkl}R_{mnkl})$
(2.8) $\displaystyle A$ $\displaystyle=$ $\displaystyle W^{+}+\frac{R}{12}I$

##  3 Main Results

###  3.1 Hyper-Kähler Structure Of Euclidean Taub-Nut Metric

We will study Euclidean Taub-NUT space and use the metric given in [4].
Euclidean Taub-NUT metric is given in coordinates as

$\tilde{g}=V\left(dx^{2}+dy^{2}+dz^{2}\right)+V^{-1}\left(dt+\Theta\right)^{2}$
(3.1)
$g_{TN}=\frac{\rho+1}{4\rho}d\rho^{2}+\rho\left(\rho+1\right)\left(\sigma_{1}^{2}+\sigma_{2}^{2}\right)+\frac{\rho}{\rho+1}\sigma_{3}^{2}$
(3.2)

where $V=1+1/2r$ and $\Theta$ is a 1-form on $\mathbb{R}^{3}$ satisfying
$d\Theta=\ast\ dV$. x, y and z are coordinates parameterizing $\mathbb{R}^{3}$
while t being another vertical coordinate. $r$ is the radial coordinate for
$\mathbb{R}^{3}$ satisfying $r^{2}=x^{2}+y^{2}+z^{2}$. It is easy to show that
the metrics in (3.1) and (3.2) are isometric by using the transformation map
$f:\left(x,y,z,t\right)\mapsto\left(\rho,\theta,\phi,\psi\right)$ defined by
the following relations

$\displaystyle x$ $\displaystyle=$
$\displaystyle\frac{\rho}{2}\sin{\theta}\cos{\phi}\ ,\ \
y=\frac{\rho}{2}\sin{\theta}\sin{\phi}\ ,\ \ z=\frac{\rho}{2}\cos{\theta},$
$\displaystyle V$ $\displaystyle=$ $\displaystyle 1+\frac{1}{\rho}\ ,\ \
\Theta=\frac{1}{2}\cos{\theta}d\phi\ ,\ \ t=\psi/2$ (3.3)

$\sigma_{1},\sigma_{2}$ and $\sigma_{3}$ given in (3.2) are left invariant
one-forms parameterizing the spaces with SU(2) isometry and they are written
in Euler angles $(\theta,\phi,\psi)$ as

$\displaystyle\sigma_{1}$ $\displaystyle=$
$\displaystyle(\sin\psi\,d\theta-\sin\theta\cos\psi\,d\phi)/2$ (3.4)
$\displaystyle\sigma_{2}$ $\displaystyle=$
$\displaystyle(\cos\psi\,d\theta+\sin\theta\sin\psi\,d\phi)/2$
$\displaystyle\sigma_{3}$ $\displaystyle=$
$\displaystyle(d\psi+\cos\theta\,d\phi)/2$

These one-forms satisfy the structure equations $d\sigma_{i}=\epsilon_{ijk}\ \
\sigma_{j}\wedge\sigma_{k}$ , where $\epsilon_{ijk}$ is the 3-dimensional
anti-symmetric tensor. Euclidean Taub-NUT metric is also given in [10]
following [3] as

$\frac{r+m}{4\left(r-m\right)}{\rm
dr}^{2}+\left(r^{2}-m^{2}\right)\left({\sigma_{1}}^{2}+{\sigma_{2}}^{2}\right)+4m^{2}\frac{r-m}{r+m}{\sigma_{3}}^{2}$
(3.5)

The metric in (3.2) can be obtained by setting $r=\rho+m$ and $m=1/2$ in
(3.5). Now that we have obtained isometries between the most common explicit
Taub-NUT metrics in literature we will continue with (3.2). As pointed out in
[4] the 2-form $\omega$ obtained with respect to the most obvious integrable
almost complex structure
$\left(J\sigma_{1}\rightarrow\sigma_{2},J\sigma_{3}\rightarrow-d\rho\left(1+\rho\right)/2\rho\right)$
is not closed. Therefore, one must find a suitable co-frame for the metric and
compatible almost complex structure so that Kähler conditions are satisfied.
We write the following orthonormal co-frame in explicit coordinates as follows

$\displaystyle e^{1}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\rho+1}{4\rho}}\left(\rho\cos{\theta}\cos{\phi}d\theta+\sin{\theta}\left(\cos{\phi}d\rho-\rho\sin{\phi}d\phi\right)\right)$
(3.6) $\displaystyle e^{2}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\rho+1}{4\rho}}\left(\rho\cos{\theta}\sin{\phi}d\theta+\sin{\theta}\left(\sin{\phi}d\rho+\rho\cos{\phi}d\phi\right)\right)$
$\displaystyle e^{3}$ $\displaystyle=$
$\displaystyle\sqrt{\frac{\rho+1}{4\rho}}\left(-\rho\sin{\theta}d\theta+\cos{\theta}d\rho\right)\
,\quad
e^{4}=\sqrt{\frac{\rho}{4\left(\rho+1\right)}}\left(\cos{\theta}d\phi+d\psi\right)$

The vector fields dual to the co-frame $\\{e^{1},e^{2},e^{3},e^{4}\\}$ are
obtained as

$\displaystyle e_{1}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\rho\left(1+\rho\right)}}\left(\cos\theta\
\cos\phi{\ \partial}_{\theta}+\rho\ \sin\theta\ \cos\phi\
\partial_{\rho}+\sin\phi\ \cot\theta\
\partial_{\psi}-\frac{\sin\phi}{\sin\theta}\partial_{\phi}\right)$
$\displaystyle e_{2}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\rho\left(1+\rho\right)}}\left(\cos\theta\
\sin\phi{\ \partial}_{\theta}+\rho\ \sin\theta\ \sin\phi\
\partial_{\rho}-\cos\phi\ \cot\theta\
\partial_{\psi}+\frac{\cos\phi}{\sin\theta}\partial_{\phi}\right)$
$\displaystyle e_{3}$ $\displaystyle=$
$\displaystyle\frac{2}{\sqrt{\rho\left(1+\rho\right)}}\left(\rho\ \cos\theta{\
\partial}_{\rho}-\sin\theta\ \partial_{\theta}\right)\ ,\quad
e_{4}=2\sqrt{\frac{\rho+1}{\rho}}\ \partial_{\psi}$ (3.7)

We choose the orientation $(e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4})$ so
that the following 2-forms become self-dual.

$\omega_{1}=e^{1}\wedge e^{2}+e^{3}\wedge e^{4}\ ,\ {\ \omega}_{2}=e^{1}\wedge
e^{4}+e^{2}\wedge e^{3}\ ,\ {\ \omega}_{3}=e^{1}\wedge e^{3}+e^{4}\wedge
e^{2}$ (3.8)

Almost complex structures that will yield the 2-forms given in (3.8) with
respect to the chosen orientation will be

$\displaystyle{\hat{J}}_{1}e_{1}=e_{2}\ ,\ {\hat{J}}_{1}e_{2}=-e_{1}\ ,\
{\hat{J}}_{1}e_{3}=e_{4}\ ,\ {\hat{J}}_{1}e_{4}=-e_{3}$ (3.9)
$\displaystyle{\hat{J}}_{2}e_{1}=e_{4}\ ,\ {\hat{J}}_{2}e_{2}=e_{3}\ ,\ \ \
{\hat{J}}_{2}e_{3}={-e}_{2}\ ,\ {\hat{J}}_{2}e_{4}={-e}_{1}$
$\displaystyle{\hat{J}}_{3}e_{1}=e_{3}\ ,\ {\hat{J}}_{3}e_{2}={-e}_{4}\ ,\
{\hat{J}}_{3}e_{3}={-e}_{1}\ ,\ {\hat{J}}_{3}e_{4}=e_{2}$

Self-dual 2-forms given in (3.8) can be written in coordinates

$\begin{split}&\omega_{1}=\frac{1}{4}\biggl{(}\cos\theta\ d\rho\wedge
d\psi+\rho\sin\theta\left(\rho\cos\theta\ d\theta\wedge d\phi-d\theta\wedge
d\psi+\sin\theta\ d\rho\wedge d\phi\right)+d\rho\wedge d\phi\biggr{)}\\\
&\omega_{2}=\frac{1}{4}\biggl{[}\left(1+\rho\right)\sin\phi\ d\theta\wedge
d\rho+\rho\cos\phi(1+\rho\ \sin^{2}\theta)\ d\theta\wedge
d\phi+\rho\cos\phi\cos\theta\left(d\theta\wedge d\psi-\sin\theta\ d\rho\wedge
d\phi\right)\\\ &\quad\ \ \ \ +\cos\phi\sin\theta\ d\rho\wedge
d\psi-\rho\sin\phi\sin\theta\ d\phi\wedge d\psi\biggr{]}\\\
&\omega_{3}=\frac{1}{4}\biggl{[}(1+\rho)\cos\phi\ d\theta\wedge d\rho-\rho\
\sin\phi(1+\rho\ \sin^{2}\theta)\ d\theta\wedge d\phi+\rho\sin\phi\
\cos\theta\ (\sin\theta\ d\rho\wedge d\phi-d\theta\wedge d\psi)\ \\\ &\quad\ \
-\sin\phi\ \sin\theta\ d\rho\wedge d\psi)-\rho\ \cos\phi\ \sin\theta\
d\phi\wedge d\psi\biggr{]}\end{split}$ (3.10)

It is a straightforward computation to check that $\omega_{1}$, $\omega_{2}$
and $\omega_{3}$ are closed. Almost complex structures are given in (3.9) with
respect to co-frame basis elements. They should be written with respect to
explicit coordinates as well. Following index equation can be written from
definition of Kähler form.

$\displaystyle\omega=\frac{1}{2}g_{\mu\nu}\ J_{\sigma}^{\mu}\ {\rm
dx}^{\sigma}\wedge{\rm dx}^{\nu}=\frac{1}{2}\omega_{\sigma\nu}\ {\rm
dx}^{\sigma}\wedge{\rm dx}^{\nu}$ (3.11) $\displaystyle
g^{\nu\alpha}g_{\mu\nu}\ J_{\sigma}^{\mu}=g^{\nu\alpha}\omega_{\sigma\nu}\
\Rightarrow\ \delta_{\mu}^{\alpha}{\ J}_{\sigma}^{\mu}={\
J}_{\sigma}^{\alpha}=g^{\nu\alpha}\omega_{\sigma\nu}$

Coefficients of the $\omega_{i}$’s ($\omega_{\sigma\nu}$) can be easily read
from (3.10). Then inverse metric coefficients ($g^{\nu\alpha}$) can be applied
in (3.11) to obtain the almost complex structures in coordinate frame.
Coefficients of the J tensors can be expressed in matrix form,

$\displaystyle J_{1}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}0&\frac{1}{\rho}&0&\cos\theta\\\
-\frac{\rho\left(1+\rho\ \sin^{2}\theta\right)}{1+\rho}&0&-\frac{\rho\
\cos\theta\sin\theta}{1+\rho}&0\\\
0&\cot\theta&0&-\frac{1+\rho{\sin}^{2}\theta}{\sin\theta}\\\
-\frac{\rho\cos\theta}{1+\rho}&0&\frac{\sin\theta}{1+\rho}&0\\\
\end{matrix}\right)$ (3.12) $\displaystyle J_{2}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}0&-\frac{\cos\phi\
\cot\theta}{\rho}&-\frac{\sin\phi}{\rho}&\frac{(1+\rho\
\sin^{2}\theta)\cos\phi}{\rho\ \sin\theta}\\\ \frac{\rho^{2}\sin
2\theta\cos\phi}{2(1+\rho)}&\frac{\sin\phi\
\cot\theta}{1+\rho}&-\frac{(1+\rho\
\sin^{2}\theta)\cos\phi}{1+\rho}&-\frac{\left(\rho(2+\rho)\sin^{2}\theta+1\right)\sin\phi}{(1+\rho)\sin\theta}\\\
\rho\ \sin\phi&\cos\phi&0&\rho\ \cos\theta\cos\phi\\\ -\frac{\rho\
\cos\phi\sin\theta}{1+\rho}&\frac{\sin\phi}{(1+\rho)\sin\theta}&-\frac{\cos\theta\cos\phi}{1+\rho}&-\frac{\cot\theta\sin\phi}{1+\rho}\\\
\end{matrix}\right)$ $\displaystyle J_{3}$ $\displaystyle=$
$\displaystyle\left(\begin{matrix}0&\frac{\sin\phi\
\cot\theta}{\rho}&-\frac{\cos\phi}{\rho}&-\frac{(1+\rho\
\sin^{2}\theta)\sin\phi}{\rho\ \sin\theta}\\\ -\frac{\rho^{2}\sin
2\theta\sin\phi}{2(1+\rho)}&\frac{\cos\phi\ \cot\theta}{1+\rho}&\frac{(1+\rho\
\sin^{2}\theta)\sin\phi}{1+\rho}&-\frac{\left(\rho(2+\rho)\sin^{2}\theta+1\right)\cos\phi}{(1+\rho)\sin\theta}\\\
\rho\ \cos\phi&-\sin\phi&0&-\rho\ \cos\theta\sin\phi\\\ \frac{\rho\
\sin\phi\sin\theta}{1+\rho}&\frac{\cos\phi}{(1+\rho)\sin\theta}&\frac{\cos\theta\sin\phi}{1+\rho}&-\frac{\cot\theta\cos\phi}{1+\rho}\\\
\end{matrix}\right)$

These almost complex structures satisfy the quaternionic relations such as
${J_{1}}^{2}={J_{2}}^{2}={J_{3}}^{2}=-\text{Id}$ ,
$J_{1}J_{2}={-J}_{2}J_{1}=J_{3}$ (others are cyclic). Lie brackets of basis
vector fields $\\{e_{1},e_{2},e_{3},e_{4}\\}$ are needed for Nijenhuis tensor
calculations.

$\left[e_{1},e_{2}\right]=\frac{2\left(\partial_{\phi}+\left(1+2\rho\right)\cos\theta\
\partial_{\psi}\right)}{\rho\left(1+\rho\right)^{2}}\ ,\ \
\left[e_{1},e_{4}\right]=-\frac{2\ \cos\phi\ \sin\theta\
\partial_{\psi}}{\rho(1+\rho)}\hfill$ $\left[e_{2},e_{4}\right]=-\frac{2\
\sin\phi\ \sin\theta\ \partial_{\psi}}{\rho(1+\rho)}\ ,\
\left[e_{3},e_{4}\right]=-\frac{2\cos\theta\
\partial_{\psi}}{\rho(1+\rho)}\hfill$
$\left[e_{1},e_{3}\right]=\frac{2\cot\theta\
\sin\phi}{\rho{(1+\rho)}^{2}}\partial_{\phi}-\frac{2\cos\phi}{\rho{(1+\rho)}^{2}}\partial_{\theta}-\frac{2\sin\phi(\sin^{2}\theta(1+2\rho)+1)}{\rho\sin\theta{(1+\rho)}^{2}}\partial_{\psi}\hfill$
$\left[e_{2},e_{3}\right]=-\frac{2\cot\theta\
\cos\phi}{\rho{(1+\rho)}^{2}}\partial_{\phi}-\frac{2\sin\phi}{\rho{(1+\rho)}^{2}}\partial_{\theta}+\frac{2\cos\phi(\sin^{2}\theta(1+2\rho)+1)}{\rho\sin\theta{(1+\rho)}^{2}}\partial_{\psi}\hfill$
(3.13)

Integrability condition for J requires Nijenhuis tensor to vanish for all
vector fields as mentioned in Theorem 1. It is now straightforward to show
that Nijenhuis tensor is vanishing for all three almost complex structures by
using the coordinate coefficients of $J_{1}$, $J_{2}$ and $J_{3}$ tensors
given in (3.12) and the brackets in (3.13). We show one example calculation
for $J_{1}$, others can be computed similarly.

$N_{1}\left(e_{1},e_{3}\right)=\left[e_{1},e_{3}\right]+J_{1}\left[J_{1}\
e_{1},e_{3}\right]+J_{1}\left[e_{1},J_{1}\ e_{3}\right]-\left[J_{1}\
e_{1},J_{1}\ e_{3}\right]\quad\ \ \ \hfill$ (3.14)
$\phantom{xxxxxxxx}=\left[e_{1},e_{3}\right]+J_{1}\left[e_{2},e_{3}\right]+J_{1}\left[e_{1},e_{4}\right]-\left[e_{2},e_{4}\right]\hfill$
$\phantom{xxxxxxxx}=\left[e_{1},e_{3}\right]-\frac{\cos\phi\ \sin
2\theta}{{(1+\rho)}^{2}}\partial_{\rho}-\frac{2\cot\theta\
\sin\phi}{\rho{(1+\rho)}^{2}}\partial_{\phi}+\frac{2\cos\phi(1+\sin^{2}\theta)}{\rho{(1+\rho)}^{2}}\partial_{\theta}\hfill$
$\phantom{xxxxxxxxxx}+\frac{2\sin\phi(1+\rho\
\sin^{2}\theta)}{\rho\sin\theta{(1+\rho)}^{2}}\partial_{\psi}+\frac{\cos\phi\
\sin 2\theta}{{(1+\rho)}^{2}}\partial_{\rho}-\frac{2\cos\phi\
\sin^{2}\theta}{\rho{(1+\rho)}^{2}}\partial_{\theta}+\frac{2\sin\phi\
\sin^{2}\theta\ (1+\rho)}{\rho\
\sin\theta{(1+\rho)}^{2}}\partial_{\psi}\hfill$
$\phantom{xxxxxxxx}=\left[e_{1},e_{3}\right]-\frac{2\cot\theta\
\sin\phi}{\rho{(1+\rho)}^{2}}\partial_{\phi}+\frac{2\cos\phi}{\rho{(1+\rho)}^{2}}\partial_{\theta}+\frac{2\sin\phi(\sin^{2}\theta(1+2\rho)+1)}{\rho\sin\theta{(1+\rho)}^{2}}\partial_{\psi}=0.\hfill$

Hence, this completes all the necessary conditions for hyper-Kähler structure,
ensuring that the explicit Euclidean Taub-NUT metric (3.2) is hyper-Kähler
with respect to the complex structures obtained. Note that an alternative way
to check the integrability is to show that J is covariantly constant $(\nabla
J=0)$ with respect to the Levi-Civita connection associated with the metric.

###  3.2 Kähler Structure Of Euclidean Kerr Metric

Kerr metric can be written using Boyer-Lindquist coordinates in the following
form [26],

$\tilde{g}_{K}=\tilde{\Xi}\ \left(\frac{{\rm
dr}^{2}}{\tilde{\Delta}}+{d\theta}^{2}\right)+\frac{\sin^{2}\theta}{\tilde{\Xi}}\left(-\alpha
dt+\left(r^{2}+\alpha^{2}\right)d\phi\right)^{2}+\frac{\tilde{\Delta}}{\tilde{\Xi}}\left(dt-\alpha\sin^{2}\theta\
d\phi\right)^{2}$ (3.15)

where $\tilde{\Delta}=r^{2}-2Mr+\alpha^{2}$ and
$\tilde{\Xi}=r^{2}+\alpha^{2}\cos^{2}{\theta}$. It describes the exterior
gravitational field of a rotating black hole with mass M and angular momentum
per unit mass of $\alpha$ which are positive real parameters. Kerr space is a
Lorentzian 4-manifold categorized under Petrov Type D class [26]. A manifold
having a metric of Lorentz signature cannot admit an almost Hermitian
structure as mentioned in Theorem 3. In this case, one can apply Wick rotation
to the metric and obtain another version of the given metric with Riemannian
signature. As a result of this transformation, solutions of the Lorentzian
Einstein equations are mapped into solutions of the Riemannian Einstein
equations. Following [15] we apply Wick rotation to (3.15) and get,

$g_{K}=\Xi\left(\frac{{\rm
dr}^{2}}{\Delta}+{d\theta}^{2}\right)+\frac{\sin^{2}\theta}{\Xi}\left(\alpha
dt+\left(r^{2}-\alpha^{2}\right)d\phi\right)^{2}+\frac{\Delta}{\Xi}\left(dt-\alpha\sin^{2}\theta
d\phi\right)^{2}$ (3.16)

Here Wick rotation is applied to both t and $\alpha$ such that $t\rightarrow
it$ , $\alpha\rightarrow i\alpha$ to yield (3.16) with
$\Delta=r^{2}-2Mr-\alpha^{2}$ and $\Xi=r^{2}-\alpha^{2}\cos^{2}{\theta}$. This
metric is known as Euclidean (Riemannian) Kerr metric. Now we can search for
Kähler forms and integrable almost complex structures. An orthonormal co-frame
for this metric can be written as,

$\\{{e}^{1},e^{2},e^{3},e^{4}\\}=\left\\{\sqrt{\frac{\Xi}{\Delta}}dr,\sqrt{\Xi}\
d\theta,\frac{\sin\theta}{\sqrt{\Xi}}\left(\alpha
dt+\left(r^{2}{-\alpha}^{2}\right)d\phi\right),\sqrt{\frac{\Delta}{\Xi}}\left(dt-\alpha{\sin}^{2}\theta\
d\phi\right)\right\\}$ (3.17)

The vector fields dual to the co-frame basis elements are

$\\{e_{1},e_{2},e_{3},e_{4}\\}=\left\\{\sqrt{\frac{\Delta}{\Xi}}\
\partial_{r},\frac{\partial_{\theta}}{\sqrt{\Xi}},\frac{\alpha\
{\sin}^{2}\theta\ \partial_{t}+\partial_{\phi}}{\sin\theta\
\sqrt{\Xi}},\frac{\left(-\alpha\
\partial_{\phi}+\left(r^{2}-\alpha^{2}\right)\partial_{t}\right)}{\sqrt{\Delta\
\Xi}}\right\\}$ (3.18)

We choose the orientation ($e^{1}\wedge e^{2}\wedge e^{3}\wedge e^{4}$) so
that the 2-form $\omega=e^{1}\wedge e^{4}+e^{2}\wedge e^{3}$ become self-dual
as we did in Taub-NUT space. And the corresponding almost complex structure is
the following

$\hat{J}e_{1}=e_{4}\ ,\quad\hat{J}e_{2}=e_{3}\ ,\quad\hat{J}e_{3}={-e}_{2}\
,\quad\hat{J}e_{4}={-e}_{1}$ (3.19)

Self-dual 2-form $\omega$ and coefficients of almost complex structure
$\hat{J}$ can be written in coordinates,

$\omega=dr\wedge dt-\alpha{\sin}^{2}\theta\ dr\wedge d\phi-\alpha\sin\theta\
dt\wedge d\theta+\left(r^{2}-\alpha^{2}\right)\sin\theta\ d\theta\wedge d\phi$
(3.20)
$J=\left(\begin{array}[]{cccc}0&-\frac{\Delta}{\Xi}&-\frac{\alpha\sin\theta}{\Xi}&0\\\
\frac{r^{2}-\alpha^{2}}{\Delta}&0&0&-\frac{\alpha}{\Delta}\\\
\alpha\sin\theta&0&0&\csc\theta\\\
0&\frac{\alpha\Delta\sin^{2}\theta}{\Xi}&\frac{\sin\theta\left(\alpha^{2}-r^{2}\right)}{\Xi}&0\\\
\end{array}\right)$ (3.21)

Taking the exterior derivative of $\omega$ we get $d\omega=2\left(r+\alpha\
\cos\theta\right)\sin\theta\ dr\wedge d\theta\wedge d\phi$, which is not
identically vanishing so the metric under consideration is not Kähler. Indeed,
there exists a closed self-dual 2-form defined on the Euclidean Kerr space as
shown in [17]. We compute the corresponding almost complex structure for that
closed 2-form as follows

$\tilde{J}=\left(\begin{array}[]{cccc}0&-\frac{\Delta}{\Xi{(r-\alpha\cos\theta)}^{2}}&-\frac{\alpha\sin\theta}{\Xi{(r-\alpha\cos\theta)}^{2}}&0\\\
\frac{r^{2}-\alpha^{2}}{\Delta{(r-\alpha\cos\theta)}^{2}}&0&0&-\frac{\alpha}{\Delta{(r-\alpha\cos\theta)}^{2}}\\\
\frac{\alpha
sin\theta}{{(r-\alpha\cos\theta)}^{2}}&0&0&\frac{csc\theta}{{(r-\alpha\cos\theta)}^{2}}\\\
0&\frac{\alpha\Delta\sin^{2}\theta}{\Xi{(r-\alpha\cos\theta)}^{2}}&\frac{\sin\theta(\alpha^{2}-r^{2})}{\Xi{(r-\alpha\cos\theta)}^{2}}&0\\\
\end{array}\right)$ (3.22)

It turns out that $\tilde{J}$ is only a scaled version of (3.21). However,
$\tilde{J}$ does not even satisfy the condition $\tilde{J}^{2}=-\text{Id}$ for
an almost complex structure, let alone the integrability condition. Therefore,
Euclidean Kerr metric fails to exhibit genuine Kählerian nature in its current
form given in (3.16).

Theorem 4 is useful for doing further analysis on Kählerity of a given space.
The Lee form associated with Kähler form $\omega$ in (3.20) is calculated by
using the formula in (2.6) as

$\xi=\frac{2\left(dr+\alpha\ \sin\theta\ d\theta\right)}{r-\alpha\
\cos\theta}=d\
\left(\text{Log}\left[C\left(r-\alpha\cos\theta\right)^{2}\right]\right)$
(3.23)

where $C$ is an arbitrary constant. The existence of an exact Lee form $\xi$
ensures that the Euclidean Kerr metric given in (3.16) is globally conformally
Kähler due to Theorem 4. Note that the obtained Lee form also satisfies (2.5).
Taking $C=1$ 111arbitrary constant $C$ could be kept in the proceeding
calculation steps but it doesn’t affect the obtained results., conformal
scaling factor can be determined from (3.23) as

$\lambda=exp(-\text{Log}\left[\left(r-\alpha\cos\theta\right)^{2}\right])={1/\left(r-\alpha\cos\theta\right)}^{2}$
(3.24)

When this conformal factor is applied to the metric in (3.16) the resulting
metric is expected to be Kähler in accordance with the Theorem 4 and its
sketch of proof given thereafter. Applying this conformal factor, we get the
scaled metric $g_{conf}=g_{K}/\left(r-\alpha\cos\theta\right)^{2}$,
corresponding co-frame and vector frame as follows

$\\{e^{1},e^{2},e^{3},e^{4}\\}=\frac{1}{\left(r-\alpha\cos\theta\right)}\left\\{\sqrt{\frac{\Xi}{\Delta}}dr,\sqrt{\Xi}\
d\theta,\frac{\sin\theta}{\sqrt{\Xi}}\left(\alpha
dt+\left(r^{2}{-\alpha}^{2}\right)d\phi\right),\sqrt{\frac{\Delta}{\Xi}}\left(dt-\alpha{\sin}^{2}\theta\
d\phi\right)\right\\}$

$\\{e_{1},e_{2},e_{3},e_{4}\\}=\left(r-\alpha\cos\theta\right)\left\\{\sqrt{\frac{\Delta}{\Xi}}\
\partial_{r},\frac{\partial_{\theta}}{\sqrt{\Xi}},\frac{\alpha\
{\sin}^{2}\theta\ \partial_{t}+\partial_{\phi}}{\sin\theta\
\sqrt{\Xi}},\frac{\left(-\alpha\
\partial_{\phi}+\left(r^{2}-\alpha^{2}\right)\partial_{t}\right)}{\sqrt{\Delta\
\Xi}}\right\\}$ (3.25)

Same orientation and almost complex structure in (3.19) will be used to
compute $\hat{\omega}$ in coordinate frame as

$\hat{\omega}=\frac{1}{\left(r-\alpha\cos\theta\right)^{2}}\left(dr\wedge
dt-\alpha{\sin}^{2}\theta\ dr\wedge d\phi-\alpha\sin\theta\ dt\wedge
d\theta+\left(r^{2}-\alpha^{2}\right)\sin\theta\ d\theta\wedge d\phi\right)$
(3.26)

One can easily verify that $\hat{\omega}$ is now closed. This is already
expected from the construction of new metric as explained in sketch of proof
of Theorem 4. Note that the coordinate-based coefficients of almost complex
structure in (3.21) does not change due to having the same factor for each
element of the scaled vector frame. We compute the Lie brackets of the basis
vector fields,

$\left[e_{1},e_{2}\right]=\frac{\alpha\sqrt{\Delta}\ (\cos\theta\
\partial_{\theta}-r\sin\theta\ \partial_{r})}{{(r+\alpha\
\cos\theta)}^{2}}\quad,\quad\left[e_{2},e_{4}\right]=\frac{r\ \alpha\
\sin\theta\left((r^{2}-\alpha^{2})\ \partial_{t}-\alpha\
\partial_{\phi}\right)}{\sqrt{\Delta}\ {(r+\alpha\ \cos\theta)}^{2}}\hfill$
$\left[e_{1},e_{3}\right]=\frac{\alpha\sqrt{\Delta}\ \cot\theta\
(\partial_{\phi}+\alpha\sin^{2}\theta\ \partial_{t})}{{(r+\alpha\
\cos\theta)}^{2}}\quad,\quad\left[e_{3},e_{4}\right]=0\hfill$
$\left[e_{1},e_{4}\right]=-\frac{\alpha^{2}\cos\theta\ \Delta\ +\alpha\ \Xi\
(M-r)}{\Delta{(r+\alpha\ \cos\theta)}^{2}}\partial_{\phi}+\frac{\alpha\
\cos\theta\ \Delta\ (r^{2}-\alpha^{2})+\Xi(\Delta
r-M(r^{2}+\alpha^{2}))}{\Delta{(r+\alpha\cos\theta)}^{2}}\partial_{t}\hfill$
$\left[e_{2},e_{3}\right]=\frac{\alpha\ r-\Xi\ \cot\theta\
\csc\theta}{{(r+\alpha\cos\theta)}^{2}}\partial_{\phi}+\frac{-r^{3}+\Xi(r+\alpha\cos\theta)+\alpha^{2}\
r}{{(r+\alpha\cos\theta)}^{2}}\partial_{t}\hfill$ (3.27)

Having obtained all Lie brackets for the elements of the vector frame, it is
straightforward to show that Nijenhuis tensor vanishes for the almost complex
structure by direct computation. We again use (2.2) for integrability check as
in Taub-NUT case.

$N\left(e_{2},e_{4}\right)=\left[e_{2},e_{4}\right]+J\left[J\
e_{2},e_{4}\right]+J\left[e_{2},J\ e_{4}\right]-\left[J\ e_{2},J\
e_{4}\right]\quad\ \ \ \hfill$ (3.28)
$\phantom{xxxxxx}=\left[e_{2},e_{4}\right]+J\left[e_{3},e_{4}\right]-J\left[e_{2},e_{1}\right]+\left[e_{3},e_{1}\right]\hfill$
$\phantom{xxxxxx}=\frac{r\ \alpha\ \sin\theta\left((r^{2}-\alpha^{2})\
\partial_{t}-\alpha\ \partial_{\phi}\right)}{\sqrt{\Delta}\ {(r+\alpha\
\cos\theta)}^{2}}+J\left(\frac{\alpha\sqrt{\Delta}\ (\cos\theta\
\partial_{\theta}-r\sin\theta\ \partial_{r})}{{(r+\alpha\
\cos\theta)}^{2}}\right)\\\ \phantom{xxxxxxxxxxx}-\frac{\alpha\sqrt{\Delta}\
\cot\theta\ (\partial_{\phi}+\alpha\sin^{2}\theta\ \partial_{t})}{{(r+\alpha\
\cos\theta)}^{2}}\hfill$ $\phantom{xxxxxx}=\frac{r\ \alpha\
\sin\theta\left((r^{2}-\alpha^{2})\ \partial_{t}-\alpha\
\partial_{\phi}\right)}{\sqrt{\Delta}\ {(r+\alpha\
\cos\theta)}^{2}}-\frac{\alpha\sqrt{\Delta}\ \cot\theta\
(\partial_{\phi}+\alpha\sin^{2}\theta\ \partial_{t})}{{(r+\alpha\
\cos\theta)}^{2}}\hfill$
$\phantom{xxxxxx}+\frac{\alpha\sqrt{\Delta}\left(\cot\theta\
\partial_{\phi}-r\ \sin\theta\left(\partial_{t}+\frac{2M\ r\
\partial_{t}-\alpha\ \partial_{\phi}}{\Delta}\right)+\alpha\ \cos\theta\
\sin\theta\ \partial_{t}\right)}{{(r+\alpha\cos\theta)}^{2}}\\\
\phantom{xxxxxxx}=0.\hfill$

Again one example calculation is explicitly shown here, others can be easily
computed by similar calculation steps. Note that $N\left(e_{1},e_{4}\right)$
and $N\left(e_{2},e_{3}\right)$ are almost trivially vanishing due to chosen
almost complex structure. Thus, after conformally scaling the Euclidean Kerr
metric in (3.16) we have shown that it becomes Kähler.

Now let’s focus our attention to alternative verification of the conformal
Kähler structure. $W^{+}$ is needed in order to check the conditions of
Theorem 5. It can be calculated for the Euclidean Kerr metric in (3.16) by
using the equation (2.8) as

$A=W^{+}+\frac{R}{12}I=\left(\begin{array}[]{ccc}-\frac{M}{(r-\alpha\cos\theta)^{3}}&0&0\\\
0&-\frac{M}{(r-\alpha\cos\theta)^{3}}&0\\\
0&0&\frac{2M}{(r-\alpha\cos\theta)^{3}}\\\ \end{array}\right)$ (3.29)

Kerr space is Ricci-flat ($R_{\mu\nu}=0$, $R=0$) so $A=W^{+}\neq 0$. It is
4-dimensional Einstein manifold and $W^{+}$ does not vanish, hence Theorem 5
can be employed here. The matrix obtained in (3.29) indeed has only one
distinct eigenvalue ($\lambda,\lambda,-2\lambda$) and
$|W^{+}|^{2}=\sum\limits_{i=1}^{3}\lambda_{i}^{2}$, where $\lambda_{i}$’s are
the eigenvalues of $W^{+}$. Then the conformal factor which transforms the
original manifold to a Kähler manifold is computed as

$\displaystyle|W^{+}|^{2/3}$ $\displaystyle=$
$\displaystyle\left(2\left(-\frac{M}{(r-\alpha\cos\theta)^{3}}\right)^{2}+\left(\frac{2M}{(r-\alpha\cos\theta)^{3}}\right)^{2}\right)^{1/3}$
$\displaystyle=$
$\displaystyle\frac{6^{1/3}M^{2/3}}{(r-\alpha\cos\theta)^{2}}$

Comparing the conformal factor in (3.2) with the one obtained from exact Lee-
form, it is seen that they are same up to a constant. The arbitrary constant
$C$ in (3.23) becomes $6^{-1/3}M^{-2/3}$ for $|W^{+}|^{2/3}$ factor . As
mentioned before choice of arbitrary constant has no effect on the results.
Therefore, we have verified the conformal Kähler structure from Weyl tensor
analysis and found that the two conformal factors are matching.

##  4 Conclusion

In this paper, we presented alternative derivations of hyper-Kähler structure
of Taub-NUT space and globally conformally Kähler structure of Kerr space via
explicit coordinate-based calculations. We have obtained isometries between
the most common explicit Taub-NUT metrics in literature and showed that
Euclidean Kerr metric is globally conformally Kähler from two different
analysis. We also showed that the conformal factor obtained from exact Lee-
form matches with the one obtained from Weyl tensor analysis. This was shown
for Kerr space, a more general proof may be needed but that can be subject of
another study. It is our expectation that deriving Kähler structures of Kerr
and Taub-NUT spaces starting directly from the commonly used metrics will
provide insights for future studies which deal with complex geometry problems
including these spaces.

## Acknowledgments

We thank M. Kalafat for useful discussions.

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