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\begin{document}

\title{\bf\Large A short survey on $\delta$-ideal CR submanifolds}
\author{\bf Toru Sasahara\\
\small \it Center for Liberal Arts and Sciences,  Hachinohe Institute of Technology\\
\small \it Hachinohe, Aomori 031-8501, JAPAN\\
\small {\it E-mail}: {\tt sasahara@hi-tech.ac.jp}}

\date{}
\maketitle

\begin{abstract}
{\footnotesize This paper surveys some of the known  results  on $\delta$-ideal
CR submanifolds in complex space forms, the nearly K\"{a}hler $6$-sphere and odd dimensional
unit spheres.
In addition,
the relationship between $\delta$-ideal CR submanifolds and critical points of
the $\lambda$-bienergy is mentioned.
Some topics on variational problem for the $\lambda$-bienergy
 are also presented.
}
\end{abstract}

{\footnotesize 2010 {\it Mathematics Subject Classification}. 53C42, 53B25.}

{\footnotesize{\it Key words and phrases.}   CR submanifolds, $\delta$-invariants, $\delta$-ideal submanifolds,
$\lambda$-biharmonic submanifolds.}

 \section{Introduction}

In submanifold theory,  it is important to establish relations between
  extrinsic and intrinsic invariants of submanifolds.

   %
In the early 1990s, the notion of  $\delta$-invariants was introduced by Chen
(see \cite{chen2}, \cite{chen3} and \cite{chen4}).
These invariants are obtained by subtracting a certain amount of sectional curvatures
from the scalar curvature.
Furthermore,
 he established pointwise optimal
 inequalities involving
 $\delta$-invariants and
 the squared mean curvature
  of
 arbitrary submanifolds in real and complex space forms.
A submanifold is   said to be {\it $\delta$-ideal} if it it satisfies an
 equality case of the inequalities everywhere.
During the last two decades,
many interesting results on $\delta$-ideal submanifolds have been obtained.

The main purpose of  this paper is to survey
some of the known  results  on $\delta$-ideal CR submanifolds in
  complex space forms, the nearly K\"{a}hler $6$-sphere and odd dimensional unit spheres.

 For a given compact almost CR manifold $M$ (with or without boundary)
 equipped with
 a compatible metric, the $\delta$-ideal CR
 immersions of $M$ minimize the  $\lambda$-bienergy
 among all isometric CR immersions of $M$.
In view of this fact,
some topics on variational problem for the $\lambda$-bienergy are also
  presented.

\section{Preliminaries}
Let $M$ be
 an $n$-dimensional  submanifold of a Riemannian manifold $\tilde M$.
 Let us denote by $\nabla$ and
 $\tilde\nabla$ the Levi-Civita connections on $M$ and
 $\tilde M$, respectively. The
 Gauss and Weingarten formulas are respectively given by
\begin{equation}
 \begin{split}
 \tilde \nabla_XY&= \nabla_XY+B(X,Y), \label{gawe}\\
 \tilde\nabla_X V&= -A_{V}X+D_XV
 \end{split}\nonumber
\end{equation}
 for tangent vector fields $X$, $Y$ and normal vector field $V$,
 where $B$, $A$ and $D$ are the second fundamental
 form, the shape operator and the normal
 connection.

 The mean curvature vector field $H$ is defined by
$H=(1/n){\rm trace}\hskip2pt B.$
 The function $|H|$ is called the  {\it mean curvature}.
 If it vanishes identically, then $M$ is called a {\it minimal submanifold}.
 In particular, if $B$ vanishes identically,  then $M$ is called a {\it totally geodesic submanifold}.

\begin{definition}[\cite{be}]
{\rm   Let $M$ be a Riemannian submanifold  of an almost Hermitian  manifold
$\tilde M$ and let $J$ be the complex  structure of $\tilde M$. A submanifold $M$
 is called a {\it CR submanifold} if there exist differentiable distributions
 $\mathcal{H}$ and  ${\mathcal H}^{\perp}$ such that
 \begin{equation}
 TM=\mathcal{H}\oplus {\mathcal H}^{\perp}, \quad \nonumber
 J\mathcal{H}=\mathcal{H}, \quad
 J{\mathcal H}^{\perp}\subset T^{\perp}M,
 \end{equation}
where $T^{\perp}M$
denotes the normal bundle of $M$.
 A CR submanifold is called a {\it K\"{a}hler submanifold} (resp. {\it totally real submanifold})
  if {\rm rank}\hskip2pt${\mathcal H}^{\perp}=0$ (resp. {\rm rank}\hskip2pt${\mathcal H}=0$).
A totally real submanifold is called a {\it Lagraingian submanifold} if
$J(TM)=T^{\perp}M$.
A CR submanifold is said to be ${\it proper}$ if {\rm rank}\hskip2pt${\mathcal H}\ne 0$
and {\rm rank}\hskip2pt${\mathcal H}^{\perp}\ne 0$.

}
\end{definition}

\section{$\delta$-invariants}
Let $M$ be an $n$-dimensional Riemannian manifold. Denote by $K(\pi)$ the sectional curvature
 of $M$ associated with a plane section $\pi\subset T_pM$, $p\in M$. For any orthonormal basis
 $\{e_1, \ldots, e_n\}$ of the tangent space $T_pM$, the scalar curvature $\tau$ at $p$ is defined
 by
 \begin{equation}
 \tau(p)=\sum_{i<j}K(e_i \wedge e_j).\nonumber
 \end{equation}
 Let $L$ be a subset of $T_pM$ of dimension $r\geq 2$ and $\{e_1, \ldots, e_r\}$
 an orthonormal basis of $L$. We define the scalar curvature $\tau(L)$ of the $r$-plane section $L$ by
 \begin{equation}
 \tau(L)=\sum_{\alpha<\beta}
K(e_{\alpha}\wedge e_{\beta}), \nonumber
\quad 1\leq\alpha,
\beta\leq r.
 \end{equation}
 For an integer $k\geq 0$, denote by $\mathcal{S}(n, k)$ the finite set which consists of unordered
$k$-tuples $(n_1, \ldots, n_k)$ of integers satisfying $2\leq n_1 \ldots, n_k<n$ and
$n_1+\ldots+n_k\leq n$.
 We denote by $\mathcal{S}(n)$ the set of $k$-tuples with $k\geq 0$ for a fixed $n$.

For each $k$-tuple $(n_1, \ldots, n_k)\in \mathcal{S}(n)$,
  the notion
 of {\it $\delta$-invariant} $\delta(n_1, \ldots, n_k)$ was introduced by  Chen \cite{chen3}
  as follows:
\begin{equation}
\delta(n_1, \ldots, n_k)(p)=\tau(p)-{\rm inf}\{\tau(L_1)+\cdots+\tau(L_k)\},\nonumber
\end{equation}
where $L_1$, \ldots, $L_k$ run over all $k$ mutually orthogonal subspaces of $T_pM$ such that
 $\dim L_j=n_j$, $j=1, \ldots, k$.

Let  $\overline{Ric}$ denote the maximum Ricci curvature function on $M$ defined by
\begin{equation} \overline{Ric}(p)={\rm max}\{S(X, X)|X\in U_pM\},\nonumber\end{equation}
where $S$ is the Ricci tensor and $U_pM$
is the unit tangent vector space of $M$ at $p$.
Then, we have $\delta(n-1)(p)=\overline{Ric}(p)$.

Let $M$ be a K\"{a}hler manifold with real dimension $2n$.
 For each $k$-tuple $(2n_1, \ldots, 2n_k)\in \mathcal{S}(2n)$,  Chen \cite{chen3} also
 introduced the notion of
  {\it complex}
$\delta$-$invariant$ $\delta^c(2n_1, \ldots, 2n_k)$, which is defined by
\begin{equation}
\delta^c(2n_1, \ldots, 2n_k)(p)=\tau(p)-{\rm inf}\{\tau(L_1^c)+\cdots+\tau(L_k^c)\},\nonumber
\end{equation}
where $L_1^c$, \ldots, $L_k^c$ run over all $k$ mutually orthogonal complex
subspaces of $T_pM$ such that
 ${\rm dim}L_j=2n_j$, $j=1, \ldots, k$.

For simplicity, we denote $\delta(\lambda, \ldots, \lambda)$ and
$\delta^c(\lambda, \ldots, \lambda)$
 by $\delta_k(\lambda)$ and $\delta_k^c(\lambda)$,
respectively, where $\lambda$ appears $k$ times.

 \section{Inequalities involving $\delta$-invariants and ideal  submanifolds}

 For each $(n_1, \ldots, n_k)\in{\mathcal S}(n)$,  let $c(n_1, \ldots, n_k)$ and
 $b(n_1, \ldots, n_k)$ be the constants given by
\begin{align}
&c(n_1, \ldots, n_k)=\frac{n^2(n+k-1-\sum_{j=1}^k n_j)}{2(n+k
-\sum_{j=1}^k n_j)},\nonumber\\
&b(n_1, \ldots, n_k)=\frac{1}{2}\Bigl(n(n-1)-\sum_{j=1}^k n_j(n_j-1)\Bigr).\nonumber
\end{align}

Chen obtained the following inequality
 for  an arbitrary submanifold in a
  real space form.
 \begin{theorem}[\cite{chen4}]\label{ingere}
Given an $n$-dimensional submanifold $M$ in  an $m$-dimensional  real space form
$R^m(\epsilon)$ of constant sectional curvature $\epsilon$, we have
 \begin{equation}
 \delta(n_1, \ldots, n_k)
 \leq c(n_1, \ldots, n_k)|H|^2
+b(n_1, \ldots, n_k)\epsilon.\label{inre}
 \end{equation}
 Equality sign of $(\ref{inre})$ holds at a point $p\in M$ for some $(n_1, \ldots, n_k)\in{\mathcal S}(n)$
if and only if there exists an orthonormal basis $\{e_1, \ldots, e_{2m}\}$ at $p$
such that $e_1, \ldots, e_{n}$ are tangent to $M$ and  the shape operators of $M$ in $R^m(\epsilon)$ at $p$ take  following forms$:$
\begin{equation}
A_{e_r}= \left(
    \begin{array}{cccc}
      A_1^r & \ldots & 0 & \\
      \vdots & \ddots & \vdots &  ${\rm \mbox{\Large 0}}$\\
      0 & \ldots & A_k^r &  \\
       &   ${\rm \mbox{\Large 0}}$ &  & \mu_r  I
    \end{array}
  \right),\label{A}
\end{equation}
$$r=n+1, \ldots, 2m,$$
where each $A_{j}^r$ is a symmetric $n_j\times n_j$ submatrix such that
\begin{equation}
{\rm trace}(A_{1}^r)=\cdots={\rm trace}(A_{k}^r)=\mu_r.\label{traceA}
\end{equation}
 \end{theorem}

 Let $\tilde M^m(4\epsilon)$ be a complex space form of complex dimension $m$ and constant
 holomorphic sectional curvature $4\epsilon$ and let $J$ be the complex
structure of $\tilde M^m(4\epsilon)$.

Let $M$ be an $n$-dimensional submanifold in $\tilde M^m(4\epsilon)$.
 For any vector $X$ tangent to $M$, we put $JX=PX+FX$, where $PX$ and $FX$ are tangential
 and normal components of $JX$, respectively. For a subspace $L\subset T_pM$ of dimension $r$,
 we set
\begin{equation}
\Psi(L)=\sum_{1\leq i<j\leq r}\<Pu_i, u_j\>^2,\nonumber
\end{equation}
 where $\{u_1, \ldots, u_r\}$ is an orthonormal basis of $L$.

 For an arbitrary  submanifold in a complex space form, we have

 \begin{proposition}[\cite{chen4}]\label{pro1}
 Let $M$ be
 an $n$-dimensional submanifold  in a complex space form $\tilde M^m(4\epsilon)$.
 Then, for mutually orthogonal subspaces $L_1$, \ldots, $L_k$ of $T_pM$ such that
 $\dim L_j=n_j$,  we have
 \begin{equation}
 \tau-\sum_{i=1}^k\tau(L_i)\leq c(n_1, \ldots, n_k)|H|^2+b(n_1, \ldots, n_k)\epsilon
+\frac{3}{2}|P|^2\epsilon
 -3\epsilon\sum_{i=1}^k\Psi(L_i).\label{gein}
 \end{equation}

The equality case of inequality $(\ref{gein})$
 holds at a point $p\in M$ if and only if there exists an orthonormal basis $\{e_1, \ldots, e_{2m}\}$
 at $p$ such that

${\rm (a)}$ $L_j={\rm Span}\{e_{n_1+\cdots+n_{j-1}+1}, \ldots,
 e_{n_1+\cdots+n_j}\}$, \quad $j=1, \ldots, k$,

${\rm (b)}$ the shape operators of $M$ in $\tilde M^m(4\epsilon)$ at $p$ satisfy {\rm (\ref{A})} and
{\rm (\ref{traceA})}.
 \end{proposition}

 Using Proposition \ref{pro1}, we obtain the following  inequalities.

\begin{proposition}[\cite{chen4}]\label{proka}
Let $M$ be a K\"{a}hler submanifold with real dimension $2n$ in a complex space form
 $\tilde M^m(4\epsilon)$. Then,
we have
 \begin{equation}
\delta^c(2n_1, \ldots, 2n_k)\leq 2\biggl(n(n+1)-\sum_{j=1}^kn_j(n_j+1)\biggr)\epsilon.\label{ka}
\end{equation}
The equality case of inequality $(\ref{ka})$ holds at a point $p\in M$ if and only if
there exists an orthonormal basis
 $\{e_1, \ldots, e_{2m}\}$ at $p$ such that $e_1, \ldots, e_{2n}$ are tangent to $M$
 and $e_{2l}=Je_{2l-1}$ $(1\leq l\leq k)$, and moreover, the shape operators of $M$ in
 $\tilde M^m(4\epsilon)$ at $p$ take
 the following forms$:$
\begin{equation}
A_{e_r}= \left(
    \begin{array}{cccc}
      A_1^r & \ldots & 0 & \\
      \vdots & \ddots & \vdots &  ${\rm \mbox{\Large 0}}$ \\
       0& \ldots & A_k^r &  \\
       & ${\rm \mbox{\Large 0}}$&  & ${\rm \mbox{\Large 0}}$
    \end{array}
  \right),\nonumber
\end{equation}
$$r=2n+1, \ldots, 2m,$$ where each $A_j^r$ is a symmetric $(2n_j)\times(2n_j)$ submatrix satisfying
${\rm trace}(A_j^r)=0$.
\end{proposition}

 \begin{proposition}[\cite{sa3}]\label{pro3}
 Let  $M$ be an $n$-dimensional CR submanifold with ${\rm rank}\hskip2pt{\mathcal H}=2h$
in
${\mathbb C}H^m(-4)$.
 Then, we have
 \begin{equation}
 \delta(n_1, \ldots, n_k)
 \leq c(n_1, \ldots, n_k)|H|^2
-b(n_1, \ldots, n_k)-3h+\frac{3}{2}\sum_{j=1}^k n_j.\label{inCH}
 \end{equation}
 Equality sign of $(\ref{inCH})$ holds at a point $p\in M$ for some $(n_1, \ldots, n_k)\in{\mathcal S}(n)$
if and only if there exists an orthonormal basis $\{e_1, \ldots, e_{2m}\}$ at $p$ such that

${\rm (a)}$ each $L_j={\rm Span}\{e_{n_1+\cdots+n_{j-1}+1}, \ldots, e_{n_1+\cdots+n_j}\}$
satisfies
$\Psi(L_j)=n_j/2$ for $1\leq j\leq k$,

${\rm (b)}$ the shape operators of $M$ in ${\mathbb C}H^m(-4)$ at $p$ satisfy {\rm (\ref{A})} and {\rm (\ref{traceA})}.
 \end{proposition}

\begin{proposition}[\cite{sa4}]\label{pro4}
Let  $M$ be an $n$-dimensional CR submanifold with ${\rm rank}\hskip2pt{\mathcal H}=2h$
in
${\mathbb C}P^m(4)$.
 Then, we have
 \begin{equation}
 \delta(n_1, \ldots, n_k)
 \leq c(n_1, \ldots, n_k)|H|^2
+b(n_1, \ldots, n_k)+3h.\label{inCP}
 \end{equation}
 Equality sign of $(\ref{inCP})$ holds at a point $p\in M$ for some $(n_1, \ldots, n_k)\in{\mathcal S}(n)$
if and only if there exists an orthonormal basis $\{e_1, \ldots, e_{2m}\}$ at $p$ such that

${\rm (a)}$ each $L_j={\rm Span}\{e_{n_1+\cdots+n_{j-1}+1}, \ldots, e_{n_1+\cdots+n_j}\}$
satisfies
$\Psi(L_j)=0$ for $1\leq i \leq k$,

${\rm (b)}$ the shape operators of $M$ in ${\mathbb C}P^m(4)$ at $p$ satisfy $(\ref{A})$ and
 $(\ref{traceA})$.
\end{proposition}

\begin{definition}{\rm
A submanifold is
said to be {\it $\delta(n_1, \ldots, n_k)$-ideal}
if it satisfies the equality case of  (\ref{inre}), (\ref{inCH}) or (\ref{inCP})  identically  for
a $k$-tuple $(n_1, \ldots, n_k)\in {\mathcal S}(n)$. Similarly,
a K\"{a}hler submanifold is said to be {\it $\delta^c(2n_1, \ldots, 2n_k)$-ideal} if it
 satisfies the equality case of (\ref{ka}) identically for
a $k$-tuple $(2n_1, \ldots, 2n_k)\in {\mathcal S}(2n)$.}
\end{definition}

For more information on $\delta$-invariants and $\delta$-ideal submanifolds,
we refer the reader to \cite{chen}.

\begin{definition}{\rm
A submanifold is said to be {\it linearly full} in $\tilde M^m(4\epsilon)$
if it  does not lie in any totally geodesic K\"ahler hypersurfaces of $\tilde M^m(4\epsilon)$.}
\end{definition}

 \section{Ideal CR submanifolds in complex hyperbolic space}

We first recall some basic definitions on hypersurfaces.
\begin{definition}{\rm Let $N$ be a submanifold in a Riemannian manifold $\tilde M$ and $UN^{\perp}$
 the unit normal bundle of $N$.
Then, for a sufficiently small $r>0$, the following mapping is an  immersion:
\begin{equation}
f_r: UN^{\perp}\rightarrow \tilde M,  \quad f_r(p, V)={\exp}_p(rV),\nonumber
\end{equation}
where exp denotes the exponential mapping of $\tilde M$. The hypersurface
$f_r(UN^{\perp})$ of $\tilde M$  is called the {\it tubular hypersurface} over $N$ with radius $r$.
If $N$ is a point $x$ in $\tilde M$, then the tubular hypersurface over $x$  is a geodesic hypersphere centered
at $x$.}
\end{definition}

\begin{definition}
{\rm For a given point $p\in\mathbb{C}H^m(-4)$, let $\gamma(t)$ be a geodesic with $\gamma(0)=p$, which is parametrized
by arch length. Denote by $S_t(\gamma(t))$ the geodesic hypersphere
centered at $\gamma(t)$ with radius $t$.
The limit of $S_t(\gamma(t))$ when $t$ tends to infinity is called a {\it  horosphere}.}
 \end{definition}

 \begin{definition}{\rm
 Let $M$ be a
  real hypersurface  in an almost Hermitian manifold and $V$ be a unit normal vector.
  A hypersurface $M$ is called a {\it Hopf hypersurface}
 if $JV$ is a principal curvature vector.}

 \end{definition}

 A real hypersurface in an almost Hermitian manifold is a proper CR submanifold with
 ${\rm rank}{\hskip 2pt}{\mathcal H}^{\perp}=1$.
 The following theorem characterizes the horosphere of $\mathbb{C}H^m(-4)$ in terms of $\delta_k(2)$.
\begin{theorem}[\cite{chen4}]\label{horo1}
Let $M$ be a $\delta_k(2)$-ideal
real hypersurface of $\mathbb{C}H^m(-4)$.
Then $k=m-1$ and $M$ is an open portion of the horosphere in  $\mathbb{C}H^m(-4)$.
\end{theorem}

\begin{remark}
{\rm The third case of  (9.5) in  \cite{chen4} does not occur, because $L_1\ldots, L_k$  are
 complex planes. Therefore, case (1) of Theorem 9.1 in \cite{chen4}
 shall be removed from the list of $\delta_k(2)$-ideal real hypersurfaces in $\mathbb{C}H^m(-4)$.}
\end{remark}

  For $\delta(2m-2)$-ideal real hypersurfaces
  in $\mathbb{C}H^m(-4)$, Chen proved the following.
\begin{theorem}[\cite{chen5}]\label{2m-2}
Let $M$ be a
real hypersurface of $\mathbb{C}H^m(-4)$.
Then $M$ is $\delta(2m-2)$-ideal if and only if $M$ is a Hopf hypersurface
with constant mean curvature given by $2\alpha/(2m-1)$, where
$A_VJV=\alpha JV$ for a unit normal vector $V$.
If $M$ has constant principal curvatures, then
$M$ is an open portion of one of the following real hypersurfaces:

{\rm (1)} the horosphere of $\mathbb{C}H^2(-4)${\rm ;}

{\rm (2)} the tubular hypersurface over totally geodesic  $\mathbb{C}H^{m-1}(-4)$
in $\mathbb{C}H^m(-4)$ with radius $r=\tanh^{-1}(1/\sqrt{2m-3})$, where $m\geq 3$.
\end{theorem}

It was proved in \cite{chen5} that
if $m=2$ in Theorem \ref{2m-2}, then
 the
 assumption of  the constancy of principal curvatures
  is satisfied.  That is to say, we have
\begin{corollary}[\cite{chen5}]
Let $M$ be a $\delta(2)$-ideal real hypersurface of $\mathbb{C}H^2(-4)$. Then
$M$ is an open portion of the horosphere.
\end{corollary}

Let ${\mathbb C}_1^{m+1}$ be the complex number $(m+1)$-space endowed with the complex coordinates
$(z_0, \ldots, z_m)$, the
pseudo-Euclidean metric given by $\tilde g=-dz_0d\bar{w_0}+\sum_{i=1}^mdz_id\bar{w_i}$ and the
standard complex structure.
For $\epsilon<0$, we put
 $H^{2m+1}_1(\epsilon)=\{z\in {\mathbb C}_1^{m+1}|\<z, z\>=1/\epsilon\}$,
 where $\< , \>$
denotes the inner product on ${\mathbb C}_1^{m+1}$ induced from $\tilde g$.

For a given $z\in H^{2m+1}_1(\epsilon)$, we put
 $[z]=\{\lambda z| \lambda\in\mathbb C, \lambda\bar{\lambda}=1\}$.
The Hopf fibration is given by
\begin{equation}  \varpi_{\{m, \epsilon\}}: H^{2m+1}_1(\epsilon)\rightarrow {\mathbb C}H^m(4\epsilon):
 z\mapsto [z].
\nonumber\end{equation}

For $\delta_k(2)$-ideal proper CR submanifolds   in ${\mathbb C}H^m(-4)$
whose codimensions are greater than one, we have the following representation formula.
\begin{theorem}[\cite{sa1}]\label{horo2}
Let $M$ be a linearly full $(2n+1)$-dimensional $\delta_k(2)$-ideal
CR submanifold in ${\mathbb C}H^m(-4)$ such that {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$, $k\geq 1$ and
$m>n+1$.
Then,  up to holomorphic isometries of ${\mathbb C}H^m(-4)$,
 the immersion of $M$ into ${\mathbb C}H^m(-4)$ is given by
 the composition $\varpi_{\{m, -1\}}\circ z$, where
 \begin{equation}
 z=\biggl(-1-\frac{1}{2}|\Psi|^2+iu, -\frac{1}{2}|\Psi|^2+iu, \Psi\biggr)e^{it},
 \label{CR1}\end{equation}
and $\Psi$ is a  $2n$-dimensional $\delta^c_n(2)$-ideal K\"{a}hler submanifold in ${\mathbb C}^{m-1}$.
\end{theorem}

Up to holomorphic isometries of ${\mathbb C}H^m(-4)$, the horosphere
 in ${\mathbb C}H^m(-4)$ is a real hypersurface defined by $\{[z]: z\in H^{2m+1}_1(-1),
 \hskip5pt
  |z_0-z_1|=1\}$ (see, for example,
 \cite{ver}).
 Hence, Theorem \ref{horo2} can be considered as an extension of Theorem
 \ref{horo1}.

As an immediate corollary of Theorem \ref{horo2}, we obtain
\begin{corollary}[\cite{cv}]
Let $M$ be a linearly full $3$-dimensional $\delta(2)$-ideal
CR submanifold in ${\mathbb C}H^m(-4)$ such that {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$
and
$m>2$.
Then,  up to holomorphic isometries of ${\mathbb C}H^m(-4)$,
 the immersion of $M$ into ${\mathbb C}H^m(-4)$ is given by
 the composition $\varpi_{\{m, -1\}}\circ z$, where $z$ is given by {\rm (\ref{CR1})} and
 $\Psi(w)$
is a  holomorphic curve in ${\mathbb C}^{m-1}$ with $\Psi^{\prime}(w)\ne 0$.
\end{corollary}

For  general
$\delta(n_1, \ldots, n_k)$-ideal  proper CR submanifolds   in ${\mathbb C}H^m(-4)$
whose codimensions are greater than one, the following classification result has been obtained.
\begin{theorem}[\cite{sa3}, \cite{sa4}]\label{cla}
Let $M$ be a linearly full $(2n+1)$-dimensional $\delta(n_1, \ldots, n_k)$-ideal
CR submanifold in ${\mathbb C}H^m(-4)$ such that {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$, $k\geq 1$ and
$m>n+1$. Then, we have
$JH\in\mathcal{H}^{\perp}$, $A_VJV=(2n/\sqrt{k(2n-k)})JV$ for $V=H/|H|$,  $DH=0$,  and moreover,
  the mean curvature is given by $$\frac{2n(k+1)}{(2n+1)\sqrt{k(2n-k)}}.$$
  If
all principal curvatures of $M$
with respect to $H/|H|$ are constant, then one of  the following two cases occurs{\rm :}

{\rm (1)}
 $M$ is locally congruent with the immersion described in Theorem {\rm \ref{horo2}}.

{\rm (2)}
$n/k\in {\mathbb Z}-\{1\}$,
$n_1=\cdots=n_k=2n/k$, and $M$ is locally congruent with the immersion
\begin{equation}
\varpi_{\{m, -1\}}\biggl(\varpi_{\{m-1, \frac{2k-2n}{2n-k}\}}^{-1}(\Psi), \sqrt{\frac{k}{2n-2k}}e^{it}\biggr),\nonumber\end{equation}
where
$\Psi$ is a $2n$-dimensional $\delta^c_k(2n/k)$-ideal K\"{a}hler submanifold in
${\mathbb C}H^{m-1}
(\frac{8k-8n}{2n-k})$.
\end{theorem}

If $n>1$, $k=1$ and $n_1=2n$ in Theorem \ref{cla}, then we have
\begin{corollary}[\cite{sa2}, \cite{sa4}]\label{2n}
Let $M$ be a linearly full $(2n+1)$-dimensional CR submanifold
 in ${\mathbb C}H^m(-4)$ such that {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$, $n>1$ and $m>n+1$.
 Then $M$ is  $\delta(2n)$-ideal if and only if $JH\in\mathcal{H}^{\perp}$,
 $DH=0$, $A_VJV=(2n/\sqrt{(2n-1)})JV$ for $V=H/|H|$, and moreover,
  the mean curvature is given by $$\frac{4n}{(2n+1)\sqrt{2n-1}}.$$
  If
all principal curvatures of $M$
with respect to $H/|H|$ are constant,   then, up to holomorphic isometries of ${\mathbb C}H^m(-4)$, the immersion of $M$ into ${\mathbb C}H^m(-4)$
is given by
\begin{equation}
\varpi_{\{m, -1\}}\Biggl(\varpi_{\{m-1, \frac{2-2n}{2n-1}\}}^{-1}(\Psi), \sqrt{\frac{1}{2n-2}}e^{it}\Biggr),
\nonumber\end{equation}
where
$\Psi$ is a  $2n$-dimensional K\"{a}hler submanifold in ${\mathbb C}H^{m-1}
(\frac{8-8n}{2n-1})$.
\end{corollary}

A hypersurface given by (2) in Theorem \ref{2m-2} can be  rewritten as follows (see, for example,
\cite[Example 6.1]{mon}):
 \begin{equation}
\varpi_{\{m, -1\}}\Biggl(H_1^{2m-1}\biggl(\frac{4-2m}{2m-3}\biggr)\times
S^1\biggl(\frac{1}{\sqrt{2m-4}}\biggr)\Biggr),
\nonumber\end{equation}
where $S^1(r)=\{z\in{\mathbb C}| z\bar{z}=r^2\}$.
Thus, Corollary \ref{2n} can be regarded as an extension of Theorem \ref{2m-2}.

\bigskip

Let $N$ be a K\"ahler hypersurface with real dimension $2n$ in a complex space form.
Let $V$ and $JV$ be normal vector fields of $N$.
Since $A_{JV}=JA_{V}$ and $JA_V=-A_VJ$ holds (cf. \cite[p.175]{kn}), there exists an
orthonormal basis $\{e_1, Je_{1}, \ldots, e_{n}, Je_{n}\}$ of $T_{p}N$ with respect to which
the shape operators $A_{V}$ and $A_{JV}$ take the following forms:
\begin{equation}
A_{V}= \left(
    \begin{array}{ccccc}
      \lambda_1 &  &  & & ${\rm \mbox{\Large 0}}$\\
      & -\lambda_1 & & & \\
      &  & \ddots &  &  \\
      &  &  & \lambda_n & \\
     ${\rm \mbox{\Large 0}}$  &   &  & & -\lambda_n
    \end{array}
  \right), \quad
A_{JV}= \left(
    \begin{array}{ccccc}
      0& \lambda_1 &  & & ${\rm \mbox{\Large 0}}$\\
      \lambda_1& 0 & & & \\
      &  & \ddots &  &  \\
      &  &  & 0 & \lambda_n\\
     ${\rm \mbox{\Large 0}}$  &   &  & \lambda_n & 0
    \end{array}
  \right).\nonumber
\end{equation}

Hence, it follows from Proposition \ref{proka}  that every K\"ahler hypersurface with
real dimension $2n$
in a complex space form is $\delta^c_k(2n/k)$-ideal for any natural number
$k$ such that  $n/k\in {\mathbb Z}$.   Accordingly, applying Theorem \ref{horo2}
yields the following.
\begin{corollary}[\cite{sa1}]\label{m=n+2}
Let $M$ be a linearly full $(2n+1)$-dimensional $\delta_k(2)$-ideal
CR submanifold in ${\mathbb C}H^{n+2}(-4)$
such that {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$ and $k\geq 1$.
Then,  up to holomorphic isometries of ${\mathbb C}H^{n+2}(-4)$,
 the immersion of $M$ into ${\mathbb C}H^{n+2}(-4)$ is given by
 the composition $\varpi_{\{n+2, -1\}}\circ z$, where
 $z$ is given by {\rm (\ref{CR1})},
and $\Psi$ is a  K\"{a}hler hypersurface in ${\mathbb C}^{n+1}$.
\end{corollary}

Similarly, we obtain the following corollary of Theorem \ref{cla}.
\begin{corollary}
Let $M$ be a linearly full $(2n+1)$-dimensional $\delta(n_1, \ldots, n_k)$-ideal
CR submanifold in ${\mathbb C}H^{n+2}(-4)$ such that {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$ and $k\geq 1$.

 If
all principal curvatures of $M$
with respect to $H/|H|$ are constant, then one of the following two cases occurs{\rm :}

{\rm (1)}
$M$ is locally congruent with the immersion described in Corollary {\rm \ref{m=n+2}}.

{\rm (2)} $n/k\in {\mathbb Z}-\{1\}$,
$n_1=\cdots=n_k=2n/k$, and $M$ is locally congruent with the immersion
\begin{equation}
\varpi_{\{n+2, -1\}}\biggl(\varpi_{\{n+1, \frac{2k-2n}{2n-k}\}}^{-1}(\Psi), \sqrt{\frac{k}{2n-2k}}e^{it}\biggr),\nonumber\end{equation}
where
$\Psi$ is a  K\"{a}hler hypersurface   in
${\mathbb C}H^{n+1}
(\frac{8k-8n}{2n-k})$.
\end{corollary}

It is natural to ask the following problem.
\begin{problem}
Find $\delta(n_1, \ldots, n_k)$-ideal
CR submanifolds with {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$
in $\mathbb{C}H^m(-4)$ such that the  principal curvatures
with respect to $H/|H|$ are not all constant.
\end{problem}

Generally,
$\delta(n_1, \ldots, n_k)$-ideal proper CR submanifolds
 in ${\mathbb C}H^m(-4)$ have the following properties.
\begin{theorem}[\cite{sa3}]
Let $M$ be a linearly full $(2n+q)$-dimensional $\delta(n_1, \ldots, n_k)$-ideal
CR submanifold in ${\mathbb C}H^m(-4)$ such that ${\rm rank}\hskip2pt{\mathcal H}^{\perp}=q$.
If $q>1$, then $M$ is minimal. If $q=1$ and $m>n+1$, then $M$ is non-minimal and
satisfies $DH=0$.
\end{theorem}

A differentiable  manifold $M$ is called an {\it almost contact manifold}  if it
admits a unit vector field $\xi$, a one-form $\eta$ and a $(1, 1)$-tensor field $\phi$ satisfying
\begin{equation}
\eta(\xi)=1, \quad \phi^2=-I+\eta\otimes\xi.\nonumber
\end{equation}
Every almost contact manifold admits a Riemannian metric $g$ satisfying
\begin{equation}
g(\phi X, \phi Y)=g(X, Y)-\eta(X)\eta(Y).\nonumber
\end{equation}
The quadruplet $(\phi, \xi, \eta, g)$ is called an {\it almost contact metric structure}.

An almost contact metric structure is called a {\it contact metric structure}
 if it satisfies
\begin{equation}
d\eta(X, Y)=\frac{1}{2}\Bigl(X(\eta(Y))-Y(\eta(X))-\eta([X, Y])\Bigr)=g(X, \phi Y).\nonumber
\end{equation}
A contact metric structure is said to be {\it Sasakian}
if the tensor field $S$ defined by
\begin{equation}
S(X, Y)=\phi^2[X, Y]+[\phi X, \phi Y]-\phi[\phi X, Y]-\phi[X, \phi Y]+2d\eta(X, Y)\xi\nonumber
\end{equation}
vanishes identically.
A manifold equipped with a Sasakian structure is called a {\it Sasakian manifold}.
We refer the reader to \cite{boy} for more information on Sasakian manifolds.

Let $M$ be a CR submanifold with {\rm rank}\hskip2pt$\mathcal{H}^{\perp}$
$=1$ in a complex space form. We
define a one-form $\eta$ by
$\eta(X)=g(U, X)$, where $U$ is a unit tangent vector field lying in
 $\mathcal{H}^\perp$, and $g$ is an induced metric on $M$.
We put $\bar U=(1/\sqrt{r})U$, $\bar\eta=\sqrt{r}\eta$ and $\bar g=rg$ for a
positive constant $r$. Then, the quadruplet
$(P, \bar U, \bar\eta, \bar g)$ defines an almost  contact structure on $M$ (cf. \cite[p.96]{dj}).

For the almost contact structure $(P, \bar U, \bar\eta, \bar g)$
on a
CR submanifold described in Theorem {\rm \ref{cla}},
we have the following.
\begin{proposition}[\cite{sa4}]
An almost contact structure $(P, \bar U, \bar\eta, \bar g)$ with $r=\sqrt{\frac{k}{2n-k}}$
on a CR submanifold in Theorem {\rm \ref{cla}}.
becomes a Sasakian structure. In particular, in the case of $(1)$, the structure is Sasakian with respect
to the induced metric.
\end{proposition}

\section{Ideal CR submanifolds in complex projective space}
 All $\delta_k(2)$-ideal
Hopf hypersurfaces of $\mathbb{C}P^m(4)$ have been determined as follows:
\begin{theorem}[\cite{chen4}]\label{claidCP}
Let $M$ be a $\delta_k(2)$-ideal
Hopf hypersurface of $\mathbb{C}P^m(4)$.
Then, one of the following three cases occurs{\rm :}

{\rm (1)} $k=1$ and $M$ is an open portion of a geodesic sphere with radius $\pi/4${\rm ;}

{\rm (2)} $m$ is odd, $k=m-1$, and $M$ is an open portion of a tubular hypersurface
with radius $r\in (0, \pi/2)$ over a totally geodesic $\mathbb{C}P^{(m-2)/2}(4)${\rm ;}

{\rm (3)} $m=2$, $k=1$, and $M$ is an open portion of a tubular hypersurface over the
complex quadric curve $Q_1:=\{[z_0,  z_1, z_2]\in \mathbb{C}P^2: z_0^2+z_1^2+z_2^2=0\}$,
 with radius $r=\tan^{-1}((1+\sqrt{5}-\sqrt{2+2\sqrt{5}})/2)=0.33311971\cdots$. Here,
 $[z_0,  z_1, z_2]$ is a homogeneous coordinate of  $\mathbb{C}P^2$.
\end{theorem}

A real hypersurface  of $\mathbb{C}P^m(4)$ is called a {\it ruled real hypersurface}
if $\mathcal{H}$ is integrable and each leaf of its maximal integral manifolds is
locally congruent to
$\mathbb{C}P^{m-1}(4)$.
For a unit normal vector $V$ of a ruled real
hypersurface $M$, the shape operator $A_V$ satisfies
\begin{equation}
A_VJV=\mu JV+\nu U \hskip5pt (\nu\ne 0), \quad
AU=\nu JV, \quad AX=0 \label{ruled}
\end{equation}
for all $X$ orthogonal to both $JV$ and $U$, where
$U$ is a unit vector orthogonal to $JV$, and $\mu$ and $\nu$
are smooth functions on $M$. Thus, all ruled real hypersurfaces of $\mathbb{C}P^m(4)$ are non-Hopf (see \cite{kim2}).

Using Proposition \ref{pro4} and (\ref{ruled}), we find that
every minimal ruled real hypersurface in  $\mathbb{C}P^m(4)$ is
$\delta_k(2)$-ideal for $1\leq k\leq m-1$. Such a hypersurface can be represented as follows:
\begin{theorem}[\cite{ada}]
A minimal ruled hypersurface of  $\mathbb{C}P^m(4)$ is  congruent to
 $\varpi\circ z$, where  $\varpi:S^{2m+1}(1)\rightarrow \mathbb{C}P^m(4)$
is the Hopf fibration and
\begin{equation}
z(s, t, \theta, w)=e^{\sqrt{-1}\theta}\Bigl(\cos{s}\cos{t}, \cos{s}\sin{t}, (\sin{s})w\Bigr)\nonumber\end{equation}
for $w\in\mathbb{C}^{m-1}$,  $|w|^2=1$,
$-\pi/2<s<\pi/2$,  $0\leq t,  \theta<2\pi$.
\end{theorem}

It seems interesting to consider the following problem.
\begin{problem}
Classify $\delta(n_1, \ldots, n_k)$-ideal non-Hopf
real hypersurfaces in $\mathbb{C}P^m(4)$.
\end{problem}

Let $M$ be an $n$-dimensional $\delta(n_1, \ldots, n_k)$-ideal CR submanifold
in ${\mathbb C}P^m(4)$. Let $L_j$ be subspaces of $T_pM$ defined in (a) of Proposition \ref{pro4}.
Define the subspace $L_{k+1}$ by
$L_{k+1}={\rm Span}\{e_{n_1+\cdots+n_k+1}, \ldots, e_{n}\}$.
It is clear that  $T_pM=L_1\oplus\cdots\oplus L_{k+1}$. We denote by $\mathcal{L}_i$ the
distribution which is   generated by $L_i$.
Then, we have the following codimension reduction theorem.
\begin{theorem}[\cite{sa4}]\label{core}
Let $M$ be an $n$-dimensional $\delta(n_1, \ldots, n_k)$-ideal CR submanifold with ${\rm rank}\hskip2pt\mathcal{H}^{\perp}=1$
in ${\mathbb C}P^m(4)$.
If $\mathcal{H}^{\perp}\subset \mathcal{L}_i$ for some $i\in\{1, \ldots, k+1\}$, then
$M$ is contained in a totally geodesic K\"{a}hler submanifold ${\mathbb C}P^{\frac{n+1}{2}}(4)$ in  ${\mathbb C}P^m(4)$.
\end{theorem}

It was proved in \cite{sa4} that
 if $\dim M=3$, then
the assumption on $\mathcal{H}^{\perp}$  in Theorem \ref{core}
holds. That is  to say, we have the following.
\begin{corollary}[\cite{sa4}]
Let $M$ be a $3$-dimensional $\delta(2)$-ideal proper CR submanifold
in ${\mathbb C}P^m(4)$.
Then,
$M$ is contained in  ${\mathbb C}P^{2}(4)$.
\end{corollary}

The following problem arises naturally.

\begin{problem}
Find  $\delta(n_1, \ldots, n_k)$-ideal
CR submanifolds with
 {\rm rank}\hskip2pt${\mathcal H}^{\perp}=1$
 in  $\mathbb{C}P^m(4)$ such that the codimensions are greater than one.
\end{problem}

\section{Ideal CR submanifolds in the nearly K\"{a}hler $6$-sphere}
Let $\mathcal O$ be  the Cayley algebra, and denote  by ${\rm Im}\hskip 2pt\mathcal O$
the purely imaginary part of
 $\mathcal O$.
 We identify ${\rm Im}\hskip 2pt\mathcal O$
  with $\mathbb R^7$ and define the exterior product $u\times v$ on it by
 \begin{equation}
 u\times v=\frac{1}{2}(uv-vu).\nonumber
 \end{equation}
 The canonical inner product on $\mathbb R^7$
 is given by $\<u ,v\>=-(uv+vu)/2$.

We define the tensor field $J$ of type $(1, 1)$ on $S^6(1)=
\{p\in{\rm Im}\hskip 2pt\mathcal O|
\<, \>=1\}$ by
\begin{equation}
JX=p\times X \nonumber
\end{equation}
for any $p\in S^6(1)$, $X\in T_pS^6(1)$.  Let $g$ be the standard metric on $S^6(1)$.
Then $(S^6(1), J, g)$ is a
nearly K\"{a}hler manifold, i.e., an almost Hermitian manifold satisfying
$(\nabla_XJ)X=0$ for any $X\in TS^6(1)$, where
$\nabla$ is the Levi-Civita connection with respect to $g$ (cf. \cite[pp.139-140]{kn}).

For  $3$-dimensional $\delta(2)$-ideal proper
CR submanifolds  in the nearly K\"{a}hler $S^6(1)$, we have the following result.
\begin{theorem}[\cite{dv}, \cite{dv2}]
Let $M$ be a $3$-dimensional $\delta(2)$-ideal proper
 CR submanifold in the nearly K\"{a}hler $S^6(1)$.
Then, $M$ is minimal and locally congruent with  the following immersion{\rm :}
\begin{equation}
\begin{split}
f(t, u, v)=(\cos t\cos u\cos v, \sin t, \cos t\sin u\cos v, \\
\cos t\cos u\sin v, 0, -\cos t\sin u\sin v, 0).\label{3cr}
\end{split}
\end{equation}
\end{theorem}

\begin{remark}{\rm A CR submanifold (\ref{3cr}) can be rewritten as
\begin{equation}
x_1^2+x_2^2+x_3^2+x_4^2+x_6^2=1, \quad x_5=x_7=0, \quad
x_3x_4+x_1x_6=0,\nonumber
\end{equation}
which implies that  it lies in $S^4(1)$.}
\end{remark}

The following theorem determines
$4$-dimensional  $\delta(2)$-ideal
proper CR submanifolds in the nearly K\"{a}hler $S^6(1)$.
\begin{theorem}[\cite{ant}, \cite{adv}]
Let $M$ be a $4$-dimensional $\delta(2)$-ideal
proper CR submanifold in the nearly K\"{a}hler $S^6(1)$.
Then, $M$ is minimal and locally congruent with  the following immersion{\rm :}
\begin{equation}
\begin{split}
f(t, u, v, w)=(\cos w\cos t\cos u\cos v, \sin w\sin t\cos u\cos v,\\
\sin 2w\sin v\cos u+\cos 2w\sin u, 0, \sin w\cos t\cos u\cos v,\\
\cos w\sin t\cos u\cos v, \cos 2w\sin v\cos u-\sin 2w\sin u).\label{4cr}
\end{split}
\end{equation}
\end{theorem}

\begin{remark}
{\rm A CR submanifold given by (\ref{4cr}) lies in $S^5(1)$.}
\end{remark}

\begin{definition}{\rm
A $2$-dimensional submanifold $N$
 of the nearly K\"{a}hler $S^6(1)$ is called an {\it almost complex curve}
if $J(T_pN)=T_pN$ for any $p\in N$.}
\end{definition}

Chen has classified $\delta_k(n_1, \ldots, n_k)$-ideal
Hopf hypersurfaces of  the nearly K\"{a}hler $S^6(1)$ as follows:
\begin{theorem}[{\cite[p.415]{chen}}]\label{hopfs6}
A Hopf hypersurface of  the nearly K\"{a}hler $S^6(1)$ is  $\delta(n_1, \ldots, n_k)$-ideal
if and only if  it is either

{\rm (1)} a totally geodesic hypersurface, or

{\rm (2)} an open part of a tubular hypersurface with radius $\pi/2$ over a non-totally geodesic
almost complex curve of $S^6(1)$.
\end{theorem}

\begin{remark}
{\rm A tubular hypersurface described in (2) of  Theorem \ref{hopfs6} is a minimal $\delta(\lambda)$-ideal hypersurface for $\lambda\in\{2, 3, 4\}$.}
\end{remark}

It is natural to consider the following problem.
\begin{problem}
Classify $4$-dimensional $\delta(2, 2)$-ideal and  $\delta(3)$-ideal
CR submanifolds    in the nearly K\"{a}hler $S^6(1)$.
\end{problem}

\section{Ideal contact CR submanifolds in  odd dimensional unit spheres}

For any point $x\in S^{2n+1}(1)\subset \mathbb{C}^{n+1}$,
we set $\xi=Jx$, where $J$  denotes the canonical
complex structure of  $\mathbb{C}^{n+1}$.
Let $g$ be the standard metric on $S^{2n+1}(1)$ and
$\eta$ be the one-form given by $\eta(X)=g(X, \xi)$.
We consider the orthogonal projection $P:T_x\mathbb{C}^{n+1}\rightarrow
T_xS^{2n+1}(1)$.
We define
a $(1, 1)$-tensor field $\phi$  on $S^{2n+1}(1)$ by
$\phi=P\circ J$.
Then, the quadruplet $(\phi, \xi, \eta, g)$ is a Sasakian structure (see, for example, \cite{blair}).

\begin{definition}[\cite{yano}]
{\rm   Let $M$ be a Riemannian submanifold  tangent to $\xi$
of a Sasakian manifold. A submanifold $M$
 is called a {\it contact CR submanifold} if  there exist
 differentiable distributions $\mathcal{H}$ and  ${\mathcal H}^{\perp}$ such that
 \begin{equation}
 TM=\mathbb{R}\xi\oplus\mathcal{H}\oplus {\mathcal H}^{\perp}, \quad \nonumber
 \phi\mathcal{H}=\mathcal{H}, \quad
 \phi{\mathcal H}^{\perp}\subset T^{\perp}M.
 \end{equation}

A contact CR submanifold is said to be ${\it proper}$ if {\rm rank}\hskip2pt${\mathcal H}\ne 0$
and {\rm rank}\hskip2pt${\mathcal H}^{\perp}\ne 0$.
}
\end{definition}

Non-minimal $\delta(2)$-ideal submanifolds in a sphere have been completely described in \cite{da}.
For minimal $\delta(2)$-ideal proper contact CR submanifolds
in $S^{2m+1}(1)$, we have the following codimension reduction theorem.
\begin{theorem}[\cite{mun}]
Let $M^n$ be a minimal $\delta(2)$-ideal proper contact CR submanifold
in $S^{2m+1}(1)$. Then $n$ is even and there exits a totally geodesic Sasakian
$S^{2n+1}(1)$ in  $S^{2m+1}(1)$ containing $M^n$ as a hypersurface.
\end{theorem}

Therefore, it is sufficient to investigate the case of hypersurfaces.
Let $N$ be a minimal surface in $S^{n}(1)$ and let $UN^{\perp}$ be its unit normal bundle.
Then, a map
\begin{equation}  F: UN^{\perp}\rightarrow S^n(1): V_p\mapsto V_p \nonumber
\end{equation}
is a minimal $\delta(2)$-ideal codimension one immersion (see \cite[Example 9.8]{chen3}).

Munteanu and Vrancken proved the following.

\begin{theorem}[\cite{mun}]
Let $M^{2n}$ be a  minimal $\delta(2)$-ideal proper contact CR hypersurface
in $S^{2n+1}(1)$. Then $M^{2n}$ can be locally considered as
the unit normal bundle of the Clifford torus
$S^{1}(1/\sqrt{2})\times S^{1}(1/\sqrt{2})\subset S^3(1)\subset S^{2n+1}(1)$.
\end{theorem}

\section{Related topics}
This section gives an account of the relationship between $\delta(n_1, \ldots, n_k)$-ideal immersions and
critical points  of  the $\lambda$-bienergy functional $E_{2,\lambda}$. Some topics
about
 variational problems for $E_{2, \lambda}$  are also presented.
\subsection{$\lambda$-bienergy functional}
Let $f:M \rightarrow N$ be a smooth map of an
 $n$-dimensional Riemannian manifold
into another Riemannian manifold.
The {\it tension field}
$\tau(f)$ of $f$ is a section
of the induced vector bundle $f^{*}TN$
defined by
\begin{equation} \tau(f)=
\sum_{i=1}^{n}\{\nabla^{f}_{e_i}df(e_i)
-df(\nabla_{e_i}e_i)\}\nonumber
\end{equation}
for a local orthonormal frame $\{e_i\}$ on $M$, where  $\nabla^f$ and $\nabla$ denote
 the induced connection and the Levi-Civita connection of $M$, respectively.
If  $f$ is an isometric immersion,  then we have
\begin{equation} \tau(f)=nH.\label{tau}
\end{equation}

A smooth map $f$ is called
a {\it harmonic map} if
it is a critical
point of the energy functional
$$
E(f)=\int_{\Omega}|df|^2dv_g
$$
over every compact domain $\Omega$ of $M$, where $dv_g$ is the volume form of $M$.
A smooth map $f$ is harmonic if and only if $\tau(f)$ vanishes identically on $M$.

\begin{definition}
{\rm For each smooth map $f$ of a compact domain $\Omega$ of $M$ into $N$,
the $\lambda$-bienergy functional  is  defined by
\begin{equation}
E_{2, \lambda}(f)=\int_{\Omega}|\tau(f)|^2dv_g+\lambda E(f).\nonumber
\end{equation}
For simplicity, we denote $E_{2, 0}(f)$ by $E_2(f)$, which is
called the bienergy functional.}
\end{definition}

Eliasson  \cite{eli}
proved that $E_{2, \lambda}$ satisfies
Condition (C) of Palais-Smale if the dimension of the domain
is 2 or 3 and the target is non-positively curved. In general,  $E_{2, \lambda}$
does not satisfy Condition (C) (see \cite{le}).

\subsection{Ideal CR immersions as
critical points of $\lambda$-bienergy functional}

Let $(M, HM, J_H, g)$ be a compact Riemannian almost CR manifold (with or without boundary) whose
CR dimension is $h$, i.e.,
a compact smooth manifold equipped with a subbundle $HM$ of $TM$ of rank $2h$
together with a bundle isomorphism
$J_H:HM\rightarrow HM$ such that $(J_H)^2=-I$, and a Riemannian metric $g$ such that
$g(X, Y)=g(J_HX, J_HY)$ for all $X$,$Y\in HM$.

An
 immersion $f$ of   $(M, HM, J_H, g)$ into $\tilde M^m(4\epsilon)$
 is called a {\it CR immersion} if $J(df(X))=df(J_H(X))$ for any $X\in HM$.

If $f$ is an isometric immersion, then $f(M)$ is a CR submanifold of $\tilde M^m(4\epsilon)$.
We denote  by
$\mathcal{ICR}(M, \tilde M^m(4\epsilon))$ the family of isometric  CR
immersions of $M$ into $\tilde M^m(4\epsilon)$.
By Proposition \ref{pro3} and \ref{pro4}, we see that a
$\delta(n_1, \ldots, n_k)$-ideal  CR immersion of $M$ into $\tilde M^m(4\epsilon)$ is a
stable critical point of $E_{2, \lambda}$
within  the class of $\mathcal{ICR}(M, \tilde M^m(4\epsilon))$.

\subsection{$\lambda$-biharmonic submanifolds and their extensions}
\begin{definition}[\cite{fel}]\label{e2}
{\rm A smooth map $f: M\rightarrow N$ is  called a
{\it $\lambda$-biharmonic map} if it is a critical point of the $\lambda$-bienergy
functional
with respect to all variations with compact support.
If $f$ is a $\lambda$-biharmonic isometric immersion, then $M$  is called a {\it $\lambda$-biharmonic submanifold} in $N$. In the case of $\lambda=0$, we simply call it a
{\it biharmonic submanifold}.}
\end{definition}

The Euler-Lagrange equation for $E_{2, \lambda}$ is given by (see \cite{ji2} and
\cite[p.515]{fel})
\begin{equation}
\tau_{2, \lambda}:=-\Delta_f(\tau(f))+{\rm trace}R^{N}(\tau(f),df)df-\lambda\tau(f)=0,\label{EL}\end{equation}
where
$\Delta_f=-\sum_{i=1}^{n}(\nabla^{f}_{e_i}\nabla^{f}_{e_i}-\nabla^{f}_{\nabla_{e_i}e_i})$ and
$R^{N}$ is
the curvature tensor  of $N$, which is defined by $$R^{N}(X, Y)Z=[\nabla^N_X, \nabla^N_Y]Z-\nabla^N_{[X, Y]}Z$$ for the Levi-Civita connection $\nabla^N$ of $N$.
For simplicity, we denote  $\tau_{2, 0}(f)$ by $\tau_2(f)$.

By decomposing the left-hand side of
(\ref{EL}) into  its tangential and normal components,  we have

\begin{proposition}[\cite{bal2}]\label{EL3}
Let $M$ be an $n$-dimensional
submanifold of $\tilde R^m(\epsilon)$. Then $M$ is $\lambda$-biharmonic
if and only if
\begin{equation}
\begin{cases}
\Delta^{D}H+{\rm trace}\hskip2pt B(\cdot, A_H(\cdot))+(\lambda-\epsilon n)H=0,\\
4{\rm trace} A_{D_{(\cdot)}H}(\cdot)+n{\rm grad}(|H|^2)=0,\nonumber
\end{cases}
\end{equation}
where $\Delta^D=-\sum_{i=1}^n\{D_{e_i}D_{e_i}-D_{\nabla_{e_i}e_i}\}$.
\end{proposition}

\begin{proposition}[\cite{fel}]\label{EL2}
Let $M$ be an $n$-dimensional
submanifold of $\tilde M^m(4\epsilon)$ such that
$JH$ is tangent to $M$. Then $M$ is $\lambda$-biharmonic
if and only if
\begin{equation}
\begin{cases}
\Delta^{D}H+{\rm trace}\hskip2pt B(\cdot, A_H(\cdot))+\{\lambda-\epsilon(n+3)\}H=0,\\
4{\rm trace} A_{D_{(\cdot)}H}(\cdot)+n{\rm grad}(|H|^2)=0.\nonumber
\end{cases}
\end{equation}
\end{proposition}

\begin{remark}
{\rm By Proposition \ref{EL2}, we see that
all  hypersurfaces with constant principal curvatures in $R^m(\epsilon)$ and  $\tilde M^m(4\epsilon)$
are $\{-|B|^2+\epsilon(m-1)\}$-biharmonic and $\{-|B|^2+2\epsilon(m+1)\}$-biharmonic, respectively.}
\end{remark}

It follows from (\ref{tau}) and (\ref{EL})  that any
minimal submanifold is $\lambda$-biharmonic.
Thus, it is interesting to investigate non-minimal
$\lambda$-biharmonic submanifolds.

\begin{remark}{\rm Let $f: M\rightarrow \mathbb{R}^n$ be an isometric immersion.
We denote  the mean curvature vector field of $M$ by $H=(H_1, \ldots, H_n)$.
 Then, it follows from  (\ref{tau}) and (\ref{EL}) that $M$ is $\lambda$-biharmonic if and only if it satisfies
 \begin{equation}  \Delta_{M}H_i=-\lambda H_i, \quad 1\leq i\leq n,\label{null}
 \end{equation}
 where $\Delta_M$ is the Laplace operator acting on $C^{\infty}(M)$.
Hence, the notion of biharmonic submanifolds in Definition \ref{e2}.
  is same as  one defined by B. Y. Chen (cf. \cite{chen9}).

 It was proved in \cite{chen8}
   that
   a submanifold $M$ satisfies (\ref{null})
 if and only if one of the following three cases occurs:

(1) $f$ satisfies $\Delta_Mf=-\lambda f$;

(2) $f$ can be written as
$f=f_0+f_1$, \quad $\Delta_Mf_0=0$, \quad$\Delta_Mf_1=-\lambda f_1$;

(3) $M$ is a biharmonic submanifold.

 An immersion described in (1) (resp. (2)) is said to be of {\it $1$-type}
 (resp. {\it null $2$-type}).
 An immersion $f: M\rightarrow \mathbb{R}^n$ is of $1$-type if and only if either $M$ is a minimal submanifold
 of  $\mathbb{R}^n$ or $M$ is a minimal submanifold of a hypersphere in $\mathbb{R}^n$
 (cf. \cite[Theorem 3.2]{chen7}). The classification of null $2$-type immersions
 is not yet complete.

 }
\end{remark}

 There exist many non-minimal biharmonic submanifolds
in a sphere or a complex projective space (see, for example, \cite{bal} and \cite{fel}).
On the other hand, the following conjecture proposed by Chen \cite{chen7} is still open.
\begin{conjecture}\label{conj}
{ Any biharmonic submanifold in  Euclidean space is minimal.}
\end{conjecture}
Several
  partial positive answers to this conjecture have been obtained (see \cite{chen9}).
  For example, Chen and Munteanu \cite{cm} proved that Conjecture \ref{conj}
  is true for  hypersurfaces which are $\delta(2)$-ideal or $\delta(3)$-ideal
  in Euclidean space of
  arbitrary dimension.

As an extension of the notion of biharmonic submanifolds,
the following notion was introduced by Loubeau and Montaldo in \cite{lm}.
\begin{definition}\label{e3}
{\rm An isometric immersion $f: M\to N$
is called a   {\it $\lambda$-biminimal\/}
if it is a critical point of
the $\lambda$-bienergy functional
with respect to all
{\it normal variations} with compact support.
Here, a normal variation means a
variation $f_t$ through $f=f_0$
such that the variational vector field
$V=df_{t}/dt|_{t=0}$ is normal to $f(M)$.
In this case, $M$ is called a {\it $\lambda$-biminimal submanifold} in $N$.
In the case of  $\lambda=0$, we simply call it {\it biminimal} submanifold.}
\end{definition}
An isometric immersion $f$ is $\lambda$-biminimal if and only if
\begin{equation}
[\tau_{2, \lambda}(f)]^{\perp}=0,\nonumber
\end{equation}
where $[\cdot]^{\perp}$ denotes the normal component of $[\cdot]$ (see \cite{lm}).
It is known that there exist ample examples of
$\lambda$-biminimal submanifolds in real and complex space forms,
which are not $\lambda$-biharmonic
(see, for example, \cite{lm}, \cite{sasa7}, \cite{sa5} and \cite{sa6}).

 In \cite{sa}, the notion of  tangentially biharmonicity for submanifolds
   was introduced as follows:
  \begin{definition}\label{e4}
{\rm  Let $f:M\rightarrow N$ be an isometric immersion. Then $M$
 is called a {\it tangentially biharmonic submanifold} in $N$
 if it satisfies
 \begin{equation}
 [\tau_{2}(f)]^{\top}=0,\label{TBE}
 \end{equation}  where $[\cdot]^\top$ denotes the tangential part of $[\cdot]$. }
 \end{definition}

\begin{example}
{\rm Let $x:M^{n-1}\rightarrow \mathbb{R}^n$ be an isometric immersion.
The normal bundle $T^{\perp}M^{n-1}$ of $M^{n-1}$ is naturally immersed in $\mathbb{R}^n
\times\mathbb{R}^n=\mathbb{R}^{2n}$ by the immersion
$f(\xi_x):=(x, \xi_x)$, which is  expressed as
\begin{equation}
f(x, s)=(x, sV)\label{nbundle}
\end{equation}
for the unit normal vector field $V$ along $x$.
We equip $T^{\perp}M^{n-1}$ with the metric induced by $f$.
If we define the complex structure $J$ on $\mathbb{C}^n=\mathbb{R}^n\times\mathbb{R}^n$
by $J(X, Y):=(-Y, X)$, then
$T^{\perp}M^{n-1}$ is a Lagrangian submanifold in $\mathbb{C}^n$
(see \cite[III.3.C]{hl}).
It was proved  in \cite{sa} that  $T^{\perp}M^2$ is a tangentially biharmonic Lagrangian submanifold in
 $\mathbb{C}^3$
 if and only if
$M^2$ is either  minimal, a part of a round sphere or a part of a circular cylinder in $\mathbb{R}^3$.}
\end{example}

\begin{remark}
{\rm For any $\lambda\in\mathbb{R}$, we have $[\tau_{2, \lambda}(f)]^{\top}
=[\tau_{2}(f)]^{\top}$.}
\end{remark}

 \begin{remark}
 {\rm  By the first variation formula for $E_2$ obtained in \cite{ji2}, we see that
 an isometric immersion $f:M\rightarrow N$ is tangentially biharmonic if and only if it
 is a critical point of $E_{2}$
with respect to all
{\it tangential variations} with compact support.
Here, a tangential variation means a
variation $f_t$ through $f=f_0$
such that the variational vector field
$V=df_{t}/dt|_{t=0}$ is tangent to $f(M)$.}
 \end{remark}

 \begin{remark}
 {\rm As described by Hilbert \cite{hil},
 the stress-energy tensor associated to a variational problem is a symmetric $2$-covariant tensor which is conservative, namely,
 divergence-free at critical points.
 The  stress-energy tensor $S_2$
for $E_{2}(f)$   was introduced by Jiang \cite{ji3} as follows:
\begin{align}
S_2(X, Y)=&\frac{1}{2}|\tau(f)|^2\<X, Y\>+\bigl<df, \nabla^{f}\tau(f)\bigr>\nonumber\\
&-\bigl<df(X), \nabla^{f}_Y\tau(f)\bigr>-
\bigl<df(Y), \nabla^{f}_X\tau(f)\bigr>.\nonumber
\end{align}
It satisfies ${\rm div}\hskip2pt S_2=-\<\tau_2(f), df\>$. Hence,
an isometric
immersion $f$ is tangentially biharmonic if and only if ${\rm div}\hskip2pt S_2=0$.
Caddeo et al. \cite{cad} called these submanifolds satisfying such a condition as
{\it biconservative submanifolds}, and moreover,
 classified biconservative surfaces in 3-dimensional real space
forms.}
\end{remark}
\begin{remark}{\rm

  Hasanis and Vlachos \cite{hav} classified
  hypersurfaces in $\mathbb{R}^4$
  satisfying (\ref{TBE}). They called  such hypersurfaces as {\it $H$-hypersurfaces}.
  Afterwards, the biharmonic ones are picked out in the class.
  As a result, the non-existence of non-minimal
  biharmonic hypersurfaces in $\mathbb{R}^4$ was proved.}
  \end{remark}

\begin{remark}
{\rm It follows from  Definition \ref{e3} and \ref{e4}
 that a map $f$ is $\lambda$-biharmonic (resp. $\lambda$-biminimal) if and only if it
is a critical point  of $E_{2}(f)$ for all
variations (resp. normal variations) with compact support and {\it fixed energy}.  Here,
$\lambda$ is the Lagrange multiplier.
}
\end{remark}

\subsection{Biharmonic ideal CR submanifolds}

For homogeneous real hypersurfaces in $\mathbb{C}P^m(4)$, namely, orbits
under some subgroups of the projective
unitary group $PU(m+1)$, we have

\begin{theorem}[\cite{iiu}]\label{clabiCP}
Let $M$ be a homogeneous hypersurface in $\mathbb{C}P^m(4)$.
Then, $M$ is non-minimal biharmonic if and only if it
is congruent to an open portion of one of the following real hypersurfaces{\rm :}

{\rm (1)} a tubular hypersurface  over $\mathbb{C}P^q(4)$ with radius
\begin{equation}
r=\cot^{-1}\biggl(\sqrt{\frac{m+2\pm\sqrt{(2q-m+1)^2+4(m+1)}}{2m-2q-1}}\biggr).\nonumber
\end{equation}

{\rm (2)} a tubular hypersurface over the Pl\"{u}cker imbedding of the complex Grassmann manifold
$Gr_2(\mathbb{C}^5)\subset \mathbb{C}P^9(4)$ with  radius $r$, where $0<r<\pi/4$ and
$t=\cot{r}$ is a unique solution of the equation
\begin{equation}  41t^6+43t^4+41t^2-15=0.\nonumber\end{equation}
In this case, $r=1.0917\cdots$.

{\rm (3)} a tubular hypersurface over the canonical imbedding of the
Hermitian symmetric space $SO(10)/U(5)\subset \mathbb{C}P^{15}(4)$
with radius $r$, where $0<r<\pi/4$ and $t=\cot{r}$
is a unique solution of the equation
\begin{equation}
13t^6-107t^4+43t^2-9=0.\nonumber\end{equation}
In this case, $r=0.343448\cdots$.
\end{theorem}

For details on the canonical imbedding of a compact Hermitian symmetric space into
$\mathbb{C}P^m(4)$, we refer the reader to  Section 4 of \cite{nt}.
\begin{remark}
{\rm Let $M$ be a real hypersurface in $\mathbb{C}P^m(4)$. Kimura \cite{kim} proved that
 $M$ is a Hopf
hypersurface with constant principal curvatures
if and only if it is homogeneous.}
\end{remark}

Combining Theorem \ref{claidCP}, Proposition \ref{EL2} and Theorem \ref{clabiCP}, we obtain
\begin{corollary}
Let $M$ be a $\delta_k(2)$-ideal non-minimal biharmonic Hopf hypersurface in $\mathbb{C}P^m(4)$.
Then, $m$ is odd and
$M$ is an open portion of  a tubular hypersurface  over $\mathbb{C}P^{(m-1)/2}(4)$ with radius
\begin{equation}
r=\cot^{-1}\biggl(\sqrt{\frac{m+2\pm 2\sqrt{m+1}}{m}}\biggr).\nonumber
\end{equation}
\end{corollary}

\begin{example}{\rm
On each CR submanifold described in Theorem \ref{cla},
there exists an orthonormal frame $\{e_1, \ldots, e_{2m}\}$ such that $e_{2r}=Je_{2r-1}$
for $r\in\{1, \ldots, n\}$, $JH\parallel e_{2n+1}\in
\mathcal{H}^{\perp}$ and the second fundamental form $B$ takes the following form:
\begin{align}
&B(e_{2r-1}, e_{2r-1})=\sqrt{\frac{k}{2n-k}}Je_{2n+1}+\phi_r\xi_r,\nonumber\\
&B(e_{2r}, e_{2r})=\sqrt{\frac{k}{2n-k}}Je_{2n+1}-\phi_r\xi_r,\nonumber\\
&B(e_{2r-1}, e_{2r})=\phi_rJ\xi_r,\nonumber\\
&B(e_{2n+1},e_{2n+1})=\frac{2n}{\sqrt{k(2n-k)}}Je_{2n+1},\nonumber\\
&B(u_i, u_j)=h(u_i, e_{2n+1})=0\quad (i\ne j),\nonumber
\end{align}
where $\phi_r$ are functions,
$\xi_r\in\nu$ and  $u_j\in L_j$ (see Lemma 7 of \cite{sa3}).
Here,  $\nu$ denotes an orthogonal complement
 of $J\mathcal{H}^{\perp}$ in $T^{\perp}M$.
Therefore, by using Proposition \ref{EL2}, we find that
all ideal CR submanifolds given in Theorem \ref{cla}
are  non-minimal $\lambda$-biharmonic submanifolds with
\begin{equation}
\lambda=-\frac{2n(2n+k^2)}{k(2n-k)}-2n-4\hskip5pt (\ne 0).\nonumber
\end{equation} }
\end{example}

The following problem seems interesting.
\begin{problem}
Classify $\delta(n_1, \ldots, n_k)$-ideal proper
CR submanifolds   in $\mathbb{C}P^m(4)$  which are non-minimal biharmonic.
\end{problem}

\begin{example}
{\rm The standard product $S^{2r+1}(1/\sqrt{2})\times S^{2s+1}(1/\sqrt{2})$  in $S^{2(r+s)+3}(1)$
 is a   biharmonic contact CR hypersurface (see \cite[Example 5.1]{yano}
 and \cite[p.92]{bal}).
 Its
 principal curvatures  are  $\{1, -1\}$ with multiplicities $\{2r+1, 2s+1\}$. We may assume that $r\geq s$.
 By Theorem \ref{ingere}, we see that
 the biharmonic hypersurface $S^{2r+1}(1/\sqrt{2})\times S^{2s+1}(1/\sqrt{2})$
 is minimal and $\delta_{2s+1}(2)$-ideal if $r=s$; otherwise it is
  non-minimal and $\delta(4s+3)$-ideal. }
  \end{example}

 Incidentally, the following conjectures proposed in \cite{bal2} remains open.
 \begin{conjecture}
 The only non-minimal biharmonic hypersurfaces in $S^{m+1}$ are the open
 parts of hyperspheres $S^{m}(1/\sqrt{2})$ or of the standard products $S^{m_1}(1/\sqrt{2})\times
 S^{m_2}(1/\sqrt{2})$, where $m_1+m_2=m$ and $m_1\ne m_2$.
 \end{conjecture}

\begin{conjecture}
Any non-minimal biharmonic submanifold in $S^n(1)$ has constant mean curvature.
\end{conjecture}

\small

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Dubai is a glamourous metropolis and host of the star-studded IRONMAN 70.3 Dubai. But this city has more to offer than superlatives.
If you're staying stateside next season, these under the radar races are still open for registration.
The capital of Catalonia boasts a unique mix of ancient and modern architecture, small cafés, tempting markets, and sun-drenched beaches.
Liz Blatchford experiences the reef in stunning Tropical North Queensland post IRONMAN Cairns.
Follow Liz Blatchford as she explores Kuranda, the Daintree and Port Douglas post IRONMAN Cairns.
In its short history Cairns Airport IRONMAN Cairns has become a must do event. Age groupers and pro's alike flock to this tropical paradise to test the limits of physical endurance.
For triathlon's number one photographer Delly Carr, it's the sunset at Busso that he looks forward to, not the sunrise.


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text:
Construction Today - March 2011 - (Page 42)
Commercial | HAROLD O'SHEA BUILDERS O'Shea is Built to Survive O'SHEA BUILDERS' RAPPORT WITH CLIENTS AND EMPLOYEES KEEP IT STRONG. BY ALAN DORICH fter 109 years, Harold O'Shea Builders has not only built construction projects, but also strong relationships with its clients, CEO Mike O'Shea says. "It is the center of our strategic plan," he says. "O'Shea is a differentiator when it comes to building lasting relationships." Based in Springfield, Ill., O'Shea Builders specializes in general contracting, remodeling and renovations, and civil services. O'Shea's great-grandfather, John O'Shea, founded the company in 1901 as a homebuilder. "He built one or two homes at a time," Mike O'Shea says. O'Shea Builders slowly evolved from its role as a homebuilder into a major commercial builder, the younger O'Shea says. Last year, the company enjoyed sales of approximately $90 million and now employs a staff of 87 workers. "We try to attract the best and brightest people for every role at O'Shea," he says. "And to keep the best and brightest, we have to work hard from a leadership standpoint to make sure everyone stays challenged and feels rewarded for their hard work." In addition, O'Shea Builders' emphasis on client relationships has allowed it to grow a strong base of loyal customers. Currently, nearly 90 percent of the company's work is for repeat clients. "Over the years, O'Shea has worked for just about every major corporation in Springfield," O'Shea says. "We've worked for them time and time again, [but] we're always trying to build new relationships." A Keeping O'Shea Focused O'Shea has been with his family's company full-time since 1986. However, "I have been coming to work [here] with my dad since I was eight years old," he recalls, noting that his earliest jobs included sweeping warehouse floors and driving trucks. "I was raised in it," he continues, adding that PHOTO: JOHN MUCHOW, MUCHOW PHOTOGRAPHY, SPRINGFIELD O'Shea Builders performed preservation and restoration work at the Cathedral of the Immaculate Conception in Springfield, Ill. « 42 CONSTRUCTION-TODAY.COM MARCH 2011 http://www.CONSTRUCTION-TODAY.COM

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The ATO forces took control over the town of Yasynuvata - Life in Ukraine. Live, @ first hand.
This and the following was reported by NSDC Information Centre spokesman Andriy Lysenko during press briefing in Kyiv.
The maps of the situation in the ATO zone are presented below.
Yesterday the ATO forces took control over the town of Yasynuvata in Donetsk oblast that is an important point of rail traffic of the region.
The liberation of the town gave possibility to to encircle the city of Dontersk from north and cut off one of important channels of supply of weapon and equipment for terrorists.
Currently the operation for detection and neutralization of remnants of terrorist groups is being carried out.
Besides, Ukrainian militants have unblocked roads that connect the settlements of Dmytrivka and Diakove.
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text:
– japoński film anime wyprodukowany w 1997 roku, oparty na serii Detektyw Conan. Polska premiera telewizyjna odbyła się 26 lutego 2012 roku. Piosenką przewodnią filmu była "Happy Birthday" śpiewana przez Kyoko.

Film miał swoją premierę 19 kwietnia 1997 roku w 13 kinach w Japonii przynosząc łączny dochód ¥174.083.700. W pierwszym tygodniu zarobił ¥8.554.500.

Fabuła
Podczas sortowania poczty w domu doktora Agasy Conan znajduje zaproszenie zaadresowane do Shinichiego od Teijiego Moriya, znanego architekta. Shinichi dzwoni do Ran używając zmieniającej głos muszki i prosi ją, aby przyszła w jego miejsce. Ran zgadza się, pod warunkiem, że Shinichi pójdzie z nią w sobotę na film.

W sobotę rano Conan odbiera telefon od dziwnego człowieka, który rzuca Shinichi'emu wyzwanie do gry. Conan przyjmuje wyzwanie, a tajemniczy dzwoniący daje mu wskazówki prowadzące do bomb ukrytych w całym Tokio. Conan odszukuje i niszczy każdą z nich. Ponieważ bomby zostają podłożone niedaleko budowli zaprojektowanych przez Teijiego, Conan dedukuje, że zamachowcem jest Teiji, który planuje zniszczyć swoje "gorsze" dzieła i stworzyć doskonały nowy budynek. Po swoim aresztowaniu Teiji ujawnia lokalizację swojej ostatniej bomby: Beika City Building – miejsce randki Shinichiego i Ran.

Bomba wybucha odcinając drogę ucieczki innym ludziom, w tym także Ran. Conan przedziera się przez walący się budynek, ale jest odcięty od Ran przez wgniecione drzwi. Używając swojej muszki i telefonu komórkowego dzwoni do Ran i prosi ją, aby poszukała bomby. Ran odnajduje ją w dużej torbie na zakupy. Aby rozbroić bombę Shinichi mówi Ran, które przewody musi przeciąć. Jednak Teiji umieścił w bombie dwa dodatkowe przewody, jeden czerwony i jeden niebieski. Jeden z nich jest pułapką, ale Shinichi nie ma pojęcia który. Mówi Ran, żeby przecięła którykolwiek z nich. Gdy przybywa ekipa ratownicza i zabiera go, Shinichi uświadamia sobie, że Teiji wiedział, że ulubionym kolorem Ran jest czerwony i tym samym kablem zapalającym jest również czerwony. W ciągu ostatnich kilku minut Ran podejmuje desperacką decyzję i przecina niebieski przewód, bo przewód czerwony reprezentuje więź łącząca ją i Shinichiego.

Obsada
 Minami Takayama – Conan Edogawa
 Kappei Yamaguchi – Shinichi Kudō
 Wakana Yamazaki – Ran Mōri
 Akira Kamiya – Kogorō Mori
 Ken'ichi Ogata – dr Hiroshi Agasa
 Chafūrin – inspektor Jūzō Megure
 Yukiko Iwai – Ayumi Yoshida
 Wataru Takagi – Genta Kojima
 Ikue Ōtani – Mitsuhiko Tsuburaya
 Naoko Matsui – Sonoko Suzuki
 Kaneto Shiozawa – inspektor Ninzaburō Shiratori
 Tarō Ishida – Teiji Moriya

Wersja polska
Wersja polska: na zlecenie telewizji AXN – SDI Media Polska
Tekst: Tomasz Potocki
Lektor: Jan Wilkans

Przypisy

Bibliografia

Linki zewnętrzne

Filmy serii Detektyw Conan
Japońskie filmy animowane
Japońskie filmy z 1997 roku
Filmy, których akcja toczy się w Tokio
TMS Entertainment

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13th Match Of World T20 (India vs Pakistan): 13th match of World T20 2014 held between India and Pakistan (IND vs PAK) on 21st March 2014. India won the match by 7 wickets with 9 balls remaining. This is a match of Group-2 in Twenty20 World Cup 2014. Amit Mishra (India) is declared as man of the match. Amit Mishra (Bowler) is bowled for 4 overs and gave 22 runs and took 2 wickets with 5.50 economy. India (IND) got 2 points and Pakistan (PAK) got zero points for the match.
Pakistan vs India (PAK vs IND): India won the toss and choose to bowl first. Pakistan scored 130 runs 7 wickets in 20 overs at 6.50 runs per over. India got the target and scored 131 runs 3 wickets in 18.3 overs at 7.08 runs per over. Thus India won the 13th match of World T20 2014 by 7 wickets with 9 balls remaining.


text:
Marlene Clark (born December 19, 1949) is an American actress, animator and fashion model. Clark is perhaps best known for her portrayals of Ganja Meda in the 1973 horror film Ganja & Hess and Janet Lawson, Lamont's girlfriend in the sitcom Sanford and Son from its fifth season in 1975 until the series conclusion in 1977.

Early life
Born on December 19, 1949, in New York City or Los Angeles (sources differ), Clark was raised in the Harlem section of New York. before her career in acting, Clark was a fashion model.

Career
Of the films Clark appeared in during the 1960s include For Love of Ivy (1968), starring Sidney Poitier, and Putney Swope (1969), which was directed by Robert Downey Sr. Clark appeared in Hal Ashby's directorial debut The Landlord (1970). Bill Gunn, who wrote the screenplay of The Landlord, then cast her in his unreleased film, Stop (1970). Clark also appeared opposite Jim Brown and Stella Stevens in Slaughter (1972). The same year, Clark was also in Night of the Cobra Woman (1972).

Clark collaborated with Gunn again when he cast her and Duane Jones in the 1973 horror film Ganja & Hess. A.H. Weiler of The New York Times wrote in his review of the film, "Miss Clark is an arresting presence as the enamored Ganja. Also, she occasionally invests an unbelievable character with style and humor." That same year, Clark also appeared in the Bruce Lee film Enter the Dragon (1973), in which she portrayed a secretary.

Clark was also in Black Mamba (1974). The same year, she appeared opposite Calvin Lockhart, Michael Gambon and Peter Cushing in The Beast Must Die (1974). In 1975, Clark was in Switchblade Sisters. Other films Clark appeared in during the 1970s included Clay Pigeon (1971), Beware! The Blob (1972) and Newman's Law (1974). On television, Clark portrayed Janet Lawson, the fiancé of Lamont Sanford in the 1970s sitcom Sanford and Son.

Personal life
In 1968, Clark married Billy Dee Williams in Hawaii. She was the stepmother of his son Corey, from his first marriage. Their marriage lasted only three years, and they officially divorced in 1971. In a 2000 interview, Clark has said she was almost raped while filming a scene for Black Mamba.

Partial filmography

For Love of Ivy (1968) – Radio Phone Girl
Midnight Cowboy (1969) – Girl at Party (uncredited)
Putney Swope (1969) – (uncredited)
Stop (1970) – Marlene
The Landlord (1970) – Marlene
Clay Pigeon (1971) – Saddle
Night of the Cobra Woman (1972) – Lena Aruza
Beware! The Blob (1972) – Mariane Hargis
Slaughter (1972) – Kim
Incident on a Dark Street (1973) – Rose
Ganja & Hess (1973) – Ganja Meda
Enter the Dragon (1973) – Secretary
The Beast Must Die (1974) – Caroline Newcliffe
Newman's Law (1974) – Edie
Black Mamba (1974) – The Witch
Lord Shango (1975) – Jenny
Switchblade Sisters (1975) – Muff
The Baron (1977) – Caroline

References

External links

1949 births
Actresses from New York City
American film actresses
American television actresses
20th-century American actresses
African-American actresses
Living people
Actresses from Los Angeles
20th-century African-American women
20th-century African-American people
21st-century African-American people
21st-century African-American women


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A feature film is a film with a full-length running time. According to the Academy of Motion Picture Arts and Sciences, American Film Institute, and British Film Institute, a feature film runs for 40 minutes or longer, while the Screen Actors Guild states that it is 80 minutes or longer. The majority of feature films are between 70 and 210 minutes long. The Story of the Kelly Gang was the first feature film based on length, and was released in Australia in 1906.
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EFG - The European Film Gateway was a 3-year project running from September 2008 to August 2011. Involving 22 partners from 16 European countries, EFG developed an online portal, which provides direct access to about 600,000 digital objects including films, photos, posters, drawings and text documents.
Discover the wealth of Europe's film heritage at www.europeanfilmgateway.eu.


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Textual Information(3) Being an old shareholder of the company and possessing shares in physical form. PAN details are not available in the records of the company.
Textual Information(4) Being an old shareholder of the company and possessing shares in physical form. PAN details are not available in the records of the company.
Textual Information(5) Being an old shareholder of the company and possessing shares in physical form. PAN details are not available in the records of the company.
Textual Information(6) Being an old shareholder of the company and possessing shares in physical form. PAN details are not available in the records of the company.
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Iyasu II or Joshua II (Ge'ez ኢያሱ; 21 October 1723 – 27 June 1755) was nəgusä nägäst (throne name Alem Sagad, Ge'ez ዓለም ሰገድ ʿAläm Sägäd, "to whom the world bows") (19 September 1730 – 27 June 1755) of Ethiopia, and a member of the Gondar branch of the Solomonic dynasty. He was the son of Emperor Bakaffa and Empress Mentewab (also known by her Baptismal name of Welete Giyorgis).
The Empress Mentewab played a major role in Iyasu's reign, perhaps against her will. Shortly after he was proclaimed Emperor, a rival claimant assaulted the Royal Enclosure for eight days, only leaving the capital Gondar when an army of 30,000 from Gojjam appeared. Although the rebels failed to penetrate its walls, nonetheless much of Gondar was left in ruins. Instead of taking the title of regent upon the succession of her underage son, Empress Mentewab had herself crowned as co-ruler, becoming the first woman to be crowned in this manner in Ethiopian history. Empress Mentewab wielded significant authority throughout the reign of her son, and well into the reign of her grandson as well.
During Iyasu II's reign, a Czech Franciscan Remedius Prutky visited his kingdom, and engaged Iyasu in talks about religion and European politics. Although he and his two companions were popular because of their medical skills, Prutky and his Catholic companion were asked to leave because of complaints from the local clergy after a year.
Despite Mentewab's counsel, Iyasu proved to be an ineffectual monarch. According to Paul Henze, Iyasu "came under criticism for devoting too much time to pleasure (he loved hunting) and for spending too many resources on embellishing the capital, paying foreign workmen, and importing luxury goods, ornaments and mirrors from Europe." Prutky, on the other hand blamed Iyasu's constrained revenues to the actions of his mother Mentewab: "Since the youthful emperor Jasu had only reached the age of eight when he ascended the throne, his mother the Queen divided out the provinces among the chief ministers in such a way that, at the time of my sojourn there, the Emperor, now over thirty years of age, saw his treasury diminished and scarcely enough for his ordinary expenses." Prutky adds that during the year Prutky was in Ethiopia (1752), the emperor was engaged in a struggle with his own sister over the revenues from Gojjam.
In a bid to gain the respect of his subjects, the Emperor Iyasu engaged in a campaign against the Kingdom of Sennar, which ended in defeat at the Battle of the Dindar River in 1738; an icon of Christ and a piece of the True Cross carried into battle were captured, and had to be ransomed for 8,000 ounces of gold. This defeat decisively ended any hope by Iyasu to prove himself competent in military affairs; as Donald Levine writes, "The subsequent subdual of Lasta, a rebel region for generations, and Iyasu's raids against tribes in the Atbara district were not sufficient to redeem that defeat or restore the force of Gondar."
During his reign two infestations of locusts afflicted the land, and an epidemic took the lives of thousands. When Abuna Krestodolos died, the treasury lacked money to pay for procurement of a new abuna. According to Edward Ullendorff, his authority "scarcely extended beyond Begemder and Gojjam; Shoa and Lasta acknowledged only a token allegiance, while in the Tigrai the long rule of the powerful Ras Mika'el had begun."
Emperor Iyasu also resented deeply the romantic liaison his mother entered into with a young member of the Imperial family. Empress Mentewab became involved with Iyasu, the son of her former sister-in-law Romanework, who was herself the sister of the late Emperor Bakaffa, and on her father's side descended in male line from another cadet line of the Solomonic dynasty. Mentewab's relationship with the much younger nephew of her late husband was considered a great scandal, and the young Prince was derisively referred to as "Melmal Iyasu", or "Iyasu the Kept". The Empress had three daughters by this Melmal Iyasu, one of whom was the beautiful Woizero Aster Iyasu who took Ras Mikael Sehul in 1769 as her third husband. Emperor Iyasu became very attached to his half-sisters, but was deeply resentful of their father. It is said that it was the Emperor himself that ordered the murder of his mother's lover by having him pushed from a cliff top near Lake Tana in 1742.
Iyasu fell seriously ill in May, 1755, and died the next month. It was generally believed that he had been poisoned by the sister of Melmal Iyasu, in revenge for her brother's death. When the Empress Mentewab sought funds from the treasury for his funeral, only a few dinars could be found. Saddened by this situation, she threatened to retire to her palace convent at Qusquam, but a group of nobles persuaded her to instead become regent for her grandson Iyoas I.
↑ Richard Pankhurst, "An Eighteenth Century Ethiopian Dynastic Marriage Contract between Empress Mentewwab of Gondar and Ras Mika'el Sehul of Tegre," in Bulletin of the School of Oriental and African Studies, 1979, p. 458.
↑ E. A. Wallis Budge, A History of Ethiopia: Nubia and Abyssinia, 1928 (Oosterhout, the Netherlands: Anthropological Publications, 1970), pp. 454f.
↑ The Royal Chronicle of his reign is translated in part by Richard K. P. Pankhurst, The Ethiopian Royal Chronicles (Addis Ababa: Oxford University Press, 1967).


text:
Mirosława Lucyna Kwiecień – polska ekonomistka, dr hab. nauk ekonomicznych, profesor nadzwyczajny Zakładu Nauk Ekonomicznych Wydziału Nauk Technicznych i Ekonomicznych Państwowej Wyższej Szkoły Zawodowej im. Witelona w Legnicy.

Życiorys
Mirosława Kwiecień od 1968 roku do 2014 była pracownikiem Uniwersytetu Ekonomicznego we Wrocławiu, na którym wcześniej uzyskała dyplom magistra (1968) i stopień naukowy doktor nauk ekonomicznych (1976). W 1992 – również na macierzystej uczelni – habilitowała się na podstawie dorobku naukowego i pracy zatytułowanej Rachunkowość jako narzędzie zarządzania. System z bazą wiedzy. Została zatrudniona na stanowisku profesora w Instytucie Rachunkowości na Wydziale Zarządzania, Informatyki i Finansów Uniwersytetu Ekonomicznego we Wrocławiu, a także w Wyższej Szkole Handlu i Rachunkowości.

Objęła funkcję profesora nadzwyczajnego w Zakładzie Nauk Ekonomicznych na Wydziale Nauk Technicznych i Ekonomicznych w Państwowej Wyższej Szkole Zawodowej im. Witelona w Legnicy.

Publikacje
 2005: Harmonizacja rachunkowości a standardy edukacji, [w:] Standardy edukacyjne rachunkowości - praktyka i stan badań
 2007: Współczesne społeczeństwo wiedzy a rachunkowość. w:WIEDZA I INNOWACJE W ROZWOJU GOSPODARKI: SIŁY MOTORYCZNE I BARIERY
 2007: Aktywa niematerialne - kapitał wiedzy - wycena
 2009: Ład korporacyjny a sprawozdanie finansowe

Przypisy

Polscy ekonomiści
Wykładowcy Uniwersytetu Ekonomicznego we Wrocławiu
Urodzeni w XX wieku


text:
Q: Ajax not redirecting to tiles view Hi I am using spring 3 mvc and tiles when I hit the URL manually or on onclick it returns the new tiles view. But through ajax it hits the controller but does not go to the view.
My spring-servlet.xml is:
<?xml version="1.0" encoding="UTF-8"?>
<beans xmlns="http://www.springframework.org/schema/beans"
    xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xmlns:p="http://www.springframework.org/schema/p"
    xmlns:mvc="http://www.springframework.org/schema/mvc" xmlns:context="http://www.springframework.org/schema/context"
    xsi:schemaLocation="http://www.springframework.org/schema/beans
http://www.springframework.org/schema/beans/spring-beans-3.0.xsd

http://www.springframework.org/schema/mvc
http://www.springframework.org/schema/mvc/spring-mvc-3.0.xsd

http://www.springframework.org/schema/context
http://www.springframework.org/schema/context/spring-context-3.0.xsd">

    <mvc:annotation-driven />
    <context:annotation-config />
    <context:component-scan base-package="in.ac.nitk.tas" />

    <bean id="viewResolver"
        class="org.springframework.web.servlet.view.UrlBasedViewResolver">
        <property name="viewClass">
            <value>
                org.springframework.web.servlet.view.tiles2.TilesView
            </value>
        </property>
    </bean>

    <bean id="ajaxViewResolver" class="org.springframework.js.ajax.AjaxUrlBasedViewResolver">
         <property name="viewClass" value="org.springframework.js.ajax.tiles2.AjaxTilesView"/>
    </bean>

    <bean id="tilesConfigurer"
        class="org.springframework.web.servlet.view.tiles2.TilesConfigurer">
        <property name="definitions">
            <list>
                <value>/WEB-INF/tiles.xml</value>
            </list>
        </property>
    </bean>
    <bean id="jacksonMessageConverter"
        class="org.springframework.http.converter.json.MappingJacksonHttpMessageConverter"></bean>
    <bean
        class="org.springframework.web.servlet.mvc.annotation.AnnotationMethodHandlerAdapter">
        <property name="messageConverters">
            <list>
                <ref bean="jacksonMessageConverter" />
            </list>
        </property>
    </bean>

</beans>

My contoller is
@Controller
@RequestMapping("/home")
public class HomeController {

    @Autowired
    private HomeService homeService;

    @RequestMapping("/login")
    public String login(@RequestBody final UserVO userVO, final ModelMap map,
            HttpSession session, HttpServletRequest req) {
               -
               -
               -
               -
               -

        return "adminView";
    }
}

js function call this
function submitLoginForm(){

    var userName =$("#userName").val();
    var password= $("#password").val();
    var userVO ={ userName: userName,
                password: password,

               };

    $.ajax({
        url : "home/login.do",
        type : "POST",
        //dataType: "html",
        contentType: "application/json; charset=utf-8",
        data: JSON.stringify(userVO),
        cache: false,    //This will force requested pages not to be cached by the browser

//        processData:false,
        success : function(data) {
          // alert(data);
        }
    });
}

My tiles.xml is
<?xml version="1.0" encoding="UTF-8"?>
 <!DOCTYPE tiles-definitions PUBLIC
       "-//Apache Software Foundation//DTD Tiles Configuration 2.0//EN"
       "http://tiles.apache.org/dtds/tiles-config_2_0.dtd">

<tiles-definitions>

 <definition name="baseLayout" template="/WEB-INF/jsp/layout/layout.jsp">
  <put-attribute name="header" value="/WEB-INF/jsp/layout/header.jsp" />
  <put-attribute name="login" value="" />
  <put-attribute name="navigation" value="/WEB-INF/jsp/layout/navigation.jsp" />
  <put-attribute name="body" value="" />
  <put-attribute name="footer" value="/WEB-INF/jsp/layout/footer.jsp" />
 </definition>

 <definition name="login" extends="baseLayout">
  <put-attribute name="login" value="/WEB-INF/jsp/layout/login.jsp" />
  <put-attribute name="body" value="/WEB-INF/jsp/layout/body.jsp" />

 </definition>

 <definition name="adminView" extends="baseLayout">
  <put-attribute name="login" value="/WEB-INF/jsp/welcome.jsp" />
  <put-attribute name="body" value="/WEB-INF/jsp/admin/adminHome.jsp" />
  <put-attribute name="navigation" value="/WEB-INF/jsp/admin/adminMenu.jsp" />
 </definition>

</tiles-definitions>

redpajama_set_name: RedPajamaStackExchange


text:
The name Lisa Sinha is synonymous with Gippsland Multicultural Services. For 25 years Ms Sinha has represented GMS's image and what the organisation stands for in terms of justice, tolerance, and acceptance of people no matter where they come from. For people who have worked with her for many years, Ms Sinha's relationship with them and the wider community transcends office hours and her job description. "As much as she's CEO you don't see a distinction between her and the workers. She's always down to earth and willing to do and get her hands dirty," GMS respite services coordinator Mersiha Grbic said. Ms Grbic landed her first job at GMS as a resettlement officer assisting refugees and newly-arrived migrants when she was 18. "She will do shredding, will go to archiving, clean up the front office, wash the dishes after the meeting, clean up the rubbish ... she puts us, her workers, first." Kelly Bull, who joined the organisation as a financial controller more than five years ago, said she knew Ms Sinha for about 18 years and worked alongside her in her previous jobs. "She's got a lot of skills than most CEO's I've met[don't]. She deliberately learnt how to do the pay so I can have leave," Mrs Bull said. Both said Ms Sinha's heart to champion the underprivileged and to be there to offer help had been with her for 25 years. "She's always there to help in any way shape or form anybody who comes to that door," Ms Grbic said. "She's incredibly fair and just had that heart to really advocate for people ... no matter which corner of society they come from whether it is aged, or women, or refugee, or whether it's children," Mrs Bull said. Ms Sinha joined GMS as its chief executive in 1993. An anti-racism campaigner, Ms Sinha championed the rights of refugees and the wider multicultural community and is passionate about social justice. She currently manages a team of eight program managers and 50 bilingual direct care workers and provide a range of services in aged care, disability and settlement. Born in Richmond, Ms Sinha comes from an Indian-New Zealand background. She worked in community housing, Aboriginal children's services, lectured in early childhood development, and was a kindergarten teacher before joining GMS. Colleagues said her passion for work was second to none as Ms Sinha, who lives outside of Latrobe Valley, would always find time to attend multicultural activities on weekends that sometimes finish late at night. Ms Grbic said GMS staff respected their boss for her work ethic and ability to empower people to take charge of their own affairs. "She's not there to fix things and it's always about empowering somebody else to do that for themselves by giving them the right tools and that, I guess, [is] what I admired most about her," she said. Mrs Bull said the 57-year-old's ability to juggle her time with her work requirements and being a single mum to her daughter, Vilani, was amazing. "One of my earliest memories of Lisa was not long after Vilani was born – like having her in a capsule in the office. I was mesmerised at how she's handled it all. I guess she's made it work and between her and Vilani they've worked it out," she said. Vilani, who is almost 18, said her mum was her role model. "I grew up being in GMS and being around diverse of people ... she's my biggest influence in life, my views of the world come from her," she said. Vilani said while her mum was busy at work she would always find time to be with her and if she had conflicts in schedule, would ask a colleague to take her to school activities. "She may not be physically organised but she's organised in her head ...she's got my back. I wouldn't be who I am without her ... she's my role model and close to my heart," she said.
https://nnimgt-a.akamaihd.net/transform/v1/crop/frm/storypad-REPuYvyMkwVN6jxvk5dHA5/0a2ef969-1db9-4405-b2be-91bb508ddf30.jpg/r0_53_256_198_w1200_h678_fmax.jpg
November 5 2018 - 12:00AM
Lisa's leadership inspires
The name Lisa Sinha is synonymous with Gippsland Multicultural Services.
For 25 years Ms Sinha has represented GMS's image and what the organisation stands for in terms of justice, tolerance, and acceptance of people no matter where they come from.
For people who have worked with her for many years, Ms Sinha's relationship with them and the wider community transcends office hours and her job description.
"As much as she's CEO you don't see a distinction between her and the workers. She's always down to earth and willing to do and get her hands dirty," GMS respite services coordinator Mersiha Grbic said.
Ms Grbic landed her first job at GMS as a resettlement officer assisting refugees and newly-arrived migrants when she was 18.
Friends and colleagues have organised a celebration of Ms Sinha's 25th anniversary as GMS chief this week. file photograph
"She will do shredding, will go to archiving, clean up the front office, wash the dishes after the meeting, clean up the rubbish ... she puts us, her workers, first."
Kelly Bull, who joined the organisation as a financial controller more than five years ago, said she knew Ms Sinha for about 18 years and worked alongside her in her previous jobs.
"She's got a lot of skills than most CEO's I've met[don't]. She deliberately learnt how to do the pay so I can have leave," Mrs Bull said.
Both said Ms Sinha's heart to champion the underprivileged and to be there to offer help had been with her for 25 years.
"She's always there to help in any way shape or form anybody who comes to that door," Ms Grbic said.
"She's incredibly fair and just had that heart to really advocate for people ... no matter which corner of society they come from whether it is aged, or women, or refugee, or whether it's children," Mrs Bull said.
Ms Sinha joined GMS as its chief executive in 1993.
An anti-racism campaigner, Ms Sinha championed the rights of refugees and the wider multicultural community and is passionate about social justice.
She currently manages a team of eight program managers and 50 bilingual direct care workers and provide a range of services in aged care, disability and settlement.
Born in Richmond, Ms Sinha comes from an Indian-New Zealand background. She worked in community housing, Aboriginal children's services, lectured in early childhood development, and was a kindergarten teacher before joining GMS.
Colleagues said her passion for work was second to none as Ms Sinha, who lives outside of Latrobe Valley, would always find time to attend multicultural activities on weekends that sometimes finish late at night.
Ms Grbic said GMS staff respected their boss for her work ethic and ability to empower people to take charge of their own affairs.
"She's not there to fix things and it's always about empowering somebody else to do that for themselves by giving them the right tools and that, I guess, [is] what I admired most about her," she said.
Mrs Bull said the 57-year-old's ability to juggle her time with her work requirements and being a single mum to her daughter, Vilani, was amazing.
"One of my earliest memories of Lisa was not long after Vilani was born – like having her in a capsule in the office. I was mesmerised at how she's handled it all. I guess she's made it work and between her and Vilani they've worked it out," she said.
Vilani, who is almost 18, said her mum was her role model.
"I grew up being in GMS and being around diverse of people ... she's my biggest influence in life, my views of the world come from her," she said.
Vilani said while her mum was busy at work she would always find time to be with her and if she had conflicts in schedule, would ask a colleague to take her to school activities.
"She may not be physically organised but she's organised in her head ...she's got my back. I wouldn't be who I am without her ... she's my role model and close to my heart," she said.
Discuss "Lisa's leadership inspires "


text:
/* WEIGHTED GRAPH */

//Create a new weighted, directed graph.
graph* _weighted_graph_new(standard_library_context*); //FINISHED

//Delete a weighted, directed graph.
bool _weighted_graph_delete(graph*); //FINISHED

//Add a new node to the graph.
graph_node* _weighted_graph_add_node(graph*, long, void*); //FINISHED

//Create an edge (graph, from, to, weight).
bool _weighted_graph_add_edge(graph*, long, long, long, graph_edge**); //FINISHED

//Create an edge with a mask (graph, from, to, weight, mask, edge).
bool _weighted_graph_add_masked_edge(graph*, long, long, long, size_t, graph_edge**); //FINISHED

//Remove a node.
bool _weighted_graph_delete_node(graph*, long); //FINISHED

//Remove an edge.
bool _weighted_graph_delete_edge(graph*, long, long); //FINISHED

//Clear the graph of levels/parents.
void _weighted_graph_clear_paths(graph*); //FINISHED

//Get the shortest path from one node to another.
bool _weighted_graph_spath(graph*, long, long, graph_path*);

//Get the shortest path from one node to another, while avoiding edges with a certain mask.
bool _weighted_graph_spath_amask(graph*, long, long, size_t, graph_path*);

//Sort the edges of each node by weight.
bool _weighted_graph_edge_sort(graph*);

//Minimum spanning tree.
//bool _weighted_graph_min_spanning(graph*, long, long);

#ifndef GENLIB_LINK_OBJECT_NCOMP

	#include "weighted_graph.c"

#endif

redpajama_set_name: RedPajamaGithub


text:
Home › Gossip
Father Of Whitney Houston's Alleged Female Lover Spills Tea About Them
Ny Magee, Eurweb
The father of Robyn Crawford, the longtime friend and rumored lover to Whitney Houston, spoke out for the first time earlier this year about his daughter's relationship with the late singer.
In an interview with Radar Online, Dennis Crawford revealed that he knew ever since Robyn and Whitney were kids that they were "girlfriends," ILOSM notes.
"I knew 40 years ago, we all knew," he shared with Radar. "I saw them together."
Mr. Crawford said that he saw the two women holding hands and kissing, and he accepted his daughter's romance with the singer.
"When she told me, there was nothing I could do about it except love her," he recalled.He also revealed that Whitney's famous cousin, singer Dionne Warwick, also knew about their relationship.
"Dionne grew up with them, and knew about it too," Dennis added.
He's not the only one who was close to the situation to speak about their relationship. Whitney's stylist, Ellin Lavar, once said (via People): "I don't think she was gay, I think she was bisexual. Robyn provided a safe place for her…in that Whitney found safety and solace."
In the 2017 documentary, "Whitney: Can I Be Me," Houston's bodyguard, Kevin Ammons, shares further intimate details about the love both Bobby Brown and Robyn had for Whitney.
"They'd battle for her affections. Bobby and Robyn had some physical altercations and there were times where he wasn't always the winner. But then Whitney would always come and pour oil over troubled waters," he adds.
After Houston's 1999 tour wrapped, Robyn left — and that's when Houston's "downfall" began, according to Lavar.
"That was the downfall of Whitney. Robyn was the person who was keeping her together."
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female lover relationship Whitney Houston


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