text:
Volume 8, Issue 32 (Fall 2020) IUESA 2020, 8(32): 1-16 | Back to browse issues page
Promoting the Habitation Worthiness, the Main Strategy of the Regenerating and Recreating Decayed and Historic Neighborhood
Zahra Nejadbahram1, Seyyed Hossein Mirzadeh *2
1- Faculty of Political Science, Islamic College of London, London, England
2- Faculty of Architecture and Urban Planning, University of Art, Tehran, Iran , hossein.mirzadeh65@gmail.com
Abstract: (248 Views)
Decayed areas and historical districts are one of the most important issues in the urban planning system of our country so that a significant budget is spent every year for the renovation and regeneration of these neighborhoods. But Experiences from last projects show that although despite the renovation of buildings and private spaces, the population of these areas has migrated to other neighborhoods of the city. This study suggests the "habitation worthiness" concept to explain the various aspects of urban life which lead people to settle in or migrate from the district. Additionally, it attends to the vitality of attentive districts. To this end, this study has used documentary study and content analysis, and also descriptive and logistic methods to review the literature and compare the evidence and concepts from studing the documents, respectively. The findings of this study show that "habitation worthiness" is a concept beyond the quality of life and should be considered in the process of urban regeneration. The "habitation worthiness" addresses all biological, physical, cultural-social, economical and environmental aspects of human habitation and provides indicators for evaluating urban projects and plans in each dimension. Consequently, ignoring any aspect is resulted flaw in the "habitation worthiness" and migrating the people resettled in these districts.
Keywords: Habitation Worthiness, Decayed Areas, Historical Neighborhoods, Urban Regeneration, Participation
Full-Text [PDF 858 kb] (58 Downloads)
Type of Study: Research | Subject: Special
Received: 2020/04/19 | Accepted: 2020/09/5 | Published: 2020/12/20 | ePublished: 2020/12/20


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text:
\section{Introduction}

An isometric action on a Riemannian manifold is of cohomogeneity
one if its orbit space is one-dimensional. Cohomogeneity one
actions are of current interest for the construction of
geometrical structures on manifolds, e.g.\ Einstein metrics and
metrics with special holonomies. The reason is that a
cohomogeneity one action can be used to reduce the system of
partial differential equations describing such a geometrical
structure to a nonlinear ordinary differential equation for which
one might be able to find explicit solutions. Given a Riemannian
manifold $M$, it is natural to find all cohomogeneity one actions
on it, perhaps just up to orbit equivalence. Two cohomogeneity one
actions on $M$ are orbit equivalent if there exists an isometry of
$M$ that maps the orbits of one action onto the orbits of the
other action. It is worthwhile to mention that the classification
problem of cohomogeneity one actions up to orbit equivalence is
equivalent to the classification problem of homogeneous
hypersurfaces up to isometric congruence. The latter one is a
classical problem is submanifold theory.

The cohomogeneity one actions on the spheres, equipped with their standard metric
of constant curvature, have been classified by Hsiang and Lawson \cite{HL}.
Remarkably, any such action is orbit equivalent to the isotropy representation of a
Riemannian symmetric space of rank two. For the other compact
symmetric spaces of rank one the classifications were obtained by
Takagi \cite{Ta} for the complex projective spaces and by Iwata \cite{Iw1},
\cite{Iw2} for the quaternionic projective spaces and the Cayley projective plane.
For simply connected
irreducible Riemannian symmetric spaces of higher rank the classification
was established by Kollross \cite{Ko}.

The methods employed by the above authors do not work for the noncompact dual
symmetric spaces. The noncompactness of the isometry group turns out to be
a subtle point. There can be uncountably many families of nonisomorphic
subgroups of the isometry group that act orbit equivalently by cohomogeneity
one. By using the classification of isoparametric hypersurfaces
on the Euclidean space $\mathbb R^n$ and the real hyperbolic
space $\mathbb R H^n$ by
Levi Civita \cite{Le}, Segre \cite{Se} and Cartan \cite{Ca}, one can obtain all
cohomogeneity one actions on these spaces up to orbit equivalence. In both cases
the orbit structure is either a Riemannian foliation, or a totally geodesic
subspace together with the distance tubes around it.
It is a general fact that a cohomogeneity one action on a symmetric
space of noncompact type, or more general on a Hadamard manifold, induces
either a Riemannian foliation, or has exactly one singular orbit and the generic orbits
are the distance tubes around it (see \cite{BB} for details and references).

In \cite{BT1} we obtained the classification, up to orbit equivalence,
of all cohomogeneity one actions on irreducible symmetric spaces of noncompact type
that induce a Riemannian foliation, that is, have no singular orbit. A
surprising consequence of this result is that the moduli space of all such
actions just depends on the rank of the symmetric space and possible duality or
triality principles on the space. In particular, on each noncompact symmetric space
of rank one this moduli space consists just of two elements. The corresponding foliations
are the horosphere foliation and a foliation with exactly one minimal leaf whose
geometry has been investigated in \cite{Be}.

The classification of all cohomogeneity one actions on irreducible
symmetric spaces of noncompact type that have a totally geodesic
singular orbit has been achieved in \cite{BT2}. It thus remains to
investigate the case of a non-totally geodesic singular orbit. As
mentioned above, in case of $\mathbb R^n$ and $\mathbb R H^n$ a singular orbit
is necessarily totally geodesic. It is remarkable that this is no
longer true for the other noncompact symmetric spaces of rank one:
the complex hyperbolic spaces $\mathbb C H^n$ ($n \geq 2$), the
quaternionic hyperbolic spaces $\mathbb H H^n$ ($n \geq 2$), and the
Cayley hyperbolic plane $\mathbb O H^2$. The first author and Br\"{u}ck
constructed in \cite{BB} many examples of cohomogeneity one
actions on these hyperbolic spaces (except for $\mathbb C H^2$) with a
non-totally geodesic singular orbit. The main result of this paper
says that, up to orbit equivalence, there are no further
cohomogeneity one actions on $\mathbb C H^n$ ($n \geq 3$), $\mathbb H H^2$ and
$\mathbb O H^2$. We also show that every singular orbit of a
cohomogeneity one action on $\mathbb C H^2$ is totally geodesic. For the
quaternionic hyperbolic space $\mathbb H H^n$, $n \geq 3$, we prove that
the set of orbit equivalence classes of cohomogeneity one actions
with a singular orbit of codimension $2$ is parametrized by the
closed interval $[0,\pi/2]$.

The results of this paper were partially obtained during a
common visit to the Mathe\-matical Research Institute Oberwolfach (Research
in Pairs programme). We would like to thank the Institute for its support and
the hospitality.
The second author was partially supported by
Grant-in-Aid for Young Scientists (B) 14740049,
The Ministry of Education, Culture, Sports, Science and Technology,
Japan.

\section{Preliminaries}

Let $M$ be a noncompact symmetric space of rank one. Then $M$ is either a
real hyperbolic space $\mathbb R H^n$, a complex hyperbolic space $\mathbb C H^n$, a
quaternionic hyperbolic space $\mathbb H H^n$, or a Cayley hyperbolic plane $\mathbb O H^2$,
where $n \geq 2$. We denote by $\mathbb F$ one of the real division algebras
$\mathbb R$, $\mathbb C$, $\mathbb H$ or $\mathbb O$, and by $\mathbb F H^n$ the corresponding hyperbolic space,
where we assume $n = 2$ if $\mathbb F = \mathbb O$.
Let $G$ be the identity component of the full isometry group of $M$, that is,
$G = SO^o(n,1),SU(n,1),Sp(n,1),F_4^{-20}$ for $\mathbb F=\mathbb R,\mathbb C,\mathbb H,\mathbb O$, respectively.
We fix a point $o \in M$ and denote by $K$ the isotropy subgroup of $G$ at $o$,
that is, $K = SO(n),S(U(n)U(1)),Sp(n)Sp(1),\hbox{\rm Spin}(9)$. Then, as a homogeneous space,
$M$ is isomorphic to $G/K$. We denote by $\hbox{\fr g}$ and $\hbox{\fr k}$ the Lie algebra of $G$
and $K$. Let $B$ be the Killing form of $\hbox{\fr g}$ and $\theta$ the Cartan involution of
$\hbox{\fr g}$ with respect to $\hbox{\fr k}$. Then $\langle X,Y \rangle = -B(X,\theta Y)$ is a positive
definite inner product on $\hbox{\fr g}$. Let $\hbox{\fr g} = \hbox{\fr k} + \hbox{\fr p}$ be the Cartan decomposition
of $\hbox{\fr g}$ induced by $\theta$. The restriction of $\langle \cdot, \cdot \rangle$ to
$\hbox{\fr p}$ induces a Riemannian metric on $G/K$ turning it into a Riemannian symmetric space
of rank one.
We normalize the Riemannian metric on $M$ so that it becomes isometric to $G/K$
with the induced metric we just described.

Let $\hbox{\fr a}$ be a maximal abelian subspace of $\hbox{\fr p}$, which is just a one-dimensional
linear subspace since the rank of $M$ is one, and
$$
\hbox{\fr g} = \hbox{\fr g}_{-2\alpha} + \hbox{\fr g}_{\alpha} + \hbox{\fr g}_0 + \hbox{\fr g}_{\alpha} + \hbox{\fr g}_{2\alpha}
$$
the corresponding restricted root space decomposition of $\hbox{\fr g}$. Note that
$\hbox{\fr g}_{-2\alpha}$ and $\hbox{\fr g}_{2\alpha}$ are trivial if $\mathbb F = \mathbb R$. Then
$$
\hbox{\fr g} = \hbox{\fr k} + \hbox{\fr a} + \hbox{\fr n}\ \ \ {\rm with}\ \ \ \hbox{\fr n} = \hbox{\fr g}_{\alpha} + \hbox{\fr g}_{2\alpha}
$$
is an Iwasawa decomposition of $\hbox{\fr g}$. The subalgebra $\hbox{\fr n}$ of $\hbox{\fr g}$ is abelian
if $\mathbb F = \mathbb R$ and two-step nilpotent otherwise. In fact, $\hbox{\fr n}$ is isomorphic to
the $(2n-1)$-dimensional Heisenberg algebra if $\mathbb F = \mathbb C$, and to a certain
generalized Heisenberg algebra if $\mathbb F \in \{\mathbb H,\mathbb O\}$
(see \cite{BTV} for more details on this). Moreover,
$\hbox{\fr z} = \hbox{\fr g}_{2\alpha}$ is the centre of $\hbox{\fr n}$ and equal to the derived subalgebra
$[\hbox{\fr n},\hbox{\fr n}]$ of $\hbox{\fr n}$. The dimension of $\hbox{\fr z}$ is equal to $1,3,7$ for $\mathbb F = \mathbb C,\mathbb H,\mathbb O$,
respectively. The subalgebra $\hbox{\fr a} + \hbox{\fr n}$ of $\hbox{\fr g}$ is solvable and $\hbox{\fr n}$ is the derived
subalgebra of $\hbox{\fr a} + \hbox{\fr n}$.

We denote by $A$ resp.\ $N$ the connected closed subgroup of $G$ with
Lie algebra $\hbox{\fr a}$ resp.\ $\hbox{\fr n}$. Then $G = KAN$ is an Iwasawa decomposition of $G$ and,
since $K$ is the isotropy subgroup of $G$ at $o$, the solvable subgroup $AN$ of $G$
acts simply transitively on $M$.
Thus $M$ is isometric to the solvable Lie group $AN$
equipped with a suitable left-invariant Riemannian metric.

We define $\hbox{\fr v} = \hbox{\fr g}_{\alpha}$. Then we can identify $\hbox{\fr v}$ with $\mathbb R^{n-1},\mathbb C^{n-1},\mathbb H^{n-1},\mathbb O$
for $\mathbb F = \mathbb R,\mathbb C,\mathbb H,\mathbb O$, respectively. More precisely, if $\mathbb F = \mathbb R$ then $\hbox{\fr v}$ is isomorphic
to $\mathbb R^{n-1}$ as a real vector space. If $\mathbb F = \mathbb C$, the K\"{a}hler structure on $\mathbb C H^n$ induces
a complex vector space structure on $\hbox{\fr v}$ so that it becomes isomorphic to $\mathbb C^{n-1}$,
and if $\mathbb F = \mathbb H$, the quaternionic K\"{a}hler structure on $\mathbb H H^n$ induces a (right) quaternionic
vector space structure on $\hbox{\fr v}$ so that it becomes isomorphic to the (right) quaternionic
vector space $\mathbb H^{n-1}$. For $\mathbb F = \mathbb O$ we simply identify $\mathbb O$ with $\mathbb R^8$, and $\hbox{\fr v}$ becomes
isomorphic to $\mathbb O$ as a real vector space.

\section{The reduction}\label{reduction}

In this section we reduce our classification problem to the problem of classifying
certain subalgebras of $\hbox{\fr a} + \hbox{\fr n}$.
We start with a general result about cohomogeneity one actions on Hadamard manifolds, i.e.,
connected, simply connected, complete Riemannian manifolds
of nonpositive curvature. Every symmetric space of noncompact type is a Hadamard manifold.
We recall that Cartan's Fixed Point Theorem states that the action of any compact
subgroup of the isometry group of a Hadamard manifold has a fixed point. We refer to
\cite{Eb} for more details on Hadamard manifolds and Cartan's Fixed Point Theorem.

\begin{prop}\label{solvable}
Let $M$ be a Hadamard manifold and $H$ a connected subgroup of
the isometry group of $M$ that acts
with cohomogeneity one on $M$ and has a singular orbit $F$. Then
there exists a connected solvable subgroup of $H$ that acts transitively on $F$.
\end{prop}

{\it Proof.} We choose a Levi-Malcev decomposition
$$
\hbox{\fr h} = \hbox{\fr h}_{ss} + \hbox{\fr h}_{solv}
$$
of the Lie algebra $\hbox{\fr h}$ of $H$
into the semidirect sum of a semisimple subalgebra $\hbox{\fr h}_{ss}$ and a solvable
ideal $\hbox{\fr h}_{solv}$. For the semisimple subalgebra $\hbox{\fr h}_{ss}$ we choose an Iwasawa decomposition
$$
\hbox{\fr h}_{ss} = \hbox{\fr h}_{ss}^{cpct} + \hbox{\fr h}_{ss}^{solv}
$$
of $\hbox{\fr h}_{ss}$ into the vector space direct sum of a compact subalgebra $\hbox{\fr h}_{ss}^{cpct}$
and a solvable subalgebra $\hbox{\fr h}_{ss}^{solv}$. Then
$$
\hbox{\fr h} = \hbox{\fr h}_{ss}^{cpct} + (\hbox{\fr h}_{ss}^{solv} + \hbox{\fr h}_{solv})\ ,
$$
where $\hbox{\fr h}_{ss}^{solv} + \hbox{\fr h}_{solv}$ is a semidirect sum of the two solvable subalgebras
so that $\hbox{\fr h}_{solv}$ is the ideal in it. Note that the semidirect sum of two
solvable Lie algebras is solvable as well. We denote by $H_{ss}^{cpct}$ and
$H^{solv}$ the connected subgroup of $H$ with Lie algebra
$\hbox{\fr h}_{ss}^{cpct}$ and $\hbox{\fr h}_{ss}^{solv} + \hbox{\fr h}_{solv}$, respectively.

By Cartan's Fixed Point Theorem, there exists a point $p \in M$ that is fixed
under the action of the compact group $H_{ss}^{cpct}$. If $p \in F$, then clearly the
solvable group $H^{solv}$ acts transitively on $F$. If $p \notin F$,
then $p$ is on a principal orbit of the $H$-action on $M$, and it follows that the
solvable group $H^{solv}$ acts transitively on this principal orbit.
Since the action of $H$ on $M$ is of cohomogeneity one, we easily see that
$H^{solv}$ acts transitively on each orbit of the $H$-action and,
in particular, also transitive on the singular orbit $F$.
This finishes the proof of Proposition \ref{solvable} \hfill \hbox{\rm q.e.d.}

\medskip
We denote by $M(\infty)$ the ideal boundary of $M$ whose points are given by
the equivalence classes of asymptotic geodesics in $M$, and equip
$\bar{M} = M \cup M(\infty)$ with the cone topology. The action of $H$ on $M$
extends canonically to an action of $H$ on $\bar{M}$.

From now on we assume that $M = \mathbb F H^n$ and that the singular orbit $F$
of the cohomogeneity one action by $H$ on $M$ is
not totally geodesic. Then none of the $H$-orbits on $M$ is totally geodesic,
and a result by Alekseevsky and Di Scala \cite{AD} implies that there exists
a unique point $x \in M(\infty)$ that is fixed under the $H$-action on $M(\infty)$.
We fix a point $o \in F$ and consider the Iwasawa decomposition
$$
\hbox{\fr g} = \hbox{\fr k} + \hbox{\fr a} + \hbox{\fr n}
$$
that is determined by $o$ and $x$. Since $H \cdot x = x$, we have
$$
\hbox{\fr h} \subset \hbox{\fr k}_x + \hbox{\fr a} + \hbox{\fr n}\ ,
$$
where $\hbox{\fr k}_x$ is the centralizer of $\hbox{\fr a}$ in $\hbox{\fr k}$. We denote by $K_x$ the
connected subgroup of $K$ with Lie algebra $\hbox{\fr k}_x$. Then we have
$K_x = SO(n-1),S(U(n-1)U(1)),Sp(n-1)Sp(1),\hbox{\rm Spin}(7)$ for $\mathbb F = \mathbb R,\mathbb C,\mathbb H,\mathbb O$,
respectively, and $H \subset K_xAN$.

By Proposition \ref{solvable} there exists a solvable subgroup $S$ of $H$ that
acts transitively on the singular orbit $F$. We denote by $\hbox{\fr s}$ the subalgebra of
$\hbox{\fr h}$ corresponding to $S \subset H$. We recall that we may write the nilpotent
subalgebra $\hbox{\fr n}$ in the form $\hbox{\fr n} = \hbox{\fr g}_{\alpha} + \hbox{\fr g}_{2\alpha}$ with some suitable
root spaces $\hbox{\fr g}_{\alpha}$ and $\hbox{\fr g}_{2\alpha}$. Since $\hbox{\fr k}_x$ centralizes $\hbox{\fr a}$,
it normalizes each root space and hence $\hbox{\fr n}$, which implies that $\hbox{\fr a} + \hbox{\fr n}$
is an ideal in $\hbox{\fr k}_x + \hbox{\fr a} + \hbox{\fr n}$. Thus the canonical projection
$$
\pi : \hbox{\fr k}_x + \hbox{\fr a} + \hbox{\fr n} \to \hbox{\fr k}_x
$$
is a Lie algebra homomorphism, and it follows that
$$
\hbox{\fr s}_c = \pi(\hbox{\fr s})
$$
is a solvable subalgebra of $\hbox{\fr k}_x$. Since every solvable subalgebra of
a compact Lie algebra is abelian, we conclude that
\begin{equation}\label{sc-abelian}
\hbox{\fr s}_c\ {\rm is\ an\ abelian\ subalgebra\ of}\ \hbox{\fr k}_x.
\end{equation}

Let
$$
\tau : \hbox{\fr k}_x + \hbox{\fr a} + \hbox{\fr n} \to \hbox{\fr a} + \hbox{\fr n}
$$
be the canonical projection and define
$$
\hbox{\fr s}_n = \tau(\hbox{\fr s})\ .
$$
It is clear that
\begin{equation}\label{dimensions}
\dim \hbox{\fr s}_n = \dim F\ .
\end{equation}
Our aim is to show that $\hbox{\fr s}_n$ is a subalgebra of $\hbox{\fr a} + \hbox{\fr n}$ and that the
orbit through $o$ of the action of the corresponding subgroup $S_n$ of $AN$ is just the
singular orbit $F$.
For each $k \in K_x$ the differential $d_ok$ of $k$ at $o$ is given by
$d_ok = \hbox{\rm Ad}(k)|(\hbox{\fr a}+\hbox{\fr n})$, where we identify $T_oM$ with $\hbox{\fr a} + \hbox{\fr n}$
by means of $M = G/K = AN$. Since the isotropy subgroup $H_o$ of $H$ at $o$
acts transitively on the unit sphere in the normal space $\nu_oF$ of $F$ at $o$,
and as $\hbox{\fr k}_x$ centralizes $\hbox{\fr a}$, we necessarily have
\begin{equation}\label{normal-in-n}
\nu_oF \subset \hbox{\fr n}
\end{equation}
and hence
\begin{equation}\label{a-in-sn}
\hbox{\fr a} \subset \hbox{\fr s}_n\ .
\end{equation}

We shall now prove that
\begin{equation}\label{derived-in-sn}
[\hbox{\fr s},\hbox{\fr s}] = \hbox{\fr s}_n \cap \hbox{\fr n} \ .
\end{equation}
Since $\hbox{\fr s} \subset \hbox{\fr s}_c + \hbox{\fr s}_n$, we have
$$
[\hbox{\fr s},\hbox{\fr s}] \subset [\hbox{\fr s}_c,\hbox{\fr s}_c] + [\hbox{\fr s}_c,\hbox{\fr s}_n] + [\hbox{\fr s}_n,\hbox{\fr s}_n]\ .
$$
The subalgebra $[\hbox{\fr s}_c,\hbox{\fr s}_c]$ is trivial since $\hbox{\fr s}_c$ is abelian according to (\ref{sc-abelian}).
Since $\hbox{\fr s}_c \subset \hbox{\fr k}_x$, $\hbox{\fr s}_n \subset \hbox{\fr a} + \hbox{\fr n}$ and $\hbox{\fr k}_x$
centralizes $\hbox{\fr a}$ and normalizes $\hbox{\fr n}$, we have
$[\hbox{\fr s}_c,\hbox{\fr s}_n] \subset [\hbox{\fr k}_x,\hbox{\fr a} + \hbox{\fr n}] \subset \hbox{\fr n}$.
Finally, since $\hbox{\fr s}_n \subset \hbox{\fr a} + \hbox{\fr n}$ and $\hbox{\fr n}$ is the derived subalgebra
of $\hbox{\fr a} + \hbox{\fr n}$, we see that $[\hbox{\fr s}_n,\hbox{\fr s}_n] \subset \hbox{\fr n}$. Altogether this implies
$[\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr n}$, which readily yields $[\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr s}_n \cap \hbox{\fr n}$.
For the converse, we fix the element $B \in \hbox{\fr a}$ for which $[B,V] = V$ and $[B,Z] = 2Z$
holds for all $V \in \hbox{\fr v} = \hbox{\fr g}_{\alpha}$ and $Z \in \hbox{\fr z} = \hbox{\fr g}_{2\alpha}$.
Because of (\ref{a-in-sn}) there exists an elements $\tilde{B} \in \hbox{\fr s}_c$
so that $\tilde{B} + B \in \hbox{\fr s}$. Let $X = V + Z \in \hbox{\fr v} + \hbox{\fr z}$ be an arbitrary
element in the orthogonal complement of $[\hbox{\fr s},\hbox{\fr s}]$ in $\hbox{\fr s}_n \cap \hbox{\fr n}$. Then there exists
a vector $\tilde{X} \in \hbox{\fr s}_c$ so that $\tilde{X} + X \in \hbox{\fr s}$, and we have
$$
0 = \langle X , [\tilde{B} + B, \tilde{X} + X] \rangle
= \langle X, [\tilde{B},\tilde{X}] + [\tilde{B},X] + [B,\tilde{X}] + [B,X] \rangle\ .
$$
Since $\hbox{\fr s}_c$ is abelian we have $[\tilde{B},\tilde{X}] = 0$. Since ${\rm ad}(\tilde{B})$ is
a skewsymmetric transformation we have $\langle X,[\tilde{B},X] \rangle = 0$. And since
$\hbox{\fr s}_c \subset \hbox{\fr k}_x$ and $\hbox{\fr k}_x$ centralizes $\hbox{\fr a}$ we have $[B,\tilde{X}] = 0$. This implies
$$
0 = \langle X,[B,X] \rangle = \langle V+Z , V+2Z \rangle =
\langle V,V \rangle + 2 \langle Z,Z \rangle\ ,
$$
and hence $V = 0 = Z$. Thus $X = 0$, which implies that the orthogonal complement of
$[\hbox{\fr s},\hbox{\fr s}]$ in $\hbox{\fr s}_n \cap \hbox{\fr n}$ is trivial. This establishes the proof of (\ref{derived-in-sn}).

Our next aim is to prove that
\begin{equation}\label{normal-in-v}
\nu_oF \subset \hbox{\fr v} = \hbox{\fr g}_\alpha\ .
\end{equation}
From (\ref{normal-in-n}) we already know that $\nu_oF \subset \hbox{\fr n}$.
If $\nu_oF \cap \hbox{\fr v} \neq 0$, we readily get $\nu_oF \subset \hbox{\fr v}$, because
$\hbox{\rm Ad}(H_o)$ acts transitively on the unit sphere in $\nu_oF$ and preserves $\hbox{\fr v}$.
Now assume that $\nu_oF \cap \hbox{\fr v} = 0$. Then (\ref{derived-in-sn})
implies that the canonical projection of $[\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr v} + \hbox{\fr z}$ onto $\hbox{\fr v}$ is
the entire space $\hbox{\fr v}$. Thus, for each $V \in \hbox{\fr v}$ there exists an element $V^\prime
\in \hbox{\fr z}$ so that $V + V^\prime \in [\hbox{\fr s},\hbox{\fr s}]$. Since $[\hbox{\fr s},\hbox{\fr s}]$ is a subalgebra, we get
$$
[V,W] = [V + V^\prime, W + W^\prime] \in [\hbox{\fr s},\hbox{\fr s}]
$$
for all $V,W \in \hbox{\fr v}$. But since $[\hbox{\fr v},\hbox{\fr v}] = [\hbox{\fr g}_{\alpha},\hbox{\fr g}_{\alpha}] = \hbox{\fr g}_{2\alpha} = \hbox{\fr z}$
this implies $\hbox{\fr z} \subset [\hbox{\fr s},\hbox{\fr s}]$ and hence $\nu_oF \subset \hbox{\fr v}$. This establishes the proof
of (\ref{normal-in-v}).

From (\ref{normal-in-v}) we see that there exists a linear subspace $\hbox{\fr v}_o$ of $\hbox{\fr v}$ so that
$\hbox{\fr s}_n = \hbox{\fr a} + \hbox{\fr v}_o + \hbox{\fr z}$. Using the Lie algebra structure of $\hbox{\fr a} + \hbox{\fr n}$, we get:
\begin{equation}\label{sn-subalgebra}
\hbox{\fr s}_n\ {\rm is\ a\ subalgebra\ of}\ \hbox{\fr a} + \hbox{\fr n}\ .
\end{equation}

Let $S_n$ be the connected subgroup of $AN$ with Lie algebra $\hbox{\fr s}_n$. Our next aim is to show
that the orbit $S_n \cdot o$ of $S_n$ through $o$ coincides with the singular orbit $F$.
For this purpose we define
$$
\hbox{\fr t} = \hbox{\fr k}_x \cap \hbox{\fr s}\ \subset \hbox{\fr s}_c\ \ {\rm and}\ \ \ \hbox{\fr s}^\prime = \mathbb R(\tilde{B} + B) + [\hbox{\fr s},\hbox{\fr s}]\ ,
$$
where $B \in \hbox{\fr a}$ and $\tilde{B} \in \hbox{\fr s}_c$ are defined as above.
Since $\hbox{\fr t} \subset \hbox{\fr s}_c$, $\tilde{B} \in \hbox{\fr s}_c$
and $\hbox{\fr s}_c$ is abelian we have $[\hbox{\fr t},\tilde{B}] = 0$. And since $\hbox{\fr t} \subset \hbox{\fr k}_x$ and $\hbox{\fr k}_x$
centralizes $\hbox{\fr a}$ we have $[\hbox{\fr t},B] = 0$. Clearly, we also have $[\hbox{\fr t},[\hbox{\fr s},\hbox{\fr s}]] \subset [\hbox{\fr s},\hbox{\fr s}]
\subset \hbox{\fr s}^\prime$ since $\hbox{\fr t} \subset \hbox{\fr s}$ and $[\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr s}$. Altogether this
implies $[\hbox{\fr t},\hbox{\fr s}^\prime] \subset \hbox{\fr s}^\prime$. Moreover, since $\hbox{\fr s}^\prime \subset \hbox{\fr s}$, we
have $[\hbox{\fr s}^\prime,\hbox{\fr s}^\prime] \subset [\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr s}^\prime$, which shows that
$\hbox{\fr s}^\prime$ is a subalgebra of $\hbox{\fr s}$. It follows that $\hbox{\fr s}^\prime$ is an ideal in $\hbox{\fr s}$ and
$\hbox{\fr s} = \hbox{\fr t} + \hbox{\fr s}^\prime$ (semidirect sum). Let $S^\prime$ be the connected subgroup of $S$ with
Lie algebra $\hbox{\fr s}^\prime$. Since $\hbox{\fr t} \subset \hbox{\fr h}_o$ we see that $S^\prime$ acts transitively on
$F$, i.e., $S^\prime \cdot o = F$.
For all $V+Z \in [\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr n} = \hbox{\fr v} + \hbox{\fr z}$ we have
\begin{equation}\label{formula}
[\tilde{B},V+Z] + V + 2Z = [\tilde{B},V+Z] + [B,V+Z] = [\tilde{B} + B,V+Z] \in \hbox{\fr s}^\prime
\end{equation}
since $\hbox{\fr s}^\prime$ is a subalgebra. But $A$ and $V+Z$ are in $\hbox{\fr s}_n$ according to
(\ref{a-in-sn}) and (\ref{derived-in-sn}), and since $\hbox{\fr s}_n$ is a
subalgebra by (\ref{sn-subalgebra}),
we have $V + 2Z = [B,V+Z] \in \hbox{\fr s}_n \cap \hbox{\fr n} = [\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr s}^\prime$
by (\ref{derived-in-sn}). By (\ref{formula}) this implies $[\tilde{B},V+Z] \in \hbox{\fr s}^\prime$.
But $\tilde{B} \in \hbox{\fr s}_c \subset \hbox{\fr k}_x$ and thus $\hbox{\rm ad}(\tilde{B})$
leaves $\hbox{\fr v}$ and $\hbox{\fr z}$ invariant, which implies that $[\tilde{B},V+Z] \in \hbox{\fr s}^\prime \cap \hbox{\fr n}
\subset [\hbox{\fr s},\hbox{\fr s}] \subset \hbox{\fr s}_n$. We thus have proved that $\hbox{\fr s}_n$ is normalized by $\tilde{B}$,
i.e., $[\tilde{B},\hbox{\fr s}_n] \subset \hbox{\fr s}_n$. Let ${\rm Exp}$ be the Lie exponential map of $\hbox{\fr g}$.
We now get
$$
F = S^\prime \cdot o \subset {\rm Exp}(\mathbb R \tilde{B})S_n \cdot o = S_n {\rm Exp}(\mathbb R \tilde{B}) \cdot o
= S_n \cdot o
$$
since ${\rm Exp}(\mathbb R \tilde{B})$ normalizes $S_n$ and ${\rm Exp}(\mathbb R \tilde{B}) \cdot o \subset
H_o \cdot o = o$. Finally, by (\ref{dimensions})
the dimensions of $F$ and $S_n$ coincide, and since both $F$ and $S_n \cdot o$ are
complete, we must have $F = S_n \cdot o$. We thus have proved:

\begin{thm}\label{mainresult}
Let $H$ be a connected subgroup of $G = I^o(\mathbb F H^n)$ that acts on $\mathbb F H^n$
with cohomogeneity one and with a non-totally geodesic singular orbit $F$.
Then there exists a unique point $x \in M(\infty)$ that is fixed under
the induced action of $H$ on $M(\infty)$.
Let $o \in F$, $K$ the isotropy group of $G$ at $o$, and $\hbox{\fr g} = \hbox{\fr k} + \hbox{\fr a} + \hbox{\fr n}$
the Iwasawa decomposition of $\hbox{\fr g}$ that is induced by $o$ and $x$.
Then there exists a subalgebra
$\hbox{\fr s}$ of $\hbox{\fr a} + \hbox{\fr n}$ of the form $\hbox{\fr s} = \hbox{\fr a} + \hbox{\fr v}_o + \hbox{\fr z}$ with some linear
subspace $\hbox{\fr v}_o$ of $\hbox{\fr v}$, so that $F$ is the orbit of the connected subgroup $S$ of $AN$
with Lie algebra $\hbox{\fr s}$.
\end{thm}

\section{The classification}

In this section we discuss the classification of cohomogeneity one actions on
noncompact symmetric spaces of rank one up to orbit equivalence. Recall that such an action
has either no singular orbit or exactly one singular orbit.

\smallskip
{\sc No singular orbit.}
In \cite{BT1} it was shown that there exist only two such actions without a singular orbit.
The first one is given by the action of the nilpotent group $N$ in an Iwasawa decomposition
$G = KAN$ of $G = I^o(\mathbb F H^n)$,
and the orbits form a horosphere foliation. The second one is given by the subgroup $S$
of $AN$ with Lie algebra $\hbox{\fr s} = \hbox{\fr a} + \hbox{\fr v}_o + \hbox{\fr z}$, where $\hbox{\fr v}_o$ is a linear subspace
of $\hbox{\fr v}$ with codimension one. The corresponding foliation has exactly one minimal leaf
and has been investigated in detail in \cite{Be}. In case of $\mathbb R H^n$ the minimal leaf
is a totally geodesic $\mathbb R H^{n-1} \subset \mathbb R H^n$.

\smallskip
{\sc Totally geodesic singular orbit.}
The cohomogeneity one actions on $\mathbb F H^n$ with a totally geodesic singular
orbit $F$ are given by:\\
\indent $M = \mathbb R H^n : F \in \{\hbox{\rm pt},\mathbb R H^1,\ldots,\mathbb R H^{n-2}\}$;\\
\indent $M = \mathbb C H^n : F \in \{\hbox{\rm pt},\mathbb C H^1,\ldots,\mathbb C H^{n-1},\mathbb R H^n\}$;\\
\indent $M = \mathbb H H^n : F \in \{\hbox{\rm pt},\mathbb H H^1,\ldots,\mathbb H H^{n-1},\mathbb C H^n\}$;\\
\indent $M = \mathbb O H^2 : F \in \{\hbox{\rm pt},\mathbb O H^1,\mathbb H H^2\}$.\\
Here, $\hbox{\rm pt}$ is a point in $\mathbb F H^n$, and the corresponding cohomogeneity one action is
just the action of the isotropy group of $I^o(\mathbb F H^n)$ at that point. More details about
this can be found in \cite{BB}.

\smallskip
{\sc Non-totally geodesic singular orbit.}
We now come to the classification of cohomogeneity one actions with a non-totally geodesic
singular orbit $F$. We will use the same notation as in the previous section.
Let $H$ be the connected component of the group of
isometries of $M$ that leave $F$ invariant. By Theorem \ref{mainresult}
there exists a unique point $x \in M(\infty)$ that is fixed under the induced action
of $H$ on $M(\infty)$. Let $o \in F$, $K$ the isotropy group of $G$ at $o$,
and $\hbox{\fr g} = \hbox{\fr k} + \hbox{\fr a} + \hbox{\fr n}$
the Iwasawa decomposition of $\hbox{\fr g}$ that is induced by $o$ and $x$.
Using again Theorem \ref{mainresult}, there exists a subalgebra
$\hbox{\fr s}$ of $\hbox{\fr a} + \hbox{\fr n}$ of the form $\hbox{\fr s} = \hbox{\fr a} + \hbox{\fr v}_o + \hbox{\fr z}$ with some linear
subspace $\hbox{\fr v}_o$ of $\hbox{\fr v}$,
so that $F$ is the orbit of the connected subgroup $S$ of $AN$
with Lie algebra $\hbox{\fr s}$.
From the construction it is clear that the identity component of $H_o$
coincides with the identity component $N_K^o(\hbox{\fr s})$ of the normalizer $N_K(\hbox{\fr s})$
of $\hbox{\fr s}$ in $K$. In order that $H$ acts with cohomogeneity
one it is therefore necessary and sufficient that the action of $N_K^o(\hbox{\fr s})$
on the normal space $\nu_oF$ is transitive on the unit sphere in $\nu_oF$.
Note that $N_K^o(\hbox{\fr s}) \subset K_x$.
Since all Iwasawa decompositions of $\hbox{\fr g}$ are conjugate to each other
under an inner automorphism
of $\hbox{\fr g}$, it therefore remains to classify all subalgebras $\hbox{\fr s}$ of $\hbox{\fr a} + \hbox{\fr n}$ of the form
$\hbox{\fr s} = \hbox{\fr a} + \hbox{\fr v}_o + \hbox{\fr z}$ with some linear subspace $\hbox{\fr v}_o$ of $\hbox{\fr v}$ such that
$N_K^o(\hbox{\fr s})$ acts transitively on the unit sphere in $\hbox{\fr v}_o^\perp$, the orthogonal
complement of $\hbox{\fr v}_o$ in $\hbox{\fr v}$. This proves the first part of the following Theorem:

\begin{thm}\label{congruency}
Let $\hbox{\fr g} = \hbox{\fr k} + \hbox{\fr a} + \hbox{\fr n}$ be the Iwasawa decomposition induced by
$o \in M$ and $x \in M(\infty)$.
\begin{enumerate}
\item[{\rm (i)}]
Let $\hbox{\fr v}_o$ be a linear subspace of $\hbox{\fr v}$
so that $\dim \hbox{\fr v}_o^\perp \geq 2$ and $N^o_{K_x}(\hbox{\fr v}_o)$ acts
transitively on the unit sphere in $\hbox{\fr v}_o^{\perp}$.
Then the connected subgroup of $G$ with Lie algebra
$N^o_{\hbox{\ffr k}_x}(\hbox{\fr v}_o) + \hbox{\fr a} + \hbox{\fr v}_o + \hbox{\fr z}$
acts on $M$ with cohomogeneity one so that the orbit
through $o$ is singular.
Furthermore, every cohomogeneity one action on $M$ with a non-totally
geodesic singular orbit can be obtained in this way
up to orbit equivalence.
\item[{\rm (ii)}]
Let $\hbox{\fr v}_o$ and $\hbox{\fr v}^\prime_o$ be linear subspaces of $\hbox{\fr v}$ as in (i),
and assume that the corresponding cohomogeneity one actions
have non-totally geodesic singular orbits.
Then, these actions are orbit equivalent if and only if
there exists an isometry $k \in K_x$ so that ${\rm Ad}(k)\hbox{\fr v}_o
= \hbox{\fr v}_o^\prime$.
\end{enumerate}
\end{thm}

{\it Proof.} It remains to prove part (ii).
The ``if''-part of the statement is obvious.
Conversely, assume that the two cohomogeneity one actions are orbit equivalent.
Then the corresponding singular orbits, say $S$ and $S^\prime$,
are congruent under an isometry $k$ of $M$.
We may assume that $k$ fixes $o$.
By construction, the normalizers $N_G(S)$ and $N_G(S^\prime)$ fix $x$,
the point at infinity that determines our Iwasawa decomposition.
Then $k$ must fix $x$ as well, since $k N_G(S) k^{-1} = N_G(S^\prime)$ and
$x$ is the unique fixed point in $M(\infty)$ of $N_G(S)$ and of $N_G(S^\prime)$.
Therefore we conclude that ${\rm Ad}(k)\hbox{\fr v}_o = \hbox{\fr v}_o^\prime$.
\hfill \hbox{\rm q.e.d.}

\medskip
We now discuss the four different hyperbolic spaces individually.

\smallskip
\fbox{$M = \mathbb R H^n$} It follows from the classification of isoparametric hypersurfaces
in $\mathbb R H^n$ by Cartan \cite{Ca} that there exist no such actions. Since a singular orbit
of a cohomogeneity one action is necessarily minimal, one can also apply a result by
Di Scala and Olmos \cite{DO} stating that every minimal homogeneous submanifold of $\mathbb R H^n$ is
totally geodesic.
The classification also follows easily from Theorem \ref{congruency}:
Assume there is a cohomogeneity one action on $\mathbb R H^n$
with a non-totally geodesic singular orbit $F$.
Theorem \ref{congruency} implies that the action is orbit equivalent to
the $H$-action induced from $\hbox{\fr h} = N^o_{\hbox{\ffr k}_x}(\hbox{\fr v}_o) + \hbox{\fr a} + \hbox{\fr v}_o$
for some suitable subspace $\hbox{\fr v}_o$ of $\hbox{\fr v}$.
But for such an $H$-action the orbit $F = H \cdot o$ is totally geodesic,
which is a contradiction.

\smallskip
\fbox{$M = \mathbb C H^n$}
In this case the K\"{a}hler structure on $\mathbb C H^n$ induces a complex structure $J$ on
$\hbox{\fr v}$ so that $\hbox{\fr v}$ is isomorphic to $\mathbb C^{n-1}$ as a complex vector subspace.
Let $\hbox{\fr v}_o$ be a linear subspace of $\hbox{\fr v}$ so that $\dim_{\mathbb R} \hbox{\fr v}_o^\perp \geq 2$.
Recall that the K\"{a}hler angle of a nonzero vector $v \in \hbox{\fr v}_o^\perp \subset \mathbb C^{n-1}$
is defined as the angle between $Jv$ and $\hbox{\fr v}_o^\perp$. In order that $N_K^o(\hbox{\fr s})$ acts
transitively on the unit sphere in $\hbox{\fr v}_o^\perp$ it is necessary that the K\"{a}hler
angle of $\hbox{\fr v}_o^\perp$ does not depend on the choice of the unit vector in $\hbox{\fr v}_o^\perp$.
We thus assume that for all nonzero vectors $v \in
\hbox{\fr v}_o^\perp$ the K\"{a}hler angle is equal to some $\varphi \in [0,\pi/2]$.
In the special case that $\varphi = 0$, $\hbox{\fr v}_o^\perp$ is a complex subspace of $\hbox{\fr v}$,
and if $\varphi = \pi/2$ then $\hbox{\fr v}_o^\perp$ is a real subspace of $\hbox{\fr v}$.
The subspaces of complex vector spaces with constant K\"{a}hler angle
have been classified in \cite{BB}. For $\varphi = 0$ we just have the complex subspaces
and for $\varphi = \pi/2$ the real subspaces, and in both cases the congruence classes
(under the action of $K_x = U(n-1)$ on $\hbox{\fr v} = \mathbb C^{n-1}$)
are parametrized by the complex resp.\ real dimension. For $\varphi \in (0,\pi/2)$ there exists
exactly one congruence class of subspaces with constant K\"{a}hler angle $\varphi$
for each dimension $0 < 2k \leq n-1$.
For any such subspace the resulting
action on $\mathbb C H^n$ is of cohomogeneity one and $F$ is a non-totally geodesic
singular orbit unless $\varphi = 0$ (then $F$ is a totally geodesic complex
submanifold). Using Theorem \ref{congruency} we therefore conclude:

\begin{thm}
The moduli space of all cohomogeneity one actions on $\mathbb C H^n$, $n \geq 2$,
with a non-totally geodesic singular orbit (up to orbit equivalence) is isomorphic to
the disjoint union
$$
\{2,\ldots,n-1\} \cup \left((0,\pi/2) \times \{2k \mid k \in \mathbb Z\ ,\ 0 < 2k < n\}\right)\ .
$$
The integer in $\{2,\ldots,n-1\}$ indicates the codimension of the singular orbit if the
normal spaces are real, and the integer in $\{2k \mid k \in \mathbb Z\ ,\ 0 < 2k < n\}$ indicates
the codimension of the singular orbit if the normal spaces have constant K\"{a}hler angle
$\varphi \in (0,\pi/2)$.
\end{thm}

\begin{cor}
Any singular orbit of a cohomogeneity one action on $\mathbb C H^2$ is
totally geodesic.
\end{cor}

Note that by this result we now have a complete classification of
the homogeneous hypersurfaces in $\mathbb C H^n$ for all $n \geq 2$. In
view of \cite{BSF}, we call a submanifold normally homogeneous if
it is homogeneous and if the slice representation acts
transitively on the unit sphere in the normal bundle. A singular
orbit of a cohomogeneity one action is clearly a normally
homogeneous submanifold. The above shows that for each $k \in
\{2,\ldots,n-1\}$ there exists, up to holomorphic congruence,
exactly one normally homogeneous submanifold $F_k$ of $\mathbb C H^n$
with real normal bundle of rank $k$, and for each $k \in
\{1,\ldots,[(n-1)/2]\}$ and each $\varphi \in (0,\pi/2)$ there
exists exactly one, up to holomorphic congruence, normally
homogeneous submanifold $F_{k,\varphi}$ of $\mathbb C H^n$ with normal
bundle of rank $2k$ and constant K\"{a}hler angle $\varphi$.

\begin{thm}
Let $M$ be a homogeneous hypersurface in $\mathbb C H^n$, $n \geq 2$.
Then $M$ is holomorphically congruent to one of the following
hypersurfaces:
\begin{itemize}
\item[(1)] a tube of radius $r \in \mathbb R_+$ around the totally
geodesic $\mathbb C H^k \subset \mathbb C H^n$ for some $k \in
\{0,\ldots,n-1\}$; \item[(2)] a tube of radius $r \in \mathbb R_+$ around
the totally geodesic $\mathbb R H^n \subset \mathbb C H^n$; \item[(3)] a
horosphere in $\mathbb C H^n$; \item[(4)] the minimal ruled real
hypersurface $S$ determined by a horocycle in a totally geodesic
$\mathbb R H^2 \subset \mathbb C H^n$, or an equidistant hypersurface to $S$;
\item[(5)] a tube of radius $r \in \mathbb R_+$ around the normally
homogeneous submanifold $F_k$ of $\mathbb C H^n$ with real normal bundle
of rank $k$, $k \in \{2,\ldots,n-1\}$; \item[(6)] a tube of radius
$r \in \mathbb R_+$ around the normally homogeneous submanifold
$F_{k,\varphi}$ of $\mathbb C H^n$ with normal bundle of rank $2k \in
\{2,\ldots,2[(n-1)/2]\}$ and constant K\"{a}hler angle $\varphi
\in (0,\pi/2)$.
\end{itemize}
\end{thm}

\smallskip
\fbox{$M = \mathbb H H^n$}
In this case the quaternionic K\"{a}hler structure on $\mathbb H H^n$ induces a quaternionic structure $\hbox{\fr J}$ on
$\hbox{\fr v}$ so that $\hbox{\fr v}$ is isomorphic to $\mathbb H^{n-1}$ as a (right) quaternionic vector subspace.
Let $\hbox{\fr v}_o$ be a linear subspace of $\hbox{\fr v}$ so that $\dim_{\mathbb R} \hbox{\fr v}_o^\perp \geq 2$.
In \cite{BB} the first author and Br\"uck
introduced the notion of a quaternionic K\"{a}hler angle, which
is defined as follows. Let $S^2$ be the two-sphere of all
almost Hermitian structures in $\hbox{\fr J}$.
For each nonzero vector $v \in \hbox{\fr v}_o^\perp$ and each $J \in S^2$ denote by
$\varphi(v,J)$ the K\"{a}hler angle of $Jv$ and $\hbox{\fr v}_o^\perp$ in the complex vector space
$(\hbox{\fr v},J)$. Since $S^2$ is compact, there exist minimum and maximum for these K\"{a}hler angles.
It was shown in \cite{BB} that for each nonzero $v$ there always exist a canonical basis $J_1,J_2,J_3$ of
elements in $S^2$ (i.e., $J_{\nu}J_{\nu+1} = J_{\nu+2} = - J_{\nu+1}J_{\nu}$, index modulo $3$)
such that $\varphi(v,J_1)$ is the minimum $\varphi_1(v)$ of these K\"{a}hler angles and $\varphi(v,J_3)$ is
the maximum $\varphi_3(v)$ of these K\"{a}hler angles. For any canonical basis with this
property the K\"{a}hler angle $\varphi_2(v) = \varphi(v,J_2)$ attains the same value. The
triple $\Phi(v) = (\varphi_1(v),\varphi_2(v),\varphi_3(v))$ of K\"{a}hler angles is called
the quaternionic K\"{a}hler angle of $\hbox{\fr v}_o^\perp$ with respect to $v$. For a cohomogeneity
one action the quaternionic K\"{a}hler angle of $\hbox{\fr v}_o^\perp$ must be independent of the
choice of the unit vector in $\hbox{\fr v}_o^\perp$.
In \cite{BB} several examples of subspaces
of $\mathbb H^{n-1}$ with constant quaternionic K\"{a}hler angle were given, but a complete
classification is still missing. The examples are as follows:

\medskip
(a) $\Phi = (0,0,0)$.
The linear subspaces of $\hbox{\fr v}$ with constant
quaternionic K\"ahler angle $\Phi =
(0,0,0)$ are the quaternionic subspaces. A linear subspace $V \subset \hbox{\fr v}$ is quaternionic
if $JV \subset V$ holds for all $J \in \hbox{\fr J}$.
For each integer $k$ with $0 < k < n$ there exists exactly one (up to orbit
equivalence) cohomogeneity one action on $\mathbb H H^n$ with a
singular orbit $F$ of real codimension $4k$ with the property that the
normal spaces of $F$ have constant quaternionic K\"{a}hler angle
$\Phi = (0,0,0)$, and $F$ is congruent to the totally
geodesic $\mathbb H H^{n-k} \subset \mathbb H H^n$.

\medskip
(b) $\Phi = (0,\pi/2,\pi/2)$.
The linear subspaces of $\hbox{\fr v}$ with constant
quaternionic K\"ahler angle $\Phi =
(0,\pi/2,\pi/2)$ are the totally complex subspaces.
A linear subspace $V \subset \hbox{\fr v}$ is totally complex if there exists
an almost Hermitian structure $J_1 \in \hbox{\fr J}$ such that $J_1V \subset V$
and $JV \subset V^\perp$ for all $J \in \hbox{\fr J}$ perpendicular to $J_1$.
For each integer $k \in \{1,\ldots,n-1\}$ there exists exactly one (up to orbit
equivalence) cohomogeneity one action on $\mathbb H H^n$ with a non-totally
geodesic singular orbit $F$ of real codimension $2k$ with the property that the
normal spaces of $F$ have constant quaternionic K\"{a}hler angle
$\Phi = (0,\pi/2,\pi/2)$.

\medskip
(c) $\Phi = (\pi/2,\pi/2,\pi/2)$.
The linear subspaces of $\hbox{\fr v}$ with constant quaternionic K\"ahler angle
$\Phi = (\pi/2,\pi/2,\pi/2)$ are the totally real subspaces.
A linear subspace $V \subset \hbox{\fr v}$ is totally real
if $JV \subset V^\perp$ holds for all $J \in \hbox{\fr J}$.
For each integer $k \in \{2,\ldots,n-1\}$ there exists exactly one (up to orbit
equivalence) cohomogeneity one action on $\mathbb H H^n$ with a non-totally
geodesic singular orbit $F$ of real codimension $k$ with the property that the
normal spaces of $F$ have constant quaternionic K\"{a}hler angle
$\Phi = (\pi/2,\pi/2,\pi/2)$.

\medskip
(d) $\Phi = (0,0,\pi/2)$.
The linear subspaces of $\hbox{\fr v}$ with constant quaternionic K\"ahler angle
$\Phi = (0,0,\pi/2)$ are the $3$-dimensional subspaces of the form
$(\hbox{\rm Im} \mathbb H)v$ for some unit vector $v \in \hbox{\fr v}$.
There exists exactly one (up to orbit
equivalence) cohomogeneity one action on $\mathbb H H^n$ with a non-totally
geodesic singular orbit $F$ of real codimension $3$ with the property that the
normal spaces of $F$ have constant quaternionic K\"{a}hler angle
$\Phi = (0,0,\pi/2)$.

\medskip
(e) $\Phi = (\varphi,\pi/2,\pi/2)$, $\varphi \in (0,\pi/2)$. The
linear subspaces of $\hbox{\fr v}$ with constant quaternionic K\"ahler
angle $\Phi = (\varphi,\pi/2,\pi/2)$, $\varphi \in (0,\pi/2)$, are
the linear subspaces with constant K\"{a}hler angle $\varphi$ in a
totally complex subspace $V$ of $\hbox{\fr v}$. Here, the K\"{a}hler angle
in $V$ is measured with respect to the almost Hermitian structure
$J_1$ as described in (b). For each integer $k \in
\{1,\ldots,[(n-1)/2]\}$ and each $\varphi \in (0,\pi/2)$ there
exists exactly one (up to orbit equivalence) cohomogeneity one
action on $\mathbb H H^n$ with a non-totally geodesic singular orbit $F$
of real codimension $2k$ with the property that the normal spaces
of $F$ have constant quaternionic K\"{a}hler angle $\Phi =
(\varphi,\pi/2,\pi/2)$.

\medskip
(f) $\Phi = (0,\varphi,\varphi)$, $\varphi \in (0,\pi/2)$. The
linear subspaces of $\hbox{\fr v}$ with constant quaternionic K\"ahler
angle $\Phi = (0,\varphi,\varphi)$, $\varphi \in (0,\pi/2)$, are
the complexifications of linear subspaces with constant K\"{a}hler
angle $\varphi$ in a totally complex subspace $\hbox{\fr w}$ of $\hbox{\fr v}$. More
precisely, let $J_2 \in \hbox{\fr J}$ be an almost Hermitian structure and
consider $\hbox{\fr v}$ as the complexification of $\hbox{\fr w}$ with respect to an
almost Hermitian structure $J_1 \in \hbox{\fr J}$ orthogonal to $J_2$, that
is $\hbox{\fr v} = \hbox{\fr w} + J_1 \hbox{\fr w}$ with a $J_2$-invariant linear subspace
$\hbox{\fr w} \subset \hbox{\fr v}$. Let $W$ be a linear subspace of the complex
vector space $(\hbox{\fr w},J_2)$ with constant K\"{a}hler angle $\varphi$.
Then the complexification of $W$ with respect to $J_1$ is a linear
subspace of $\hbox{\fr v}$ with constant quaternionic K\"ahler angle $\Phi
= (0,\varphi,\varphi)$. For each integer $k \in
\{1,\ldots,[(n-1)/2]\}$ and each $\varphi \in (0,\pi/2)$ there
exists exactly one (up to orbit equivalence) cohomogeneity one
action on $\mathbb H H^n$ with a non-totally geodesic singular orbit $F$
of real codimension $4k$ with the property that the normal spaces
of $F$ have constant quaternionic K\"{a}hler angle $\Phi =
(0,\varphi,\varphi)$.

\medskip
We conjecture that each cohomogeneity one action on $\mathbb H H^n$ with
a non-totally geodesic singular orbit is orbit equivalent to one of
these examples.
This is true for $n=2$, and for the case that the singular orbit has
codimension $2$.

\begin{thm}
The moduli space of all cohomogeneity one actions on $\mathbb H H^2$
with a non-totally geodesic singular orbit (up to orbit equivalence) is isomorphic to
the set $\{2,3\}$.
The number $k \in \{2,3\}$ parametrizes the unique (up to
orbit equivalence) cohomogeneity one
action on $\mathbb H H^2$ with a non-totally geodesic
singular orbit of codimension $k$.
\end{thm}

{\it Proof.} In case of $\mathbb H H^2$ the quaternionic vector space $\hbox{\fr v}$ has
quaternionic dimension one. It is easy to see that every $2$-dimensional
subspace of a one-dimensional quaternionic subspace has constant
quaternionic K\"{a}hler angle $\Phi = (0,\pi/2,\pi/2)$, and every
$3$-dimensional
subspace of a one-dimensional quaternionic subspace has constant
quaternionic K\"{a}hler angle $\Phi = (0,0,\pi/2)$. The result then
follows from (b) and (d) above. Note that codimension $4$ occurs for
quaternionic K\"{a}hler angle $\Phi = (0,0,0)$, which leads to a totally
geodesic singular orbit.
\hfill \hbox{\rm q.e.d.}

\begin{thm}
The moduli space of all cohomogeneity one actions on $\mathbb H H^n$, $n > 2$,
with a non-totally geodesic singular orbit with codimension $2$
(up to orbit equivalence) is isomorphic to the closed interval $[0,\pi/2]$.
The number $\varphi \in [0,\pi/2]$ parametrizes the unique (up to
orbit equivalence) cohomogeneity one
action on $\mathbb H H^n$ with a non-totally geodesic
singular orbit of codimension $2$ for which
the normal spaces have constant quaternionic K\"{a}hler angle
$\Phi = (\varphi,\pi/2,\pi/2)$.
\end{thm}

{\it Proof.} Every $2$-dimensional subspace of $\hbox{\fr v}$ has
constant quaternionic K\"{a}hler angle
$\Phi = (\varphi,\pi/2,\pi/2)$ for some $\varphi \in [0,\pi/2]$. The result
then follows from (b), (c) and (e) above. \hfill \hbox{\rm q.e.d.}

\smallskip
\fbox{$M = \mathbb O H^2$}
In \cite{BB} the first author and Br\"uck
classified all subspaces $\hbox{\fr v}_o$ of $\hbox{\fr v} = \mathbb R^8$ for which there exists a
subgroup of $K_x = \hbox{\rm Spin}(7)$ that acts transitively on the unit
sphere in $\hbox{\fr v}_o^\perp$. In fact, any subspace $\hbox{\fr v}_o$ of $\hbox{\fr v}$
with dimension $k \in \{1,2,4,5,6\}$ has this property, but there
are no $3$-dimensional subspaces with this property.
We denote by $G_k^+(\mathbb R^8)$ the Grassmann manifold of oriented $k$-planes
in $\mathbb R^8$, and by $G_k(\mathbb R^8)$ the Grassmann manifold of (unoriented)
$k$-planes in $\mathbb R^8$. It is clear that $G_k^+(\mathbb R^8)$ is a two-fold
covering of $G_k(\mathbb R^8)$, and that there is a natural isomorphism between
the Grassmann manifolds of $k$- and $(8-k)$-planes.

The Lie group $\hbox{\rm Spin}(7)$ acts on $\mathbb R^8$ by its irreducible $8$-dimensional
spin representation. This naturally induces actions of $\hbox{\rm Spin}(7)$ on
$G_k^+(\mathbb R^8)$ and $G_k(\mathbb R^8)$. For $k=1$,
it was proved by Borel \cite{Bo} that
$\hbox{\rm Spin}(7)$ acts transitively on $G_1^+(\mathbb R^8) = S^7$ and that $S^7 =
\hbox{\rm Spin}(7)/G_2$. For $k=2$ we also have a transitive action,
so that $G_2^+(\mathbb R^8) = \hbox{\rm Spin}(7)/U(3)$ (see e.g.\
\cite{Br}), and hence also $G_6^+(\mathbb R^8) = \hbox{\rm Spin}(7)/U(3)$.
Also for $k=3$ the action is transitive, and we have
$G_3^+(\mathbb R^8) = \hbox{\rm Spin}(7)/SO(4)$ (see e.g.\ \cite{On}), and thus also
$G_5^+(\mathbb R^8) = \hbox{\rm Spin}(7)/SO(4)$.

The action of $\hbox{\rm Spin}(7)$ on $G_4^+(\mathbb R^8)$ is not transitive,
but of cohomogeneity one (see \cite{Br} and \cite{HaL} for details).
One singular orbit of this action consists of the so-called
Cayley 4-planes in $\mathbb O$ introduced by Harvey and Lawson \cite{HaL}.
The submanifolds of $\mathbb O$ all of whose tangent spaces are Cayley 4-planes
are so-called Cayley submanifolds of $\mathbb O$ and provide a beautiful
example of a calibrated geometry. This singular orbit is isomorphic
to $\hbox{\rm Spin}(7)/(SU(2)^3/\mathbb Z_2)$, and the second singular orbit consists just
of the Cayley 4-planes with opposite orientation. This can also be seen in
the following way. Let $V \in G_3^+(\mathbb R^8)$ be an oriented $3$-plane in $\mathbb R^8$.
We know from the above that $Spin(7)$ acts transitively on $G_3^+(\mathbb R^8)$ and
the isotropy group at $V$ is some $SO(4) \subset Spin(7)$. There is a
unique unit vector $\xi$ in the orthogonal complement $V^\perp$ of $V$ in
$\mathbb R^8$ so that the $4$-plane $V \oplus \mathbb R\xi$ is a Cayley 4-plane. Then
$V \oplus \mathbb R(-\xi)$ is the same $4$-plane with opposite orientation.
The action of $SO(4)$ on the unit sphere $S^4$ in $V^\perp$ is the
standard action determined by the two fixed points $\pm\xi$. The principal
orbits are the $3$-spheres in $S^4$ with center $\xi$. Each such orbit parametrizes
in a canonical way a set of oriented $4$-planes in $\mathbb R^8$ containing the
$3$-dimensional subspace $V$. We now turn to the induced action of $Spin(7)$ on the
Grassmannian $G_4(\mathbb R^8)$ of unoriented $4$-planes in $\mathbb R^8$.
This action is clearly of cohomogeneity one as well. The two
singular orbits on $G_4^+(\mathbb R^8)$ become identified under the two-fold
covering map $G_4^+(\mathbb R^8) \to G_4(\mathbb R^8)$, and provide one singular orbit
of the action. The second singular orbit in $G_4(\mathbb R^8)$
is the projection of the principal orbit on $G_4^+(\mathbb R^8)$
containing $4$-planes of both orientations. This orbit contains the $4$-planes that
are constructed from the unique totally geodesic principal orbit of the
$SO(4)$-action on $S^4 \subset V^\perp$. The second singular orbit in
$G_4(\mathbb R^8)$ is therefore a $2$-fold subcovering of any principal orbit
and thus has the same dimension as the principal orbits.

From Theorem \ref{congruency}
it is clear that if the action of $\hbox{\rm Spin}(7)$ is transitive on
$G_k(\mathbb R^8)$, then all cohomogeneity one actions constructed from a
$k$-dimensional subspace of $\hbox{\fr v}$ are orbit equivalent. In the case
$k=4$, the cohomogeneity one actions induced from a $4$-dimensional
subspace of $\hbox{\fr v}$ up to orbit equivalence are in one-to-one
correspondence with the orbits of the action of $\hbox{\rm Spin}(7)$ on $G_4(\mathbb R^8)$.
Altogether this now implies:

\begin{thm}
The moduli space of all cohomogeneity one actions on $\mathbb O H^2$
with a non-totally geodesic singular orbit (up to orbit equivalence) is isomorphic to
the disjoint union
$$
\{2,3,6,7\} \cup (\{4\}\times [0,1])\ .
$$
The number $k \in \{2,3,6,7\}$ parametrizes the unique (up to
orbit equivalence) cohomogeneity one
action on $\mathbb O H^2$ with a singular orbit of codimension $k$.
The set $\{4\} \times [0,1]$ parametrizes
the cohomogeneity one actions on $\mathbb O H^2$ with a
singular orbit of codimension $4$ (up to orbit equivalence).
\end{thm}

The above result says that for each $k \in \{2,3,6,7\}$ there
exists exactly one, up to isometric congruence, normally
homogeneous submanifold $F_k$ of $\mathbb O H^2$ with normal bundle of
rank $k$, and for each $\varphi \in [0,1]$ there exists exactly
one, up to isometric congruence, normally homogeneous submanifold
$F_{4,\varphi}$ of $\mathbb O H^2$ with normal bundle of rank $4$. We now
have a complete classification of the homogeneous hypersurfaces in
the Cayley hyperbolic plane.

\begin{thm}
Let $M$ be a homogeneous hypersurface in $\mathbb O H^2$. Then $M$ is
isometrically congruent to one of the following hypersurfaces:
\begin{itemize}
\item[(1)] a geodesic hypersphere of radius $r \in \mathbb R_+$ in $\mathbb O
H^2$; \item[(2)] a tube of radius $r \in \mathbb R_+$ around the totally
geodesic $\mathbb O H^1 \subset \mathbb O H^2$; \item[(3)] a tube of radius $r
\in \mathbb R_+$ around the totally geodesic $\mathbb H H^2 \subset \mathbb O H^2$;
\item[(4)] a horosphere in $\mathbb O H^2$; \item[(5)] the minimal
homogeneous hypersurface $S$ in $\mathbb O H^2$, or an equidistant
hypersurface to $S$; \item[(6)] a tube of radius $r \in \mathbb R_+$
around the normally homogeneous submanifold $F_k$ of $\mathbb O H^2$ with
normal bundle of rank $k$, $k \in \{2,3,6,7\}$; \item[(7)] a tube
of radius $r \in \mathbb R_+$ around the normally homogeneous submanifold
$F_{4,\varphi}$ of $\mathbb O H^2$ with normal bundle of rank $4$ and
$\varphi \in [0,1]$.
\end{itemize}
\end{thm}

redpajama_set_name: RedPajamaArXiv


Strata+Hadoop London – May 6, 2015 – Cloudera, the leader in enterprise analytic data management powered by Apache Hadoop™, together with Confluent, is leading an initiative to create an open source API compatibility testing framework for advancing adoption of Apache Kafka™ by users, independent software vendors (ISVs), original equipment manufacturers (OEMs), and solution providers. The challenge for today's developers is the incremental risk, costs, and time associated with building products and applications that rely on the Kafka APIs and protocols, and making sure that the underlying platform implements those interfaces correctly. By having access to free tools for validating compatibility with those specs, development can be significantly streamlined.
This framework will be released in open source as part of the Apache Kafka project.
"This initiative, in conjunction with others we are bringing to market, reflects our commitment at Cloudera to preserve and accelerate the adoption of open source and open standards that create the foundation for the big data platform of the future," said Mike Olson, Chief Strategy Officer at Cloudera. "Customers benefit greatly from open standards such as Kafka, which allow for more data to be consumed, and at a faster rate, into next-generation workloads, thus transforming data into business value more quickly than legacy applications, and being able to automatically validate compatibility will help ensure that work vendors do never interferes with compatibility with the open source project."
Kafka, currently maintained at the Apache Software Foundation and initially designed and largely developed by Confluent employees, acts as a real-time, fault tolerant, highly scalable messaging system. Kafka exploits new data infrastructures made possible by massively parallel commodity clusters. Its fundamental ability to make high volume data available as a real-time stream for consumption is extremely beneficial for systems with very different requirements—from batch systems like Hadoop, to real-time systems that require low-latency access, and to stream-processing engines that transform the data streams as they arrive.
"Open standards have always been key to accelerating new technology adoption in our industry. Oracle ships and supports the entire Cloudera suite on Oracle Big Data Appliance and is committed to open standards like Apache Kafka, Apache Flume and Apache Sqoop. We welcome this effort."
"StreamSets, like Cloudera, is completely dedicated to furthering the adoption of open, multi-vendor standards as important building blocks in tomorrow's Big Data platform. Apache Kafka, which is an important component of the StreamSets offering, enables the ingestion of more data, more rapidly, into next-generation analytic workloads, so we support this Open Source Compatibility Framework initiative as a means of accelerating its uptake across the ecosystem."


text:
NASW Foundation inducts seven new Pioneers
by Paul Pace
Social workers who have blazed trails, challenged concepts and grown to become great mentors are among the latest inductees to the NASW Social Work Pioneers® program.
Supported by the NASW Foundation, the Pioneers represent role models for future generations of social workers striving to improve society.
The newest inductees are:
Iris Carlton-La Ney
Carlton-La Ney is a professor in the School of Social Work at the University of North Carolina at Chapel Hill. She has taught social work for 36 years at three North Carolina universities.
Her career has focused on issues involving African-Americans and she has contributed to the historical influence of African-American communities on the social work profession.
Carlton-La Ney has written multiple book chapters and articles on African-American aging, social work in the American south, African-American breast cancer survivors, and training African-American social workers.
One of her seminal publications, "African American Social Work Pioneers' Response to Need," published in "Social Work" in 1998, has been required reading in accredited baccalaureate social work programs across the country.
Carlton-La Ney has received the Bell Ringer Image Award from Bennett College and the Feminist Scholar Honoree Award from the Women's Council of the Council on Social Work Education.
The University of Maryland-Baltimore School of Social Work identified her as one of America's Most Influential Social Workers and in 2012 selected her as one of the school's 50 Heroes for Justice.
Yvonne Chase
Chase's leadership in governmental, nonprofit, private sector and university-based efforts has served as a role model for other social workers.
The assistant professor at the University of Alaska Anchorage, Department of Human Services, has taken some of the toughest jobs in human services, and has created and developed new human services.
She is a well-known speaker on topics related to the Indian Child Welfare Act. Her governmental roles span child welfare, juvenile justice, children's mental health addictions, prisons, and other sectors.
She has served in top leadership roles and has proven herself to be a true social worker in her support of those who are disenfranchised and marginalized. She has been a nationally recognized leader building more effective tribal state agreements and has helped states increase their collaborative child welfare practices with tribes. In Alaska, she created the first comprehensive report on the state mental health system.
At NASW, she has served as president of the NASW Alaska Chapter and has been a member of the Delegate Assembly and national board of directors, among other NASW positions.
Chase is a former director on the NASW Assurance Services board, serves on the board of the newly formed Assurance Services RRG, and is currently chairwoman of the NASW Chapter/National Modernization Task Force. She also serves as chairwoman of the Gosnell Scholarship Committee.
Chase received the 2010 Shining Star award, presented by the National Association of Black Elected Officials.
Sister Ann Patrick Conrad
Conrad recently retired as an associate professor at the National Catholic School of Social Service at The Catholic University of America. She continues to be active as a fellow at the CUA Institute for Policy Research and Catholic Studies and in other campus activities. She has served in many roles at the school, including dean, chairwoman of the Master of Social Work Program, director of admissions and assistant to the director of field instruction.
Conrad aided disaster victims while in high school and later used her experience from that time to provide social services to families through Catholic Charities during disaster-relief efforts.
As an educator, Conrad became one of the first and leading social workers to introduce ethics into the curriculum and developed a course on Theories of Social Justice.
Conrad has also been an active participant in NASW and has frequently been invited as a presenter on professional ethics, social service outreach, social justice and human rights.
Wayne Duehn
Duehn is professor emeritus with the School of Social Work at the University of Texas at Arlington.
His contributions in journals and with social work researchers in the field of sexuality marked an important transition for disseminating new information for students and seasoned professionals.
His investigation into child sexual abuse, child welfare investigation, and treatment was new and unexplored. His work from the past 30 years has served as an example of how one person can create a vision and passion for changing the field and community with respect to understanding child sexual abuse.
Duehn was a research associate at the Masters and Johnson Institute and he was co-author of "Beyond Sexual Abuse:The Healing Power of Adoptive Families," which is an outgrowth of an ongoing educational program of the Three Rivers Adoption Council in Pittsburgh, Pa. This project is designed to develop educational materials to assist adoptive families in parenting the sexually abused child.
Most recently, Duehn developed a child abuse-prevention program for the Department of Defense Dependents Schools, which has been implemented worldwide.
In 1969, he was first awarded with a "career teaching award" with NIMH and he has been the recipient of the UTA School of Social Work Fernando Torgerson award for teaching excellence four times.
David Epperson
Epperson, who died in 2011, was a dean emeritus and professor emeritus at the University of Pittsburgh School of Social Work.
He is remembered for a career devoted to social work values, education, racial equality, and social and civic improvement, but also as a man who forged social work connections both nationally and internationally.
Epperson was the only African-American dean of the University of Pittsburgh School of Social Work and the longest-serving dean of social work in the country at 29 years. Under Epperson's leadership, the school's enrollment more than tripled.
His work in Pittsburgh's community development organizations, foundations, and racial-equality efforts was complemented by his more than 50 years of working with the YMCA at the local, national and international levels and increasing the diversity and advocacy within the organization.
For 20 years he served on the international committee of the YMCA as chairman of its Office for Africa and developing exchange programs with YMCAs in Kenya, Uganda and Ethiopia.
He worked to improve racial relations and fight racial disparities in employment, housing and education and designed and ran Pittsburgh's poverty program. Epperson was a member of the board of directors for both the Council on Social Work Education and the National Association of Deans and Directors.
Salome Raheim
Raheim is dean and professor at the University of Connecticut School of Social Work and the first African-American to be appointed as dean.
She has provided diversity and cultural-competence training and consultation to schools, universities, human service organizations and businesses throughout the U.S. and on four continents.
Her national leadership positions include serving as a fellow in the New York Academy of Medicine Leadership in Aging Academy.
In 2013, she was one of 150 leading social work educators from across the country who attended a White House briefing on the expanding role of social work in today's changing health care environment, focusing on the Patient Protection and Affordable Care Act.
When Raheim was the director of the School of Social Work at the University of Iowa, she was responsible for increasing the number of faculty of color. The Cultural Competence Initiative, which she led, enhanced diversity within the university. The initiative received the 2002 Catalyst Award, given for innovative programs, policies and activities that enhance diversity within the university.
In 2011 and 2013, the NAACP named her one of the "100 Most Influential Blacks in Connecticut."
Michael Sheridan
Sheridan is a nationally recognized scholar of religion, spirituality and social work and an associate tenured professor at the National School of Social Service at Catholic University. As director of Catholic University's Center for Spirituality and Social work, she focuses on coordinating spirituality and social work.
In past years, she has taught multiple courses at Catholic University, the University of Windsor and Virginia Commonwealth University, focusing on research, spirituality and social work. Sheridan is currently an educational consultant and trainer for the National Institute of Health on social justice issues in medicine.
With an interest in issues of injustice within the correctional system, Sheridan has worked in maximum-security prisons as a group facilitator, focusing on the children of incarcerated parents. She continues to provide leadership in relating religion and spirituality in social work practice.
She was a founding member of the CSWE Religion and Spirituality Clearing House where she is committed to promoting social workers' knowledge, values and skills for ethical and effective practice that takes into account the diverse religions and spiritual perspectives among clients and communities.
Sheridan's efforts have been noted to move the profession into areas that have the potential to affect client progress and arrays of coping skills.

redpajama_set_name: RedPajamaCommonCrawl


text:
Q: Why is it not possible to loop a call execute to get multiple responses inside my IntentService via Retrofit? Why is it not possible to loop a call execute to get multiple responses inside my IntentService via Retrofit?
Please see my code:
public class UpdateAgendaService extends IntentService {

    public static final int STATUS_RUNNING = 0;
    public static final int STATUS_FINISHED = 1;
    public static final int STATUS_ERROR = 2;
    private Agenda agenda;
    public UpdateAgendaService() {
        super(UpdateAgendaService.class.getName());
    }

    @Override
    protected void onHandleIntent(Intent intent) {

        final ResultReceiver receiver = intent.getParcelableExtra("receiver");
        String[] dateWeek  = intent.getStringArrayExtra("dateWeek");
        if (dateWeek != null) {
            receiver.send(STATUS_RUNNING, Bundle.EMPTY);

            Bundle bundle = new Bundle();
            try {
                //Why is this not possible?
                List<Agenda> agendaList = getAgendaList(dateWeek);
                receiver.send(STATUS_FINISHED, bundle);
                }
            } catch (Exception e) {

                /* Sending error message back to activity */
                bundle.putString(Intent.EXTRA_TEXT, e.toString());
                receiver.send(STATUS_ERROR, bundle);
            }
        }
        Log.d(Utilities.TAG, "Service Stopping!");
        this.stopSelf();

    }

    private List<Agenda> getAgendaList(String[] upcomingWeekdates){
        List<Agenda> agendaList = null;
        for (int i = 0; i < upcomingWeekdates.length; i++) {
            String weekDay = upcomingWeekdates[i];
            agendaList.add(getAgenda(weekDay));
        }
        return agendaList;
    }
    private Agenda getAgenda(String date) {
        Agenda agenda = null;
        ApiService apiService = new QardioApi().getApiService();

        Call<Agenda> call = apiService.getAgenda(date);
        try {
            agenda = call.execute().body();
        } catch (IOException e) {
            e.printStackTrace();
        }
        return agenda;

    }
}

So the situation is that I have an API which has a url: http//:myapi.com/[date] which when called via retrofit gives me a JSON response of the Agenda(Events) for that specific day. What I want to do is to show the agenda(events) for the upcoming week, that is why I did it via loop given a String array of dates for the upcoming week. Imagine kind of like the Eventbrite app.
What I am doing wrong? I read somewhere that I should do this via JobQueue/Eventbus, should I be doing that? But I am a bit hesitant because I don't want to be using any more third party libraries. However, if that is the last case scenario I will probably just go with that then.

A: Nevermind guys. That was because of a very stupid mistake i made.
I just changed:
List<Agenda> agendaList = null;

to
List<Agenda> agendaList = new ArrayList<>();

redpajama_set_name: RedPajamaStackExchange


text:
Home » Agri-Business » UK and EU agree 'in principle' on rules regarding Northern Ireland following Brexit
UK and EU agree 'in principle' on rules regarding Northern Ireland following Brexit
Dec 8, 2020, 2:55pm
The EU and UK have today (Tuesday December 8) agreed 'in principle' the rules surrounding the implementation of the Protocol on Northern Ireland which forms part of the Brexit Withdrawal Agreement.
The co-chairs of the EU-UK Joint Committee – European Commission Vice-President Maros Sefcovic and the UK Chancellor of the Duchy of Lancaster, Michael Gove – yesterday held a political meeting to address the outstanding issues related to the implementation of the Withdrawal Agreement.
The protocol protects the Good Friday (Belfast) Agreement.
Statement of agreement
In a statement Sefcovic said that following "intensive and constructive work" over the past weeks by the EU and the UK, the two co-chairs can now announce their agreement in principle on all issues, in particular with regard to the Protocol on Ireland and Northern Ireland.
An agreement in principle has been found in the following areas, amongst others:
Border control posts / entry points specifically for checks on animals, plants and derived products;
Export declarations;
The supply of medicines;
The supply of chilled meats, and other food products to supermarkets;
A clarification on the application of state aid under the terms of the protocol.
The parties have also reached an agreement in principle with respect to the decisions the Joint Committee has to take before January 1, 2021.
This concerns the practical arrangements regarding the following:
The EU's presence in Northern Ireland when UK authorities implement checks and controls under the protocol;
Determining criteria for goods to be considered 'not at risk' of entering the EU when moving from Great Britain to Northern Ireland;
The exemption of agricultural and fish subsidies from state aid rules;
The finalisation of the list of chairpersons of the arbitration panel for the dispute settlement mechanism so that the arbitration panel can start operating as of next year;
Correction of errors and omissions in an annex of the protocol.
As a result of the agreed solutions, the UK will withdraw certain clauses in the UK Internal Market Bill which compromised the Northern Ireland protocol, and not introduce any similar provisions in its Taxation Bill.
This agreement in principle will now be subject to respective internal procedures in the EU and in the UK.
Once this is done, a fifth regular meeting of the EU-UK Joint Committee will be called to formally adopt them. This will take place in the coming days and before the end of the year.
Agri-Business EU northern ireland protocol UK


No meetings July and August of 2019. LLL Canada--Who we are and what we do: https://www.lllc.ca/thursdays-tipla-leche-league-canada-who-we-are%E2%80...Our meetings are discussion groups, with four general monthly topics that repeat throughout the year. The topics are 1. Advantages of Breastfeeding, 2. The Baby Arrives; the Family and the Breastfed Baby, 3. The Art of Breastfeeding and Avoiding Difficulties, and 4. Nutrition and Weaning. You can start attending at any time; you do not have to wait for meeting #1. There is always time to have breastfeeding questions answered, no matter what the topic is. A call or email to a Leader can often answer your questions and concerns. If you leave a message for a Leader and your call isn't returned within 24 hours, please phone again or try any other Leader. Contact information for LLL Chatham Leaders is on the following webpage: https://www.lllc.ca/lllc-chatham.
We meet from 10am until 11:30am, on the fourth Thursday of each month, except during June, November and December, when we meets earlier in the month. Babies are always welcome! BUS ROUTES (change in the summer): 1C Eastbound runs every 15 minutes, and crosses Forest Glade Dr. at Wildwood Drive, two blocks from the meeting. The Ottawa 4 Eastbound runs every 20minutes, and crosses Lauzon Rd. at Forest Glade Drive, less than one block from the meeting. Lauzon 10 Southbound from Tecumseh Mall, leaves every 35 minutes and goes right by the meeting place. The Crosstown 2 and the 1C connect with the Lauzon 10 at Tecumseh Mall.
To see all 2018 meeting dates, download the 2018 Meeting Dates from the bottom of the list of attachments, at the top of the page.
Come in the main entrance to the building, which is closest to the street. Park in the visitors spots to the right, or behind the building. The entrance from the side parking lot is only for the daycare.
The Windsor evening meeting, same location, is the fourth Wednesday at 7 pm, except in June, November and December when it meets earlier in the month.


text:
What's Next for DDoS Attacks?
on September 18, 2013 |
DDoS Attack Specialist Defend Against DDoS Denial of Service Denial of Service Attack DoS Attacks Stop DDoS Stop DDoS Attacks Stop DoS
Sept. 18 marks the one-year anniversary of Izz ad-Din al-Qassam Cyber Fighters' first announcement about distributed-denial-of-service attacks to be waged against the U.S. financial services industry
This self-proclaimed hacktivist group, which U.S. government officials have suggested is being backed by the Iranian government, has for the last 12 months targeted the online banking platforms of nearly every top 100 U.S. banking institution. The group has claimed it's attacking U.S. banks because of outrage over a YouTube movie trailer deemed offensive to Muslims.
The group's attacks against banks for the last several months have been unsuccessful at taking sites down. And its Phase 4 campaign is in a lull. Still, experts caution banking institutions against letting their guards down. And they warn that the government, media, healthcare and energy sectors could be among the next targets. That's because banking institutions have enhanced their defenses, so other sectors are easier targets.
Some experts, including McAfee Labs and Arbor Networks, expect that al-Qassam could join forces with the Syrian Electronic Army, a collective of attackers that supports Syrian President Bashar al-Assad, to attack a variety of U.S. websites. Experts urge organizations to update their DDoS defenses.
Botnet Remains Strong
Scott Hammack CEO of DDoS-mitigation provider Prolexic, says that despite the recent lull in al-Qassam's attacks, there's no indication that the group's botnet, known as Brobot, is waning.
"We have validated thousands of infected web servers that can potentially participate in future campaigns. We still don't see the scale of the attacks we saw five months ago, but we are definitely seeing a lot of probing," he says, which suggests more attacks are on the way.
DDoS attacks over the last five months have not been as large, but that should not be misinterpreted to mean Brobot has been retired, Hammack says. "Maybe Iran, with their new leaders, are saying 'Cool off a little bit.' But the gun is still loaded. They still have the arsenal at their disposal; they just haven't fired it in a while."
If al-Qassam were to unite with other cybergroups, such as the Syrian Electronic Army, it could mark a new era of cyberwarfare against the U.S., experts say.
"We have to realize this is cyberterrorism," says Ashley Stephenson, CEO of Corero Network Security. "The disruption, the publicity, the nuisance, the investment these banks have to make … the success of terrorism is not just the act itself, but the amount it costs the victims [for defenses]," he says.
Banking institutions and those in other sectors have to continually enhance and update their DDoS defenses, he says, "so that no one needs to panic on the day of an attack"
al-Qassam's attacks have served as a training ground for other attackers, says Dan Holden of DDoS-mitigation firm Arbor Networks.
"In terms of DDoS in general, we will see more DNS amplification attacks. It's not that difficult," he says.
A DNS amplification attack relies on a much larger list of DNS servers to amplify the attack. "The attackers are going to have to get better and bigger to take anyone down," Holden says.
That's because DDoS defenses have improved across the board, although some industries, such as banking, are further along in their mitigation strategies than others, Holden explains.
The real question about al-Qassam's future is not how its attacks might be waged, but rather who will be the target, he says.
Holden says he doesn't see the attacks themselves changing; but the targets will change. "I would be surprised if they continue attacking the banks," he says. "At this point, what is the point? Something's got to change."
Cyber-attackers Unite?
Because the Syrian Electronic Army and al-Qassam both have waged DDoS attacks to gain attention for their social and political causes, some experts expect them to join forces.
While al-Qassam has focused on banks, the SEA's primary targets have been media and government sites, according to McAfee Labs.
Now, DDoS experts, including McAfee Labs, argue both groups have similar interests in taking down U.S. sites, and by joining forces, they could use Brobot for a renewed purpose.
One industry expert, who asked not to be named, says the distinction between the SEA and al-Qassam has increasingly blurred.
"Isn't the Syrian Army likely the same guys as al-Qassam? And if you look at the geopolitical stuff, the two align," this expert says. "They're holding off attacks because they're waiting to see what happens with Syria, frankly. And the banks are defending well, so they will move on to a new target. Besides, using the movie as an excuse for attacks is not effective or really being believed anymore."
"The Syrian Electronic Army has said quite clearly that if the U.S. does anything [as far as military strikes] they are coming after us," Arbor Networks' Holden says. "And if there is any sympathy for that, it's a great excuse for QCF [al-Qassam Cyber Fighters] to repurpose and retool their botnet for something else. They could jump onboard there. This is the perfect excuse to change the story about attacking because of the video. This is the perfect cover."
If forces do unite, Holden says other critical infrastructure sectors, beyond banking, will likely be targeted, possibly defense contractors, rather than government itself.
"I don't foresee government being a target, but, instead, a weaker vertical," Holden says. "If anything government-focused were to be attacked, I would think it would be government contractors that would be somehow associated with Syria."
Holden says e-commerce sites also could be prime targets. "[Those sites] are obviously related to capitalism, like the banks, and the money lost would be huge," he says. "The impact would be very real, given how much commerce occurs over the Internet."
Media sites could be another target, Holden says.
Regulatory Oversight
Because of ongoing DDoS threats, more regulatory and legislative oversight related to how critical infrastructure industries address DDoS risks is likely on the way, says Corero's Stephenson.
"The attacks have heightened the need for guidance or legislation, whether that's from the SEC [Securities and Exchange Commission] or an international agency like the European Commission," he says. The need for more regulation and cross-industry collaboration has been highlighted by al-Qassam's attacks, Stephenson adds.
DDoS attacks have become a part of everyday business, and all sectors should be sharing threat intelligence, Stephenson says
"One of the things I took away from this last year is that the banks really have learned a lot," he says. "This type of cyberthreat is now business as usual. This is going to be a continuous threat and an ongoing risk of doing business online, and I don't think we're going to win here by keeping attack information secret."
The more organizations disclose about the attacks they suffer, the stronger defenses can be built, Stephenson says. "That's where the vendors come in," he says. "When they have an attack that is defeated, they can put the information together and pass that on to the authorities. A year ago, that wasn't happening like it is today," and there's always room for improvement.
Source: http://www.bankinfosecurity.com/whats-next-for-ddos-attacks-a-6074


Hello and thank you for visiting our website. A-Cap Cabinetry is a small company that specializes in custom cabinetry, cabinet refacing, and custom built furniture. By being a small company we are able to keep our prices low. We have much less over head than our competitors, and in return we are able to pass the savings on to you. The only draw back to this is that A-Cap Cabinetry can only take on so many projects per year. At A-Cap Cabinetry we strive for perfection every time. We only use the best materials, and always try to buy American made products. Our motto is a kitchen for every budget. By that we mean good solid professionally built and installed cabinetry for every budget. At A-Cap Cabinetry we have no high pressure sales people, so don't be afraid to give us a call.
So why do we call ourselves specialists?
There are two main reason why we call ourselves specialists. The first is because we have more than ten years exclusively dedicated to cabinetry and cabinet refacing, and the second is that A-Cap Cabinetry holds a C6 cabinet specialty license with the state of California. We are not a general contractor, or a counter top company that also does cabinetry.
Everyone knows what a person is talking about when they say new cabinetry, but what is cabinet refacing? Refacing is the simple procedure of removing your old doors, sanding down your old cabinet boxes and frames, and applying a new laminate in the color of your choice. Then all we need to do is install your new doors, new drawer boxes, and new drawer guides to give your kitchen a great new look and feel. With a reface almost anything is possible. We can change a cabinet that has doors into a cabinet that has drawers or the opposite. We can even add some new cabinets to an existing kitchen and make them blend in as if they were always there. Because we specialize in Refacing we are always up to date on all the newest products and techniques that the industry has to offer.
There are many different reasons to reface. One could be your happy with the layout of your cabinetry and just want to refresh the look of them. Another maybe you just put in a new shinny granite counter top and now you want your cabinets to look just as good; however The number one reason to reface is because with cabinet refacing you can get a Beautiful kitchen for half the cost of new cabinetry.
Don't forget to check out page two with styles and colors !!!


Queenstown, New Zealand is a beautiful part of the world ( brief look ). If you would also like to familiarize yourself with the area click here for maps.
If you have not traveled to New Zealand before then it would be wise to research the climate before embarking on your trip. Here are some general guidelines to help you out for now that includes guidelines on local amenities and activities.
Check the Queenstown Web Page for comprehensive information on what's happening in Queenstown for this time of year. There are many more details here about visiting Queenstown than you can imagine!
Want to find out about how much money you will need to take with you? Here are some helpful hints on costs.
If your interested in investments in New Zealand speak with New Zealand Settler Investments about possible investments and immigration to the country.


The Australian Competition and Consumer Commission (ACCC) has received 300 reports about a new immigration scam targeting migrants with more than $150,000 reported lost.
The scam sees people pretending to be from the Department of Immigration targeting migrants and visa holders, claiming there are problems with their immigration paperwork or visa status and they need to pay a fee to correct the problem and avoid deportation. These scammers often glean personal information from social media, making the demands seem more legitimate.
"Recently these scammers have been going one step further, threatening the arrest of loved ones or claiming they have already been arrested or detained. They demand payment through wire transfers or iTunes gift cards.
"Scammers may try to pressure migrants by calling incessantly and harassing them, even threatening to send the police to their houses. Simply hang up and do not respond. If you give your money to a scammer you will never see it again," said ACCC Deputy Chair Delia Rickard.
You can find out more information about the scam and how to protect yourself on the Australian Government's SCAM WATCH website.


Get one of the properties from the last received radio packet.
In addition to a number, string, or name-value pair, the packet received also contains other information about the transmission of the packet. You can get this additional information by selecting a property from the packet.
signal strength: the strength of the radio signal when the packet was received.
serial number: the serial number of the board sending the packet.
time: the time when the packet was sent.
serial number: the value is the serial number of the board sending the packet.
time: the value is the system time, in microseconds, of the sender at the time when the packet was sent.


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Paul Norell (* 11. Februar 1952 in London, England) ist ein neuseeländischer Filmschauspieler britischer Herkunft.

Leben
Paul Norell wurde in England geboren und lebt heute in Neuseeland.

Er ist vor allem durch seine Nebenrolle (König der Toten) im mit elf Academy Awards ausgezeichneten Fantasyfilm Der Herr der Ringe: Die Rückkehr des Königs von 2003 bekannt. Davor wirkte er in neuseeländischen Fernsehserien wie Hercules und Xena mit. Paul Norell gehörte neben Noel Appleby, Martyn Sanderson, Stephen Ure und Craig Parker zu einer Gruppe von 15 Schauspielern, die im Zuge des Erfolges der Trilogie die Produktionsgesellschaft auf Beteiligung am Gewinn durch Merchandising verklagten, mit der Klage jedoch scheiterten.

Weblinks

Einzelnachweise

Filmschauspieler
Neuseeländer
Brite
Geboren 1952
Mann

redpajama_set_name: RedPajamaWikipedia


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Ukrainian Ambassador Thanks CBC, Bulls for Support Efforts
The Durham Bulls hosted a Ukraine Night on Wednesday, August 24, 2022, on Ukrainian Independence Day. The proceeds of various efforts throughout the evening went to support relief efforts in Ukraine.
Capitol Broadcasting recently received a letter of thanks from Oksana Markarova, Ambassador of Ukraine to the United States, for efforts the company, and particularly the Durham Bulls, made to support the people of Ukraine.
"From turning the American Tobacco Historic District in Durham into a Ukraine Support zone, to flying blue and yellow colors alongside America's stars and stripes, to hosting Ukraine Night at the iconic Durham Bulls Athletic Park to celebrate the Ukraine and its 31st Independence Anniversary – the people of Ukraine are grateful for your support!" wrote Ambassador Markarova.
In May under the umbrella "CBC Cares," Capitol Broadcasting hosted "CBC Stands with Ukraine" across all of the company's campuses and highlighted agencies and organizations working to support the people of Ukraine. WRAL.com provided a list of vetted resources through which the Triangle community can support Ukraine without being scammed.
That CBC effort sparked an idea at the Durham Bulls, which resulted in a special night at the ballpark. Ukraine Night at the Bulls stemmed from CBC's initial involvement and desire for the divisions to raise awareness for Ukraine.
The Bulls came up with the idea of hosting a Ukraine Night to support the people of Ukraine. They already had a game schedule on Wednesday, August 24, which coincided with Ukrainian Independence Day.
Fans who purchased a department store gift card of any kind and donated to Ukrainanians in the Carolinas on game day received a voucher for a ticket for the game. The gift cards went directly to support Ukrainian refugees in the Triangle.
DBBC Director of Marketing & Communications Michael Ward explained how the event came to pass.
"We formed a relationship with Olena Kozlova-Pates very early last spring as she is the President of Ukrainians In The Carolinas support group that raises awareness and helps provide resources to refugees who have fled the country," he said.
The Bulls set the special night for the game between the Bulls and the Nashville Sounds at the DBAP.
"We did a Ukraine + Bulls shirt design that sold out within hours of a pre-sale," said Ward. "On the night, we had a Ukrainian choir sing their National Anthem, themed promotions, donated seats to Ukrainian refugees living in the Triangle, video board messages and much more. The stadium was decorated in Blue and Gold to show solidarity with Ukraine as well. It was a wonderful night. Proceeds from our shirt sale and ticket sales went to support relief efforts in Ukraine."
In early October, Corp President & COO Jimmy Goodmon received the letter from Ambassador Markarova thanking CBC for supporting Ukraine and its fight for freedom, particularly mentioning the Bulls efforts.
"Together with Ukrainians in the Carolinas, a North Carolina non-profit that has assisted the people of Ukraine over the last eight years and who had the privilege of assisting with the August 24th Ukraine Night at the Durham Bulls event, I would like to extend my sincer appreciation to your terrific team who made Ukraine Night possible!" she wrote. "I would like to especially thank Michael Ward for leading this effort, as well as Leslie Martin, Mike Birling, Peter Wallace, Chip Allen, Patrick Norwood, Andrew Green and Paxton Rembis for an unforgettable event celebrating Ukraine and its courageous people, especially welcoming and supporting Ukrainian refugees to the Triangle!"
Find Out More about "CBC Stands with Ukraine":
Capcom: CBC Cares Ukraine Wraps Five-Day Focus, Resources Remain Available
Capcom: CBC Cares Ukraine Continues with Highlights from CBC Divisions
Thanks to DBBC's Michael Ward for these Capcom photos & this Capcom video.
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The worker who is always engaged in work against the injunctions of the scripture, who is materialistic, obstinate, cheating and expert in insulting others, and who is lazy, always morose and procrastinating is said to be a worker in the mode of ignorance.
In the scriptural injunctions we find what sort of work should be performed and what sort of work should not be performed. Those who do not care for those injunctions engage in work not to be done, and such persons are generally materialistic. They work according to the modes of nature, not according to the injunctions of the scripture. Such workers are not very gentle, and generally they are always cunning and expert in insulting others. They are very lazy; even though they have some duty, they do not do it properly, and they put it aside to be done later on. Therefore they appear to be morose. They procrastinate; anything which can be done in an hour they drag on for years. Such workers are situated in the mode of ignorance.
One who does what should not be done is ayuktah. One who acts according his nature, so that whatever occurs in his mind he does rather than according to the instruction of the guru is prakrtah. One who offends others is naiskrtikah.
The jnanis must perform the sattvika tyaga described here, and must take shelter of the knowledge fixed in sattvika karma, must perform sattvika work, and must become a sattvika doer. This is the sannyasa of the jnani. This is my realization on this topic, how I have extracted this meaning of the section.
Not only these three items—the knowledge, action (bhakti yoga) and doer (bhakta)-are beyond the gunas, but rather everything related to bhakti is beyond the gunas according to the philosophy of bhakti.
Thus, for the devotees who are beyond the gunas, things related to bhakti such as knowledge, action, faith, residence and the happiness derived from it, are all beyond the gunas. For the jnanls who are in the mode of sattva, things related to jnana are all in the mode of sattva. Everything related to karmis acting in the mode of rajas is rajasic. For the unrestrained person in the mode of tamas, everything related to their action is tamasic. This should be understood by looking over the contents of the Gita. It has also been stated in the fourteenth chapter that even the jnani, at the last stage, after giving up jnana, attains a position beyond the gunas only by the power of pure bhakti, which remains at that point.
18.28 'Ayukta' is the doer who is unqualified for acts enjoined by the Sastras; the meaning is that he is engaged in perverse acts; who is 'unrefined', means one uninstructed; who is 'stubborn', means one who is not disposed to act; who is 'depraved' means one who has the taste for black magic etc; who is dishonest is one who is treacherous; who is 'indolent' is one who is not inclined to carry out actions undertaken; who is 'despondent' is one given to excessive despondency; and one who is 'dilatory', is a person who, while engaged in black magic, etc., pays malevolent attention to produce long-standing evil to others — such a doer is declared to be Tamasika. Thus, has been told the threefold division in terms of the Gunas of the knowledge about the work that ought to be performed, and about the agent of work. Now, Sri Krsna describe s the threefold division of Buddhi and Dhrti (fortitude) on the basis of Gunas. These faculties give the determinate knowledge of all realities in existence and of all ends of human life (Purusarthas).
The activity that is required to be performed at an auspicious time but is delayed and hence missed the mark also indicates the meaning of dirgha- sutri which includes indolence and incompetence. Although procrastination is accurate it also includes indolence, for even if one is inspired to perform prescribed Vedic activities there is some impediment and obstacles in accomplishing them. This is the result of obscuration of vision and lack of cognizance of the Supreme Lord Krishna's paramount position. So even one with the potential for great achievements, due to bewilderment, despondency and laziness fails to reap the benefits for not performing the proper activity at the proper time. Such a one is undeniably situated in tama guna the mode of ignorance.
18.28 The agent who is ayuktah, unsteady; prakrtah, naive, of very unrefined intelligence, like a child; stabdhah, unbending like a staff-he does not bend down to anyone; sathah, deceitful, cunning, hiding his own powers; naiskrtikah, wicked, given to destroying the livelihood of others; alasah, lazy, not inclined even to his own duties; visadi, morose, ever in a mood of dejection; and dirghasutri, procrastinating, postponing duties for long, [Ast. adds here, 'sarvada mandasvabhavah, always slow by nature'.-Tr.] not accomplishing even in a month what is to be done today or tomorrow;-one who is such, he ucyate, is said to be; tamasah, possessed of tamas.
18.26-28 Muktasangah etc. upto Tamasa ucyate. He who does not make speech of egoism : He who does not claim 'I am the agent' i.e., he who is different from the one who claims so by natural inclination, or claims as such with an intention that 'I should do so', or claims so in an efficient manner. This nini (suffix employed in anahamvadi) does not prohibit for a Yogin, the speech 'I do' under the influence of the cover of the mundane life. Who is overpowered by joy and grief : i.e., at the time of success and failure [respectively]. Wickedness : cruelty.
ayuktaḥ — not referring to the scriptural injunctions; prākṛtaḥ — materialistic; stabdhaḥ — obstinate; śaṭhaḥ — deceitful; naiṣkṛtikaḥ — expert in insulting others; alasaḥ — lazy; viṣādī — morose; dīrgha-sūtrī — procrastinating; ca — also; kartā — worker; tāmasaḥ — in the mode of ignorance; ucyate — is said to be.