text:
\section*{Introduction}
In the study of almost hypercomplex manifolds the Hermitian
metrics are well known. The parallel study of almost hypercomplex
manifolds with skew-Hermitian metrics is in progress of
development~\cite{GrMaDi1},~\cite{GrMaDi2}.

Let $(M,H)$ be an almost hypercomplex manifold, i.e. $M$ is a
$4n$-dimen\-sional differentiable manifold and $H$ is a triple
$(J_1,J_2,J_3)$ of anticommuting almost complex structures, where
$J_3=J_1 \circ J_2$ \cite{So},\cite{AlMa}.

A standard hypercomplex structure for all $x(x^i,y^i,u^i,v^i)\in
T_pM$, $p\in M$ is defined in \cite{So} as follows
\begin{equation}\label{14}
\begin{array} {l}
J_1x(-y^i,x^i,v^i,-u^i),\quad J_2x(-u^i,-v^i,x^i,y^i),\quad
J_3x(v^i,-u^i,y^i,-x^i).
\end{array}
\end{equation}

Let us equip $(M,H)$ with a pseudo-Riemannian metric $g$ of
signature $(2n,2n)$ so that
\begin{equation}\label{1}
g(\cdot,\cdot)=g(J_1\cdot,J_1\cdot)=-g(J_2\cdot,J_2\cdot)=-g(J_3\cdot,J_3\cdot).
\end{equation}
We called such metric a \emph{pseudo-Hermitian metric} on an
almost hypercomplex manifold \cite{GrMaDi1}. It generates a
K\"ahler 2-form $\Phi$ and two pseudo-Hermitian metrics $g_2$ and
$g_3$ by the following way
\begin{equation}\label{G}
\Phi:=g(J_1\cdot,\cdot),\qquad g_2:=g(J_2\cdot,\cdot),\qquad
g_3:=g(J_3\cdot,\cdot).
\end{equation}
The metric $g$ ($g_2$, $g_3$, respectively) has an Hermitian
compatibility with respect to $J_1$ ($J_3$, $J_2$, respectively)
and a skew-Hermitian compatibility with respect to $J_2$ and $J_3$
($J_1$ and $J_2$, $J_1$ and $J_3$, respectively).

On the other hand, a quaternionic inner product $<\cdot,\cdot>$ in
$\mathbb{H}$ generates in a natural way the bilinear forms $g$,
$\Phi$, $g_2$ and $g_3$ by the following decomposition:
$<\cdot,\cdot>=-g+i\Phi+jg_2+kg_3$.

The structure $(H,G):=(J_1,J_2,J_3;g,\Phi,g_2,g_3)$ is called a
\emph{hypercomplex pseu\-do-Hermit\-ian structure} on $M^{4n}$ or
shortly a \emph{$(H,G)$-structure} on $M^{4n}$. The manifold
$(M,H,G)$ is called an \emph{almost hypercomplex pseudo-Hermitian
manifold} or shortly an \emph{almost
$(H,G)$-manifold}~\cite{GrMaDi1}.

The basic purpose of the recent paper is to construct explicit
examples of the $(H,G)$-manifolds of the lowest dimension at $n=1$
and to characterize them.

The following structural $(0,3)$-tensors  play basic role for the
characterization of the almost $(H,G)$-ma\-nifold
\[
F_\alpha (x,y,z)=g\bigl( \left( \nabla_x J_\alpha
\right)y,z\bigr)=\bigl(\nabla_x g_\alpha\bigr) \left( y,z \right),
\quad \alpha=1,2,3,
\]
where $\nabla$ is the Levi-Civita connection generated by $g$.

It is well known \cite{AlMa}, that the almost hypercomplex
structure $H=(J_\alpha)$ is a hypercomplex structure if the
Nijenhuis tensors
\[
N_\alpha(X,Y)=
    \left[X,Y \right]
    +J_\alpha\left[X,J_\alpha Y \right]
    +J_\alpha\left[J_\alpha X,Y \right]
    -\left[J_\alpha X,J_\alpha Y \right]
\]
vanish for each $\alpha=1,2,3$. Moreover, one $H$ is hypercomplex
iff two of $N_\alpha$ are zero.

Since $g$ is a Hermitian metric with respect to $J_1$, we use the
classification of the almost Hermitian manifolds given in
\cite{GrHe}. According to it the basic class of these manifolds of
dimension 4 are the class of almost K\"ahler manifolds $\mathcal{AK}=\mathcal{W}_2$
and the class of Hermitian manifolds $\mathcal{H}=\mathcal{W}_4$. The class of the
$\mathcal{AK}$-manifolds are defined by condition $d \Phi = 0$ or
equivalently $\mathop{\makebox{\huge $\sigma$}} \limits_{x,y,z}
F_1(x,y,z)=0$. The class of the Hermitian 4-manifolds is
determined by $N_1=0$ or
\[
\begin{array}{rl}
F_1(x,y,z)=\frac{1}{2} &\left[
g(x,y)\theta_1(z)-g(x,z)\theta_1(y)\right.\\[3pt]
&\left. -g(x,J_1y)\theta_1(J_1z)+g(x,J_1z)\theta_1(J_1y) \right]
\end{array}
\]
where $\theta_1(\cdot)=g^{ij}F_1(e_i,e_j,\cdot)=\delta\Phi(\cdot)$
for any basis $\{e_i\}_{i=1}^{4}$, and $\delta$ -- the
co\-de\-ri\-vative.

On other side, the metric $g$ is a skew-Hermitian one with respect
to $J_2$ and $J_3$. A classification of all almost complex
manifolds with skew-Hermitian metric (Norden metric or B-metric)
is given in \cite{GaBo}. The basic classes are:
\[
\begin{array}{l}
\begin{array}{rl}
\mathcal{W}_1: F_\alpha(x,y,z)=\frac{1}{4} &
    \left[
    g(x,y)\theta_\alpha(z)
    +g(x,z)\theta_\alpha(y)\right.\\[3pt]
    &\left
    +g(x,J_\alpha y)\theta_\alpha(J_\alpha z)
    +g(x,J_\alpha z)\theta_\alpha(J_\alpha y)\right],\\[3pt]
\end{array}
\\[3pt]
\mathcal{W}_2: \mathop{\makebox{\huge $\sigma$}} \limits_{x,y,z}
F_\alpha(x,y,J_\alpha z)=0,\qquad
\mathcal{W}_3: \mathop{\makebox{\huge $\sigma$}} \limits_{x,y,z}
F_\alpha(x,y,z)=0,
\end{array}
\]
where \(\theta_\alpha(\cdot)=g^{ij}F_\alpha (e_i,e_j,\cdot)\),
\(\alpha =2,3\), for an arbitrary basis \(\{e_i\}_{i=1}^{4}\).

We denote the main subclasses of the respective complex manifolds
by $\mathcal{W}(J_\alpha)$, where $\mathcal{W}(J_1):=\mathcal{W}_4(J_1)$ \cite{GrHe}, and
$\mathcal{W}(J_\alpha):=\mathcal{W}_1(J_\alpha)$ for $\alpha=2,3$ \cite{GaBo}.

In the end of this section we recall some known facts from
\cite{GrMaDi1} and \cite{GrMaDi2}.

A sufficient condition an almost $(H,G)$-manifold to be an
integrable one is following

\begin{thm} \label{t31}
Let $(M,H,G)$ belongs to $ \mathcal{W}(J_\alpha) \bigcap \mathcal{W}(J_\beta)$. Then
$(M,H,G)$ is of class $\mathcal{W}(J_\gamma)$ for all cyclic permutations
$(\alpha, \beta, \gamma)$ of $(1,2,3)$.
\end{thm}

A pseudo-Hermitian manifold is called a
\emph{pseudo-hyper-K\"ahler manifold} (denotation $(M,H,G) \in
\mathcal{K}$), if $F_\alpha=0$ for every $\alpha=1,2,3$, i.e. the manifold
is K\"ahlerian with respect to each $J_\alpha$ (denotation
$(M,H,G) \in \mathcal{K}(J_\alpha)$).

\begin{thm}\label{t33}
If $(M,H,G) \in \mathcal{K}(J_\alpha)\bigcap \mathcal{W}(J_\beta)$ $(\alpha\neq\beta
\in \{1,2,3\})$ then $(M,H,G) \in \mathcal{K}$ .
\end{thm}
As $g$ is an indefinite metric, there exists isotropic vector
fields $X$ on $M$. Following \cite{GRMa} we consider the
invariants
\[
\norm{\nabla J_\alpha}^2= g^{ij}g^{kl}g\bigl( \left( \nabla_{e_i}
J_\alpha \right) e_k, \left( \nabla_{e_j} J_\alpha \right) e_l
\bigr), \qquad \alpha=1,2,3,
\]
where $\{e_i\}_{i=1}^{4}$ is an arbitrary basis of $T_pM$, $p\in
M$.

\begin{defn}
An $(H,G)$-manifold is called: (i) isotropic K\"ahlerian with
respect to $J_\alpha$
    if $\norm{\nabla J_\alpha}^2=0$ for some $\alpha\in\{1,2,3\}$;
(ii) isotropic hyper-K\"ahlerian
    if it is isotropic K\"ahlerian with respect to every $J_\alpha$ of $H$.
\end{defn}

\begin{thm}\label{t34}
Let $M$ be an $(H,G)$-manifold of class $\mathcal{W}=\bigcap_\alpha
\mathcal{W}(J_\alpha)$ $(\alpha=1,2,3)$ and $\norm{\nabla J_\alpha}^2$
vanishes for some $\alpha=1,2,3$. Then $(M,H,G)$ is an isotropic
hyper-K\"ahler manifold, but it is not pseudo-hyper-K\"ahlerian in
general.
\end{thm}

A geometric characteristic of the pseudo-hyper-K\"ahler manifolds
according to the curvature tensor $R=[\nabla ,\nabla ] -
\nabla_{[\ ,\ ]}$ induced by the Levi-Civita connection is given
in \cite{GrMaDi2}.
\begin{thm}\label{R=0}
Each pseudo-hyper-K\"ahler manifold is a flat pseudo-Riemann\-ian
manifold with signature $(2n,2n)$.
\end{thm}

\section{The two known examples of almost $(H,G)$-manifolds}

\subsection{A pseudo-Riemannian spherical manifold with
$(H,G)$-struc\-ture}

Following \cite{Wo} we have considered in \cite{GrMaDi1} and
\cite{GrMaDi2} a pseudo-Riemannian spherical manifold $S_2^4$ in
pseudo-Euclid\-ean vector space $\mbox{\w R}_2^5$ of type $(--+++)$. The
structure $H$ is introduced on $\tilde S_2^4=S_2^4\setminus
\{(0,0,0,0,\pm 1)\}$ as in (\ref{14}) and the pseudo-Riemannian
metric $g$ is the restriction of the inner product of $\mbox{\w R}_2^5 $ on
$\tilde{S}_2^4$. Therefore $\tilde{S}_2^4$ admits an almost
hypercomplex pseudo-Hermitian structure. The corresponding
manifold is of the class $\mathcal{W}(J_1)$ but it does not belong to $\mathcal{W}$
and it has a constant sectional curvature $k=1$.
Moreover, we established that the considered manifold is
conformally equivalent to a flat $\mathcal{K}(J_1)$-manifold, which is not
a
$\mathcal{K}$-manifold and $(\tilde{S}_2^4,H,G)$ is an Einstein manifold.

\subsection{The Thurston manifold with $(H,G)$-structure}

In \cite{GrMaDi1} we have followed the interpretation of Abbena
\cite{Ab} of the Thurston manifold. We have considered a
4-dimensional compact homogenous space $L\slash\Gamma$, where $L$
is a connected Lie group and $\Gamma$ is the discrete subgroup of
$L$ consisting of all matrices whose entries are integers. We have
introduced the almost hypercomplex structure
$H=\left(J_\alpha\right)$ on $T_EL$ as in (\ref{14}) and we
translate it on $T_AL$, $A\in L$, by the action of the left
invariant vector fields. The $J_\alpha$ are invariant under the
action of $\Gamma$, too. By analogy we have defined a left
invariant pseudo-Riemannian inner product in $T_EL$. It generates
a pseudo-Riemannian metric $g$ on $M^4=L$. Then the generated
4-manifold $M$ is equipped with a suitable $(H,G)$-structure and
$(M,H,G)$ is a $\mathcal{W} (J_1)$-manifold but it does not belong to the
class $\mathcal{W}$.

\section{Engel manifolds with almost $(H,G)$-structure}

In the next two examples we consider
$M=\mathbb{R}^4=\left\{(x^1,x^2,x^3,x^4)\right\}$ with a basis \(
\begin{array}{c}
\left\{e_1=\ddx{1},e_2=\ddx{2}+x^1\ddx{3}+x^3\ddx{4},e_3=-\ddx{3},e_4=-\ddx{4}\right\}
\end{array}
\) and an Engel structure $\mathcal{D}={\rm span} \{e_1,e_2\}$, i.e. an
absolutely non-integrable regular two-dimensional distribution on
$TM$ \cite{GRMa}.

\subsection{Double isotropic hyper-K\"ahlerian structures but
neither hypercomplex nor symplectic}

At first we use the introduced there a pseudo-Riemannian metric
and almost complex structures given by
\begin{equation}\label{41}
\begin{array}{c}
g=(dx^1)^2+\{1-(x^1)^2-(x^3)^2\}(dx^2)^2-(dx^3)^2   \\[3pt]
-(dx^4)^2-2x^1dx^2dx^3+2x^3dx^2dx^4,
\\[3pt]
J:\; Je_1=e_2,\;Je_2=-e_1,\;Je_3=e_4,\;Je_4=-e_3,
\\[3pt]
J':\; J'e_1=e_2,\;J'e_2=-e_1,\;J'e_3=-e_4,\;J'e_4=e_3.
\end{array}
\end{equation}

It is given in \cite{GRMa} that $(J,g)$ and $(J',g)$ are a pair of
indefinite almost Hermitian structures which are isotropic
K\"ahler but neither complex nor symplectic.

It is clear that \( \left\{e_i\right\}_{i=1}^4 \) is an
orthonormal $(++--)$-basis. We accomplish the introduction of an
$(H,G)$-structure on $M$ by
\[
J_1:=J';\;
J_2:\;J_2e_1=e_3,\;J_2e_2=e_4,\;J_2e_3=-e_1,\;J_2e_4=-e_2;\;
J_3:=J_1J_2.
\]

By direct computations we verify that the constructed manifold is
an $(H,G)$-manifold and it is isotropic hyper-K\"ahlerian but not
K\"ahlerian and not integrable  with non-vanishing Lie forms with
respect to any $J_\alpha\ (\alpha=1,2,3)$.

\begin{rem}
If we define $J_1$ as $J$ instead of $J'$ then the kind of example
is not changed. So we receive a pair of almost $(H,G)$-structures
corresponding to the given almost complex structures.
\end{rem}

The non-zero components of the curvature tensor $R$ and the basic
linear invariant of the almost Hermitian manifold $(M,J_1,g)$ are
given in \cite{GRMa} by
\[
\begin{array}{c}
R_{1221}=\frac{3}{4},\;
R_{1331}=-R_{2142}=-R_{2442}=-R_{3143}=R_{3443}=\frac{1}{4},\;
R_{2332}=1;
\\[3pt]
\normq{F_1}=0,\quad \normq{N_1}=8,\quad\tau=0,\quad\tau^*_1=-2,
\end{array}
\]
where the following denotations are used for
$\varepsilon_a=\normq{e_a}$
\[
\begin{array}{c}
\normq{F_1}=\normq{\nabla\Phi}=\sum_{a,b,c=1}^4 \varepsilon_a
\varepsilon_b \varepsilon_c F_1(e_a,e_b,e_c)^2,\\[3pt]
\normq{N_1}=\sum_{a,b=1}^4 \varepsilon_a \varepsilon_b
\normq{N_1(e_a,e_b)},\\[3pt]
 \tau=\sum_{a,b=1}^4 \varepsilon_a
\varepsilon_b R(e_a,e_b,e_b,e_a),\quad
\tau^*_1=\frac{1}{2}\sum_{a,b=1}^4 \varepsilon_a \varepsilon_b
R(e_a,J_1e_a,e_b,J_1e_b).
\end{array}
\]
We get the corresponding linear invariants with respect to $J_2$
and $J_3$:
\[
\begin{array}{lll}
&\normq{F_2}=0,\qquad \normq{N_2}=0,\qquad&\tau^*_2=0;
\\
&\normq{F_3}=0,\qquad \normq{N_3}=-8,\quad&\tau^*_3=0,
\end{array}
\]
where $\tau^*_\alpha=\sum_{a,b=1}^4 \varepsilon_a \varepsilon_b
R(e_a,e_b,J_\alpha e_b,e_a);\;\alpha=2,3$.

\subsection{Double isotropic hyper-K\"ahlerian structures which
are non-integrable but symplectic}

Now we consider the same Engel manifold
$(M=\mathbb{R}^4,\mathcal{D})$ but let the pseudo-Riemannian
metric and the pair of almost complex structures be defined by
other way: \cite{GRMa}
\[
\begin{array}{c}
g=(dx^1)^2-\{1-(x^1)^2+(x^3)^2\}(dx^2)^2+(dx^3)^2\\[3pt]
-(dx^4)^2-2x^1dx^2dx^3+2x^3dx^2dx^4,
\\[3pt]
J:\; Je_1=e_3,\;Je_2=e_4,\;Je_3=-e_1,\;Je_4=-e_2,
\\[3pt]
J':\; J'e_1=e_3,\;J'e_2=-e_4,\;J'e_3=-e_1,\;J'e_4=e_2.
\end{array}
\]

In this case \( \left\{e_i\right\}_{i=1}^4 \) is an orthonormal
basis of type $(+-+-)$. It is shown that $(M,J,g)$ and $(M,J',g)$
are a pair of isotropic K\"ahler almost K\"ahler manifolds with
vanishing linear invariants.

We accomplish the introduced almost complex structures to almost
hypercomplex structures on $M$ by using the following way: we set
the given $J$ (resp. $J'$) as $J_1$ (resp. $J'_1$), then we
introduce $J_2$ (resp. $J'_2$) by
\begin{equation}\label{42h}
\begin{array}{c}
J_2:\;J_2e_1=e_2,\;J_2e_2=-e_1,\;J_2e_3=-e_4,\;J_2e_4=e_3;\\[3pt]
J'_2:\;J'_2e_1=e_2,\;J'_2e_2=-e_1,\;J'_2e_3=e_4,\;J'_2e_4=-e_3
\end{array}
\end{equation}
and finally we set $J_3:=J_1J_2$ (resp. $J'_3:=J'_1J'_2$).

 It is easy to check that $H=(J_\alpha)$ and $H'=(J'_\alpha)$
 together with $g$ generate a pair of almost hypercomplex
 pseudo-Hermitian structures on $M$.

 We characterize the both received $(H,G)$-manifolds as isotropic
 hyper-K\"ahler but not K\"ahler manifolds
 and not integrable manifolds with non-vanishing Lie forms
 with respect to any $J_\alpha$.
 Moreover, they have the following linear invariants:
\[
\begin{array}{c}
\normq{N_1}=0,\;\; \normq{N_2}=-\normq{N_3}=8,\;\;
\normq{F_\alpha}=0,\;\; \tau=\tau^*_\alpha=0 \;\; (\alpha=1,2,3).
\end{array}
\]

\section{Real spaces with almost $(H,G)$-structure}
\subsection{Real semi-space with almost $(H,G)$-structure}

Let us consider the real semi-space
$\mathbb{R}_+^4=\left\{\left(x^1,x^2,x^3,x^4\right),\; x^i\in
\mathbb{R},\; x^1>0 \right\}$ with the basis given by
\(
\begin{array}{c}
\left\{e_1=x^1\ddx{1},e_2=x^1\ddx{2},e_3=x^1\ddx{3},e_4=x^1\ddx{4}\right\}.
\end{array}
\)
It is clear that this basis is orthonormal of type $(++--)$ with
respect to the pseudo-Riemannian metric \(
g=\left\{(dx^1)^2+(dx^2)^2-(dx^3)^2-(dx^4)^2\right\}/(x^1)^2. \)
We introduce an almost hypercomplex structure $H=(J_\alpha)$ as
follows
\begin{equation}\label{43h}
\begin{array}{l}
J_1:\;J_1e_1=e_2,\;J_1e_2=-e_1,\;J_1e_3=e_4,\;J_1e_4=-e_3;\\[3pt]
J_2:\;J_2e_1=e_3,\;J_2e_2=-e_4,\;J_2e_3=-e_1,\;J_2e_4=e_2;\quad J_3=J_1J_2\\[3pt]
\end{array}
\end{equation}
and we check that $H$ and $g$ generates an almost
$(H,G)$-structure on $\mathbb{R}_+^4$.

We verify immediately that $H$ is integrable and the obtained
hypercomplex pseudo-Hermit\-ian manifold $(\mathbb{R}_+^4,H,G)$
belongs to the class $\mathcal{W}=\bigcap_\alpha \mathcal{W}(J_\alpha)$  but it is
not isotropic K\"ahlerian with respect to $J_\alpha$
$(\alpha=1,2,3)$.

By direct computations we obtain for the curvature tensor that
$R=-\pi_1$, i.e. the manifold has constant sectional curvatures
$k=-1$ and it is an Einstein manifold. Moreover, the linear
invariants are
\[
\begin{array}{c}
\normq{N_\alpha}=0,\quad
2\normq{F_1}=4\normq{\theta_1}=-\normq{F_\beta}=-\normq{\theta_\beta}=16,\\[3pt]
\tau=-3\tau^*_1=-12,\; \tau^*_\beta=0,\;
\end{array}
\]
where $\alpha=1,2,3;\ \beta=2,3$; and $(\mathbb{R}_+^4,H,G)$ is
conformally equivalent to a pseudo-hyper-K\"ahler manifold by the
change $\bar{g}=(x^1)^2g$.

\subsection{Real quarter-space with almost $(H,G)$-structure}

Let the real quarter-space
\[
M=\mathbb{R}_+^2\times
\mathbb{R}_-^2=\left\{\left(x^1,x^2,x^3,x^4\right),\; x^i\in
\mathbb{R},\; x^1>0,\; x^3>0 \right\}\] be equipped with a
pseudo-Riemannian metric
\[
\begin{array}{c}
g=\frac{1}{(x^1)^2}\left\{(dx^1)^2+(dx^2)^2\right\}
-\frac{1}{(x^3)^2}\left\{(dx^3)^2+(dx^4)^2\right\}.
\end{array}
\]
Then the basis \(
\left\{e_1=x^1\ddx{1},e_2=x^1\ddx{2},e_3=x^3\ddx{3},e_4=x^3\ddx{4}\right\}
\) is an orthonormal one of type $(++--)$. We introduce an almost
hypercomplex structure $H=(J_\alpha)$ $(\alpha=1,2,3)$ as in the
previous example by (\ref{43h}).

The received almost $(H,G)$-manifold is a
$\mathcal{K}(J_1)$-manifold and an isotropic hyper-K\"ahler
manifold. As a corollary, $N_1=0$, $F_1=0$, $\theta_1=0$ and hence
$\normq{N_1}=\normq{F_1}=\normq{\theta_1}=0$. For the $J_\alpha$
$(\alpha =2,3)$ the Nijenhuis tensors $N_\alpha$, the tensors
$F_\alpha$, and the Lie forms $\theta_\alpha$ are non-zero
(therefore $H$ is not integrable), but the linear invariants
$\normq{N_\alpha}$, $\normq{F_\alpha}$  and
$\normq{\theta_\alpha}$ vanish.

The non-zero components of the curvature tensor are given by
$R_{1221}=-R_{3443}=-1$. For the Ricci tensor we have
$\rho_{ii}=-1$ $(i=1,...,4)$. Therefore the basic non-zero
sectional curvatures are $k(e_1,e_2)=-k(e_3,e_4)=-1$ and the
scalar curvatures $\tau$, $\tau^*_\alpha$ $(\alpha =1,2,3)$ are
zero.

\section{Real pseudo-hyper-cylinder with almost
$(H,G)$-structure}

Let $\mathbb{R}^5_2$ be a pseudo-Euclidean real space with an
inner
product $\langle\cdot,\cdot\rangle$ of signature
$(+++--)$. Let us consider a pseudo-hyper-cylinder defined by
\[
S:\; (z^2)^2+(z^3)^2-(z^4)^2-(z^5)^2=1,
\]
where $Z\left(z^1,z^2,z^3,z^4,z^5\right)$ is the positional vector
at $p\in S$. We use the following parametrization of $S$ in the
local coordinates $\left(u^1,u^2,u^3,u^4\right)$ of $p$:
\[
\begin{array}{c}
Z=Z(u^1,\ \cosh u^4 \cos u^2,\ \cosh u^4 \sin u^2,\ \sinh u^4 \cos
u^3,\ \sinh u^4 \sin u^3).
\end{array}
\]

We consider a manifold on the surface
$\tilde{S}=S\setminus\{u^4=0\}$. Then the basis
\( \left\{e_1=\partial_{1},\ e_2=\frac{1}{\cosh u^4}\partial_{2},\
e_3=\frac{1}{\sinh u^4}\partial_{3},\ e_4=\partial_{4}\right\} \)
of $T_p\tilde{S}$ at $p\in \tilde{S}$ is an orthonormal basis of
type $(++--)$ with respect to the restriction $g$ of
$\langle\cdot,\cdot\rangle$ on $\tilde{S}$. Here and further
$\partial_{i}$ denotes $\frac{\partial Z}{\partial u^i}$ for $i=1,...,4$;

We introduce an almost hypercomplex structure by the following way
\begin{equation}\label{45h}
\begin{array}{l}
J_1:\;J_1e_1=e_2,\;J_1e_2=-e_1,\;J_1e_3=-e_4,\;J_1e_4=e_3;\\[3pt]
J_2:\;J_2e_1=e_3,\;J_2e_2=e_4,\;J_2e_3=-e_1,\;J_2e_4=-e_2;\quad J_3=J_1J_2\\[3pt]
\end{array}
\end{equation}
and check that $H=(J_\alpha)$ and the pseudo-Riemannian metric $g$
generate an almost $(H,G)$-structure on $\tilde{S}$.

By straightforward calculations with respect to $\{e_i\}$
$(i=1,...,4)$ we receive that the almost $(H,G)$-manifold
$\tilde{S}$ is not integrable with non-zero Lie forms regarding
any $J_\alpha$ of $H$ and it has the following linear invariants:
\[
\begin{array}{l}
\normq{N_1}=2\normq{F_1}=2\normq{\nabla
J_1}=8\normq{\theta_1}=-8\tanh^2 u^4;\\[3pt]
\normq{N_2}=-8\coth^2 u^4,\quad \normq{\theta_2}=\left(2\tanh
u^4+\coth u^4\right)^2,\\[3pt]
\normq{F_2}=\normq{\nabla J_2}=4\left(2\tanh^2 u^4+\coth^2
u^4\right);\\[3pt]
\normq{N_3}=-8\left(\tanh u^4-\coth u^4\right)^2,\quad
\normq{\theta_3}=\left(\tanh
u^4+\coth u^4\right)^2,\\[3pt]
\normq{F_3}=\normq{\nabla J_3}=4\left(\tanh^2 u^4+\coth^2
u^4\right).
\end{array}
\]

The non-zero components of the curvature tensor and the
corresponding Ricci tensor and scalar curvatures are given by
\[
\begin{array}{c}
R_{2332}=-1,\quad R_{2442}=-\tanh^2 u^4,\quad R_{3443}=\coth^2
u^4\\[3pt]
\rho_{22}=1+\tanh^2 u^4,\quad \rho_{33}=-1-\coth^2 u^4,\quad
\rho_{44}=-\tanh^2 u^4-\coth^2 u^4\\[3pt]
\tau=2\left(1+\tanh^2 u^4+\coth^2 u^4\right),\quad
\tau^*_\alpha=0,\quad \alpha=1,2,3.
\end{array}
\]

Hence $(\tilde{S},H,G)$ has zero associated scalar curvatures and
$H$ is a non-integrable structure on it.

\section{Complex surfaces with almost $(H,G)$-structure}

The following three examples concern several surfaces
$S^2_\mathbb{C}$ in a 3-dimensional complex Euclidean space
$\left(\mathbb{C}^3,\langle\cdot,\cdot\rangle\right)$. It is well
known that the decomplexification of $\mathbb{C}^3$ to
$\mathbb{R}^6$ using the $i$-splitting, i.e.
$(Z^1,Z^2,Z^3)\in\mathbb{C}^3$, where $Z^k=x^k+iy^k$ $(x^k,y^k\in
\mathbb{R})$, is identified with $(x^1,x^2,x^3,y^1,y^2,y^3)\in
\mathbb{R}^6$. Then the multiplying by $i$ in $\mathbb{C}^3$
induces the standard complex structure $J_0$ in $\mathbb{R}^6$.
The real and the opposite imaginary parts of the complex Euclidean
inner product $\Re\langle\cdot,\cdot\rangle$ and
$-\Im\langle\cdot,\cdot\rangle$ are the standard skew-Hermitian
metrics $g_0$ and $\tilde{g}_0=g_0(\cdot,J_0\cdot)$ in
$(\mathbb{R}^6,J_0,g_0,\tilde{g}_0)$, respectively. So, the
natural decomplexification of an $n$-dimensional complex Euclidean
space is the $2n$-dimensional real space with a complex
skew-Hermitian structure $(J_0,g_0,\tilde{g}_0)$.

\subsection{Complex cylinder with almost $(H,G)$-structure}

Let $S^2_\mathbb{C}$ be the cylinder in
$\left(\mathbb{C}^3,\langle\cdot,\cdot\rangle\right)$ defined by
$(Z^1)^2+(Z^2)^2=1$. Let the corresponding surface $S^4$ in
$(\mathbb{R}^6,J_0,g_0,\tilde{g}_0)$ be parameterized as follows
\[
\begin{array}{rl}
S^4:\; Z=Z(&\cos u^1 \cosh u^3,\ \sin u^1 \cosh u^3,\ u^2,\\[3pt]
&\sin u^1 \sinh u^3,\ -\cos u^1 \sinh u^3,\ u^4).
\end{array}
\]
Then the local basis $\left\{\partial_1,...,\partial_4\right\}$ is
orthonormal of type $(++--)$ and it generates the metric
$g=(du^1)^2+(du^2)^2-(du^3)^2-(du^4)^2$ on $S^4$. The almost
hypercomplex structure $H$ is determined as usually by (\ref{14}).
It is easy to verify that the received $(H,G)$-manifold is a flat
pseudo-hyper-K\"ahler manifold.

\subsection{Complex cone with almost $(H,G)$-structure}

Now let $S^2_\mathbb{C}$ be the complex cone in
$\left(\mathbb{C}^3,\langle\cdot,\cdot\rangle\right)$ determined
by the equation
$(Z^1)^2+(Z^2)^2-(Z^3)^2=0$
. Then we consider the corresponding 4-dimensional surface $S$ in
$(\mathbb{R}^6,J_0,g_0,\tilde{g}_0)$ by the following
parametrization of $Z$:
\[
\begin{array}{l}
(u^1 \cos u^2 \cosh u^4-u^3 \sin u^2 \sinh u^4,\ u^1 \sin
u^2 \cosh u^4+u^3 \cos u^2 \sinh u^4,\ u^1,\\[3pt]
\;u^1 \sin u^2 \sinh u^4+u^3 \cos u^2 \cosh u^4,-u^1 \cos u^2
\sinh u^4+u^3 \sin u^2 \cosh u^4,\ u^3).
\end{array}
\]
Further we consider a manifold on
$\tilde{S}=S\setminus\{0,0,0,0,0,0\}$, i.e. we exclude the plane
$u^1=u^3=0$ from the domain of $S$ which maps the origin. Then the
derived metric $g$ on $\tilde{S}$ has the following non-zero
components regarding $\left\{\partial_k\right\}$:
\[
g_{11}=-g_{33}=2,\quad g_{22}=-g_{44}=(u^1)^2-(u^3)^2,\quad
g_{24}=g_{42}=2u^1u^3.
\]
We receive the following orthonormal basis of signature $(++--)$:
\[
\begin{array}{c}
\left\{e_1=\frac{1}{\sqrt{2}}\partial_{1},\
e_2=\lambda\partial_{2}+\mu\partial_{4},\ e_3=\frac{1}{\sqrt{2}}\partial_{3},\
e_4=-\mu\partial_{2}+\lambda\partial_{4}\right\},
\end{array}
\]
where
$\lambda=u^1/\{(u^1)^2+(u^3)^2\}$, $\mu=u^3/\{(u^1)^2+(u^3)^2\}$.
We introduce a structure $H$ as in (\ref{14}). It is easy to check
that $H$ and $g$ generate an almost $(H,G)$-structure on
$\tilde{S}$. By direct computations we get that the received
$(H,G)$-manifold is a flat hypercomplex manifold which is
K\"ahlerian with respect to $J_1$ but it does not belong to
$\mathcal{W}(J_2)$ or $\mathcal{W}(J_3)$ and the Lie forms
$\theta_2$ and $\theta_3$ are non-zero. The corresponding linear
invariants are given by
\[
\begin{array}{c}
\normq{F_2}=\normq{\nabla
J_2}=2\normq{\theta_2}=16\left\{\mu^2-\lambda^2\right\},\\[3pt]
\normq{F_3}=\normq{\nabla
J_3}=2\normq{\theta_3}=4\left\{\mu^2-\lambda^2\right\}.
\end{array}
\]

\subsection{Complex sphere with almost $(H,G)$-structure}

In this case let $S^2_\mathbb{C}$ be the unit sphere in
$\left(\mathbb{C}^3,\langle\cdot,\cdot\rangle\right)$ defined by
$(Z^1)^2+(Z^2)^2+(Z^3)^2=1$. After that we consider the
corresponding 4-surface $S$ in
$(\mathbb{R}^6,J_0,g_0,\tilde{g}_0)$ with the following
parametrization of
$Z(x^1,x^2,x^3,y^1,y^2,y^3)$:
\[S:\;
\begin{array}{l}
x^1=\cos u^1 \cos u^2 \cosh u^3 \cosh u^4-\sin u^1 \sin u^2 \sinh u^3 \sinh u^4, \\[3pt]
x^2=\cos u^1 \sin u^2 \cosh u^3 \cosh u^4+\sin u^1 \cos u^2 \sinh u^3 \sinh u^4, \\[3pt]
x^3=\sin u^1 \cosh u^3, \\[3pt]
y^1=\cos u^1 \sin u^2 \cosh u^3 \sinh u^4+\sin u^1 \cos u^2 \sinh u^3 \cosh u^4, \\[3pt]
y^2=-\cos u^1 \cos u^2 \cosh u^3 \sinh u^4+\sin u^1 \sin u^2 \sinh u^3 \cosh u^4, \\[3pt]
y^3=-\cos u^1 \sinh u^3.
\end{array}
\]
Further we consider a manifold on $\tilde{S}=S\setminus\{0,0,\pm
1,0,0,0\}$, i.e. we exclude the set $u^1=\pm\pi/2,u^3=0$ from the
domain $(-\pi,\pi)^2\times\mathbb{R}^2$ of $S$ which maps the pair
of "poles".

The induced metric on $\tilde{S}$ has the following non-zero local
components:
\[
\begin{array}{c}
g_{11}=-g_{33}=1,\quad g_{22}=-g_{44}=\cos^2 u^1 \cosh^2
u^3-\sin^2 u^1 \sinh^2
u^3,\\[3pt]
g_{24}=g_{42}=2\sin u^1\cos u^1 \sinh u^3\cosh u^3.
\end{array}
\]
Further we use the following orthonormal basis of signature
$(++--)$:
\[
\begin{array}{c}
\left\{e_1=\partial_{1},\ e_2=\lambda\partial_{2}+\mu\partial_{4},\ e_3=\partial_{3},\
e_4=-\mu\partial_{2}+\lambda\partial_{4}\right\},
\end{array}
\]
where
$\lambda=\frac{\cos u^1 \cosh u^3}{\cos^2 u^1+\sinh^2 u^3}$,
$\mu=\frac{\sin u^1 \sinh u^3}{\cos^2 u^1+\sinh^2 u^3}$.
We introduce a structure $H$ as in (\ref{14}) and we verify that
$H$ and $g$ generate an almost $(H,G)$-structure on $\tilde{S}$.
By direct computations we get that $(\tilde{S},H,G)$ is a
$\mathcal{K}(J_2)$-manifold of pointwise constant totally real
sectional curvatures
\[
\begin{array}{c}
\nu=\frac{\sinh^2 2u^3-\sin^2 2u^1}{4(\cos^2 u^1+\sinh^2
u^3)^4},\quad \nu^*_2=\frac{\sin 2u^1\sinh 2u^3}{2(\cos^2
u^1+\sinh^2 u^3)^4},
\end{array}
\] where
$\nu:=\frac{R(x,y,y,x)}{\pi_1(x,y,y,x)}$,
$\nu^*_2:=\frac{R(x,y,y,J_2x)}{\pi_1(x,y,y,x)}$ for a basis
$\{x,y\}$ of any non-de\-ge\-ne\-rate totally real section
$\sigma$ (i.e. $\sigma\perp J_2\sigma$). $(\tilde{S},J_2,g)$ is an
almost Einstein manifold since its Ricci tensor is $\rho=2(\nu g-
\nu^*_2 g_2)$. But, the Nijenhuis tensors and the Lie forms
corresponding to other two almost complex structures $J_1$ and
$J_3$ are non-zero. Beside that, we receive the following linear
invariants:
\[
\begin{array}{c}
\tau=8\nu,\qquad \tau^*_1=0,\qquad \tau^*_2=8\nu^*_2,\qquad \tau^*_3=0,\\[3pt]
\normq{N_1}=2\normq{\nabla J_1}=8\normq{\theta_1}=-32\nu,\;
-\normq{N_3}=2\normq{\nabla J_3}=8\normq{\theta_3}=32\nu.
\end{array}
\]

\section{Lie groups with almost $(H,G)$-structure}

The next two examples are inspired from an example of a locally
flat almost Hermitian surface constructed in \cite{TrVa}. Let
$\mathcal{L}$ be a connected Lie subgroup of
$\mathcal{GL}(4,\mathbb{R})$ consisting of matrices with the
following non-zero entries
\[
\begin{array}{c}
a_{11}=a_{22}=\cos u_{1},\quad a_{12}=-a_{21}=\sin
u_{1},\\[3pt]
a_{13}=u_{2},\quad a_{23}=u_{3},\quad a_{33}=1,\quad a_{44}=\exp
u_{4}
\end{array}
\]
for arbitrary $u^{1},u^{2},u^{3},u^{4}\in \mathbb{R}$.

The Lie algebra of $\mathcal{L}$ is isomorphic to the Lie
subalgebra of $\mathfrak{gl}(4;\mathbb{R})$ generated by the
matrices $X_1$, $X_2$, $X_3$, $X_4$ with the the following
non-zero entries:
\[
(X_1)_{13}=(X_2)_{12}=-(X_2)_{21}=(X_3)_{23}=(X_4)_{44}=1.
\]

\subsection{A Lie group as a complex manifold but non-hypercomplex one}
For the first recent example let us substitute the following
pseudo-Riemannian $g$ for the metric on $\mathcal{L}$ used there:
$g(X_i,X_j)=\varepsilon_a \delta_{ij}$, where $1\leq i,j\leq 4$;
$\varepsilon_1=\varepsilon_2=-\varepsilon_3=-\varepsilon_4=1$.
Further we introduce an $\mathcal{L}$-invariant almost
hypercomplex structure $H$ on $\mathcal{L}$ as in (\ref{14}).
Then, there is generated an almost $(H,G)$-structure on
$\mathcal{L}$ and the received manifold is complex with respect to
$J_2$ but non-hypercomplex and the Lie forms do not vanish. The
non-zero components of the curvature tensor $R$ is determined by
$R_{1221}=R_{1331}=-R_{2332}=1$ and the linear invariants are the
following:
\[
\begin{array}{c}
\normq{N_1}=2\normq{\nabla J_1}=8\normq{\theta_1}=-\normq{\nabla
J_2}=-2\normq{\theta_2}=\normq{N_3}=-8,\\[3pt]
\normq{\nabla J_3}=12\normq{\theta_3}=12,\qquad
\tau=-\tau^*_1=2,\quad \tau^*_2=\tau^*_3=0.
\end{array}
\]
\subsection{A Lie group as a flat K\"ahler manifold but non-hypercomplex one}
For the second example we use the following pseudo-Riemannian $g$
on $\mathcal{L}$: $g(X_i,X_j)=\varepsilon_a \delta_{ij}$, where
$1\leq i,j\leq 4$;
$\varepsilon_1=-\varepsilon_2=\varepsilon_3=-\varepsilon_4=1$. We
actually substitute only the type of the signature: $(+-+-)$ for
$(++--)$ of the basis $\{X_1,X_2,X_3,X_4\}$. Then we introduce $H$
by the following different way:
\[
\begin{array}{llll}
J_1X_1=X_3,\; &J_1X_2=X_4,\; &J_1X_3=-X_1,\; &J_1X_4=-X_2,\\[3pt]
J_2X_1=-X_4,\; &J_2X_2=X_3,\; &J_2X_3=-X_2,\; &J_2X_4=X_1,\qquad
J_3=J_1J_2.
\end{array}
\]
Therefore we obtain that the constructed $(H,G)$-manifold is flat
and K\"ahler\-ian with respect to $J_1$ but regarding $J_2$ and
$J_3$ it is not complex and the structural tensors have the form
$F_2(X,Y,Z)=-\theta_2(J_3X)g(Y,J_3Z)$,
$F_3(X,Y,Z)=-\theta_3(J_2)g(Y,J_2Z)$. The non-zero linear
invariants for $\beta=2,3$ are the following: \(
\begin{array}{c}
-\normq{N_\beta}=2\normq{\nabla
J_\beta}=2\normq{F_\beta}=8\normq{\theta_\beta}=8.
\end{array}
\)

\section*{Acknowledgement} The first author is supported by
the Matsumae International Foundation's fellowship under the
guidance of the second author.


The legend of King Mahabali is the most popular and the most fascinating of all legends behind Onam. Onam celebrates the visit of King Mahabali to the state of Kerala every year. The festival is celebrated with fervour as King Mahabali is greatly respected by his subjects. King Mahabali is also popularly called Maveli and Onathappan.
The story goes that the beautiful state of Kerala was once ruled by an Asura (demon) king, Mahabali. The King was greatly respected in his kingdom and was considered to be wise, judicious and extremely generous. It is said that Kerala witnessed its golden era in the reign of King Mahabali. Everybody was happy in the kingdom, there was no discrimination on the basis of caste or class. Rich and poor were equally treated. There was neither crime, nor corruption. People did not even lock their doors, as there were no thieves in that kingdom. There was no poverty, sorrow or disease in the reign of King Mahabali and everybody was happy and content.
It may be noted Mahabali was the son of Veerochana and grandson of Prahlad, the devout son of demon King Hiranyakashyap. Mahabali had a son called Bana, who became a legendary king in his own right and became popular as Banraj in central Assam. Mahabali belonged to the Asura (demon) dynasty but was an ardent worshiper of Lord Vishnu. His bravery and strength of character earned him the title of "MahabaliChakravathy" or Mahabali - the King of Kings.
Looking at the growing popularity and fame of King Mahabali Gods became extremely concerned and jealous. They felt threatened about their own supremacy and began to think of a strategy to get rid of the dilemma.
To curb the growing reign of Mahabali and maintain their own supremacy, Aditi, the mother of Gods seeked help of Lord Vishnu (the preserver in the Hindu trinity) whom Mahabali worshiped.
It was said Mahabali was very generous and charitable. Whenever anybody approached him for help or requested for anything he always granted. To test the King, Lord Vishnu disguised himself as a dwarf and a poor Brahmin called Vamana. He came to the Kingdom of Mahabali, just after Mahabali performed his morning prayers and was preparing to grant boons to Brahmins.
Disguised as Vamana, Vishnu said he was a poor Brahmin and asked for a piece of land. The generous King said, he could have as much land as he wanted. The Brahmin said that he just wanted as much land as could be covered by his three steps. The King was surprised to hear but agreed.
A learned adviser of the King, Shukracharya sensed that Vamana was not an ordinary person and warned the King against making the promise. But, the generous King replied that it would be a sin for a King to back on his words and asked the Brahmin to take the land. The King could not imagine that the dwarf Brahmin was Lord Vishnu himself.
Just as King Mahabali agreed to grant the land, Vamana began to expand and eventually increased himself to the size of cosmic proportions. With his first step the Brahmin boy covered the whole of earth and with the other step he covered the whole of the skies. He then asked King Mahabali where is the space for him to keep his third foot.
The King realised that he was no ordinary Brahmin and his third step will destroy the earth. Mahabali with folded hands bowed before Vamana and asked him to place his last step on his head so that he could keep the promise. The Brahmin placed his foot on the head of the King, which pushed him to patala, the nether world. There the King requested the Brahmin to reveal his true identity. Lord Vishnu then appeared before the King in his person. The Lord told the King that he came to test him and the King won the test. King Mahabali was pleased to see his lord. Lord Vishnu also granted a boon to the King.
The King was so much attached with his Kingdom and people that he requested that he be allowed to visit Kerala once in a year. Lord Vishnu was moved by the Kings nobility and was pleased to grant the wish. He also blessed the King and said even after losing all his worldly possessions, the King would always be loved by Lord Vishnu and his people.
It is the day of the visit of King Mahabali to Kerala that is celebrated as Onam every year. The festival is celebrated as a tribute to the sacrifice of King Mahabali. Every year people make elaborate preparations to welcome their King whom they affectionately call Onathappan. They wish to please the spirit of their King by depicting that his people are happy and wish him well. The second day, Thiruvonam is the biggest and the most important day of this festival. It is believed that King Mahabali visits his people on the second day.
Onam celebrations are marked in Trikkakara, a place 10 km from Kochi (Cochin) on the Edapally- Pookattupadi road. Trikkara is said to be the capital of the mighty King Mahabali. A temple with a deity of 'TrikkakaraAppan' or 'Vamanamurthy' who is Lord Vishnu himself in disguise is also located at this place. Nowhere else in Kerala can one find a deity of 'Vamanamurthy'. This fascinating legend is artistically depicted at the Suchindram Temple in Kanyakumari district of Tamil Nadu.


When it comes to kitchenware and recipe trends you might not be afraid to be adventurous and try new things but our recent research suggests that certain kitchen gadgets and ingredients have us a little confused, not just with how to incorporate them into our cooking, but even with how to pronounce them!
To find out just how many people struggle with the pronunciation of certain words, we surveyed the nation, asking people to choose the correct way to pronounce some of the more confusing kitchen and food related words. It turns out we might all not be quite as clued up as we think we are.
One of the more confusing items proved to be a trivet – a silicone tray that protects surfaces from heat damage – with people calling it a dartboard and a tea strainer, whilst a huller created merriment too – a small steel tool used to remove the stems from strawberries, but some people mistook this item for a tea bag squeezer or a pair of tongs!
Our research indicates that the general public might need to brush up on their food and kitchenware knowledge, despite their love of cooking and food.
Find out how much of a kitchen enthusiast you really are and test your knowledge with our fun quiz below.


text:
Kevin Rowland (I)
Soundtrack | Actor | Composer
Kevin Rowland was born on August 17, 1953 in Wolverhampton, West Midlands, England.
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Before achieving fame in Dexys Midnight Runners, he was a member of an art-rock band called Lucy And The Lovers, who were heavily influenced by Deaf School. The band turned into a punk rock outfit called The Killjoys.
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The Lee and Kin Show got to sit and talk with Teejay about his brand Its Just Paint, art, social media, and more. Make sure you follow @ItsJustPaint on instagram.
Music, entertainment and business are like Lee's heartbeats and the air he breathes. They are his passion and every day he shows that.


This a great t-shirt. I'm sure that in 1985, plutonium is available at every corner drug store, but in 1955, it's a little hard to come by. Marty, I'm sorry, but I'm afraid you're stuck here. Never mind that now, never mind that now. No, I refuse to except the responsibility. It's taken me almost thirty years and my entire family fortune to realize the vision of that day.
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A 10 minute HIIT workout is more effective for cardio training than a 30 minute jog? Check out our blog post on High Intensity Interval Training.


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Every Thursday Evening at a neat little spot called "Little Tokyo" in Stephens City, VA. The music gets underway around 8 pm, and runs until 11 or so.
If you like fast grilled Hibachi, perhaps a little saki, or some of those big 22 oz japanese beers, come on out and kick back while I host a night of good music and good friends. I've invited some very special musicians to join me on stage, and this will be the best week yet.
Little Tokyo is a very open and relaxed atmosphere, and a great spot to chill with plenty of room, and a positive vibe.
No Cover Charge ~ All Ages Show ~ Music Starts at 8 PM.
- Come on out and enjoy yourselves!
By the way, this event can also be viewed at the Face Book Events Page if you want to check it out over there.
Directions: Little Tokyo is easy to find, just take the Stephens City exit from I-81 (exit 307) then get on 277 (Fairfax Pike) heading east. Go 1.3 miles, passing thru the light at Warrior Drive. Take the very next left into the Sherando shopping plaza and you'll see Little Tokyo straight back. Call 540-868-0000 for details and directions.