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Last major update issued on August 9, 2012 at 05:30 UTC.
The geomagnetic field was quiet to minor storm on August 8. Solar wind speed at SOHO ranged between 321 and 431 km/s under the influence of weak CME effects.
Solar flux measured at 20h UTC on 2.8 GHz was 133.3 (decreasing 32.1 over the last solar rotation). The Potsdam WDC planetary A index was 11 (STAR Ap - based on the mean of three hour interval ap indices: 10.9). Three hour interval K indices: 52221212 (planetary), 41312311 (Boulder).
Region 11535 [N19W56] decayed slowly and quietly.
Region 11537 [N12W29] was quiet and stable.
Region 11538 [S23W35] was quiet and stable.
Region 11539 [S20W15] gained a few spots and was quiet.
Region 11540 [S26W08] decayed slightly and was quiet.
Region 11541 [S17W03] was quiet and stable.
Region 11542 [S14E51] has polarity intermixing and is likely to produce further low level C class flares. A minor M class flare is possible.
Region 11543 [N20E59] displayed little change and remained quiet.
S1842 [N17W43] developed slowly and quietly.
New region S1849 [S10E10] emerged with a few spots.
New region S1850 [S31E07] emerged quickly and has many spots. If the region develops further minor M class flares will become possible.
August 6-8: No obviously Earth directed CMEs were observed in LASCO and STEREO imagery.
A small trans equatorial coronal hole (CH526) was in an Earth facing position on August 7. CH536 nearly closed on August 8 due to coronal expansion following a filament eruption in the suthwest quadrant. A coronal hole (CH527) in the southern hemisphere will likely rotate into an Earth facing position on August 9-10. A coronal hole (CH528) in the northern hemisphere could rotate into an Earth facing position on August 10-11.
The geomagnetic field is expected to be quiet to unsettled on August 9. Quiet to active conditions are possible on August 10-11 (due to effects from CH526) and on August 12-14 (due to effects from CH527 and CH528).


text:
\section{Introduction and Results}
An isometric action of a compact Lie group on a Riemannian
manifold $M$ is called {\it polar} if it admits a {\it section},
i.e.~a connected submanifold $\Sigma$ of $M$ that meets all
orbits perpendicularly at each point of intersection. If the
section is flat, the action is called {\it hyperpolar}.

In this paper, $M=G/K$ will denote a Riemannian symmetric space
of compact type. As the classification of hyperpolar actions on
irreducible symmetric spaces of compact type \cite{Kollross}
shows, all examples of such actions of cohomogeneity at least two
are orbit equivalent to the so-called {\it Hermann actions},
i.e.~actions of symmetric subgroups of $G$. Recall that a
subgroup $H\subset G$ is called {\it symmetric} if there exists an
involutive automorphism $\sigma:{\mathfrak g}\to {\mathfrak g}$ with fixed point
algebra ${\mathfrak h}$.

In Section \ref{sec_proof}, we prove the following theorem.
\begin{Thm2}  Let $H\subset G$ be a symmetric subgroup, $p\in M$ regular
and $v,w\in \nu_pHp$. Then the tangent space $T_pHp$ is an
invariant subspace of the curvature operator $R_v(x)=R(x,v)v$ and
the restriction of $R_v$ to $T_pHp$ commutes with the shape
operator $A_w$ of $Hp$.
\end{Thm2}

Therefore the curvature and shape operators of $Hp$ can be
simultaneously diagonalized. The eigenspaces of the curvature
operators are given by the root spaces of $M$; more precisely, a
coarser version of the root space decomposition obtained by
regarding only the restrictions of the roots to the tangent space
of the section is relevant here -- see Section
\ref{sec_RootSpaceDecomp}. As a corollary of the above theorem,
we obtain that for singular orbits, the restricted curvature
operator $R_v$ commutes with the shape operator $A_w$ if $v$ and
$w$ lie in the same section.

In Section \ref{sec_EV_ComInv}, we restrict ourselves to the case
where $H$ can be conjugated in such a way that the involutions
corresponding to $H$ and $K$ commute\footnote{Note that in this
case the triple $(G,H,K)$ is called a {\it symmetric triad} in
\cite{ConlonTop}.}, which is possible except in a few cases
\cite{Conlon}. We can completely determine the eigenspaces of the
shape operators in terms of the restrictions of the roots
(Theorem \ref{thm_ShapeOp}), thereby generalizing \cite{Verhoczki}
where the case $H=K$ is treated.

In the general case, which is treated in Section
\ref{sec_EV_General}, we can show how the eigenvalues of $A_v$
change if the normal direction $v$ is varied (Proposition
\ref{prop_ShapeOpGeneral}).

Using the methods of \cite{CNV}, where the case $H=K$ is treated,
we calculate in Section \ref{sec_Appl} the volumes of the
principal orbits; furthermore we reprove a Weyl-type integration
formula for actions of Hermann type (\cite{F-J}, which is a
generalization of Theorem I.5.10 of \cite{Helgason_GGA}) using
our calculations of the shape operators.

We would like to remark that, with slight modifications, our
results are also true in the noncompact case, the only difference
being some sign changes and some replacements of trigonometric
functions by hyperbolic ones. Nevertheless, for better
readability, we will present the proofs only for the compact
case. Note that in the noncompact case, $H$ can always be
conjugated in such a way that the two involutions commute, see
\cite{Berger}, Lemma 10.2. The shape operators in the noncompact
case are also calculated in \cite{Koike}, but in a completely
different way.

\section{Preliminaries}
Let $M=G/K$ be a Riemannian symmetric space of compact type and
set $p=eK$. Then $G$ is a semisimple compact Lie group, and we
assume the metric on $M$ to be induced by the Killing form of
$G$. The Lie algebra ${\mathfrak g}$ can be identified with the Lie algebra
of Killing vector fields on $M$, with the bracket being the
negative of the bracket of the Killing vector fields. Considering
the Cartan decomposition $${\mathfrak g}={\mathfrak k}\oplus {\mathfrak m}$$ and using the
identification of ${\mathfrak g}$ with the Killing vector fields, we have
\begin{equation}{\mathfrak k}=\{X\in {\mathfrak g}\mid X(p)=0\}\quad\text{and}\quad {\mathfrak m}=\{X\in
{\mathfrak g}\mid (\nabla X)_{p}=0\},\label{eqn_G_Killing}
\end{equation}see \cite{Sakai}, Lemma 6.8. The Killing vector fields in
${\mathfrak m}$ are those induced by transvections along geodesics through
$p$.

If $X,Y,Z\in {\mathfrak m}$, we can express the curvature of $M$ at the
point $p$ by
\begin{equation}
R(X(p),Y(p))Z(p)=-[[X,Y],Z](p). \label{eqn_CurvSymmSpace}
\end{equation}
Note that this equality remains valid if we assume only two of
the Killing vector fields to be induced by transvections -- if
e.g.~$X\in {\mathfrak g}$ is arbitrary, this follows from
$[[{\mathfrak k},{\mathfrak m}],{\mathfrak m}]\subset [{\mathfrak m},{\mathfrak m}]\subset {\mathfrak k}$.

Let now ${\mathfrak a}\subset {\mathfrak m}$ be a maximal abelian subalgebra, denote
the set of restricted roots by $\Delta$ and a choice of positive
roots by $\Delta^+$. Then the corresponding root space
decomposition of $M$ is
\begin{equation}
\label{eqn_RootSD_aIntr} {\mathfrak k}={\mathfrak z}_{\mathfrak k}({\mathfrak a})\oplus \sum_{\alpha\in
\Delta^+} {\mathfrak k}_\alpha\quad \text{ and }\quad {\mathfrak m}={\mathfrak a}\oplus
\sum_{\alpha\in \Delta^+} {\mathfrak m}_\alpha,
\end{equation}
where
\begin{equation}
{\mathfrak k}_\alpha=\{X\in {\mathfrak k}\mid {\rm ad}_W^2(X)=-\alpha(W)^2X \text{ for all
}W\in {\mathfrak a}\} \label{eqn_K_Alpha}
\end{equation}
and
\begin{equation}
{\mathfrak m}_\alpha=\{X\in {\mathfrak m}\mid {\rm ad}_W^2(X)=-\alpha(W)^2X \text{ for all
}W\in {\mathfrak a}\}.\label{eqn_M_Alpha}
\end{equation}
We call $X\in {\mathfrak k}_\alpha$ and ${\mathfrak m}_\alpha$ {\it related} if
$[W,X]=-\alpha(W)Y$ and $[W,Y]=\alpha(W)X$ for all $W\in {\mathfrak a}$ (see
\cite{Loos}, p.~61). For any $X\in {\mathfrak k}_\alpha$ there exists a
related vector $Y\in {\mathfrak m}_\alpha$, and vice versa; in particular,
the vector spaces
${\mathfrak k}_\alpha$ and ${\mathfrak m}_\alpha$ are isomorphic.\\

For $v\in T_pM$, the {\it curvature operator} $R_v$ is defined to
be the endomorphism of $T_pM$ given by $R_v(u)=R(u,v)v$.

The {\it shape operator} $A_\xi:T_pN\to T_pN$ of a submanifold
$N\subset M$ in the normal direction $\xi\in \nu_pN$ is defined
as $A_\xi x=-(\nabla_x \xi)^T$; with this choice of sign, a
Jacobi field $J$ along the normal geodesic $\gamma$ in direction
$\xi$ is an $N$-Jacobi field if and only if $J(0)\in T_pN$ and
$J'(0)+A_\xi J(0)\in \nu_pN$.

Let $H\subset G$ act on $M=G/K$. If $p\in M$ is regular, the fact
that the slice representation at $p$ is trivial implies that we
can extend normal vectors to well-defined $H$-equivariant normal
vector fields on $Hp$. If the $H$-action is polar, these are
automatically parallel with respect to the normal connection, see
\cite{PalaisTerng}, Theorem 5.6.7. For any such vector field
$\xi$, we thus get
\begin{equation}\label{Eqn_ShapeOpEquivNVF}
A_{\xi(p)}x=-\nabla_x\xi.
\end{equation}
The exponential map of $G$ will be denoted by $\exp$ and the one
of $M$ by ${\rm Exp}$. For parallel translation along a curve
$\gamma$, we will write
${}_\gamma||_{t_0}^{t_1}:T_{\gamma(t_0)}M\to T_{\gamma(t_1)}M$.

\section{Proof of the Main Theorem}\label{sec_proof}
In this section, we will first prove the following theorem about
the principal orbits of Hermann actions; the singular orbits will
be dealt with in Corollary \ref{Cor_CommuteSingular}.

\begin{Thm}\label{Thm_Commute} Let $H\subset G$ be a symmetric subgroup,
$p\in M$ regular and $v,w\in \nu_pHp$. Then the tangent space
$T_pHp$ is an invariant subspace of the curvature operator
$R_v(x)=R(x,v)v$ and the restriction of $R_v$ to $T_pHp$ commutes
with the shape operator $A_w$ of $Hp$.
\end{Thm}
First we need a lemma.
\begin{Lemma}\label{Lem_H_Xi_Commute}
Let $X\in {\mathfrak h}$ and $\xi$ be an $H$-equivariant normal vector
field on $Hp$. Then $[X,\xi]=0$.
\end{Lemma}
\begin{proof} Let $\gamma(t)={\rm Exp}(t\xi(p))$. Then we have
\begin{align*}
\nabla_{X(p)}\xi&=\left.\frac{\nabla}{ds}\right|_{s=0}\xi(\exp(sX)\cdot
p)=\left.\frac{\nabla}{ds}\right|_{s=0}\left.\frac{d}{dt}\right|_{t=0}\exp(sX)\cdot\gamma(t)\\
&=\left.\frac{\nabla}{dt}\right|_{t=0}\left.\frac{d}{ds}\right|_{s=0}\exp(sX)\cdot
\gamma(t)=\left.\frac{\nabla}{dt}\right|_{t=0}X(\gamma(t))=\nabla_{\xi(p)}X.
\end{align*}
\end{proof}

\begin{proof}[Proof of Theorem \ref{Thm_Commute}] The invariance of
$T_pHp$ under $R_v$ follows from the fact that for any $x\in
T_pHp$ and any $u\in\nu_pHp$, we have
$$\left<R(x,v)v,u\right>=-\left<R(v,u)v,x\right>=0$$ because the action is
polar.

Without loss of generality, we may assume that $p=eK$. Note that
then ${\mathfrak t}:={\mathfrak h}^\bot\cap {\mathfrak m}\cong \nu_pHp$ is an abelian
subalgebra of ${\mathfrak g}$.

Let $\vartheta\in {\mathfrak t}$ be the Killing vector field on $M$ with
$\vartheta(p)=v$ and $(\nabla \vartheta)(p)=0$ and let $\xi$ be
the $H$-equivariant normal vector field on $Hp$ with $\xi(p)=w$.
Furthermore, let $\xi'\in {\mathfrak t}$ be the Killing vector field with
$\xi'(p)=w$ and $(\nabla\xi')(p)=0$ and set $g(t)=\exp(t\xi(p))$.
For $X\in {\mathfrak h}$, we have
\begin{align*}
R_v(A_w&(X(p)))\overset{(\ref{Eqn_ShapeOpEquivNVF})}{=}-R(\nabla_{X(p)}
\xi,v)v\overset{\ref{Lem_H_Xi_Commute}}{=}-R(\nabla_{\xi(p)} X,v)v\\
&=-\nabla_{\xi(p)}(R(X,\vartheta)\vartheta)+R(X(p),\underbrace{\nabla_{\xi(p)}\vartheta}_{=0})v+R(X(p),v)\underbrace{\nabla_{\xi(p)}
\vartheta}_{=0}\\
&\!\!\!\!\!\!\!\!\!\!\!
\text{(}\nabla R=0\text{ on a symmetric space)}\\
&=-\left.\frac{\nabla}{dt}\right|_{t=0}
R(X(g(t)p),\vartheta(g(t)p))\vartheta(g(t)p)\\
&=-\left.\frac{\nabla}{dt}\right|_{t=0}d(g(t))(R({\rm Ad}_{g(t)^{-1}}X,{\rm Ad}_{g(t)^{-1}}\vartheta){\rm Ad}_{g(t)^{-1}}\vartheta)\\
&=-\left.\frac{\nabla}{dt}\right|_{t=0}d(g(t))
R({\rm Ad}_{g(t)^{-1}}X,\vartheta(p))\vartheta(p)\\
&\!\!\!\!\!\!\!\!\!\!\!
\text{(}{\mathfrak t}\text{ is abelian)}\\
&=-\left.\frac{\nabla}{dt}\right|_{t=0}{}_\gamma||_0^t
R({\rm Ad}_{g(t)^{-1}}X,\vartheta(p))\vartheta(p)\\
&=-\left.\frac{d}{dt}\right|_{t=0}R({\rm Ad}_{g(t)^{-1}}X,\vartheta(p))\vartheta(p)\\
&=\left.\frac{d}{dt}\right|_{t=0}[[{\rm Ad}_{g(t)^{-1}}X,\vartheta],\vartheta](p)\\
&\!\!\!\!\!\!\!\!\!\!\! \text{(}\vartheta\text{ is induced by
transvections; see
(\ref{eqn_CurvSymmSpace}))}\\
&= [[[X,\xi'],\vartheta],\vartheta](p)\\
&=[[[X,\vartheta],\xi'],\vartheta](p)+[[X,\underbrace{[\xi',\vartheta]}_{=0}],\vartheta](p)\\
 &=[[[X,\vartheta],\vartheta],\xi'](p)\\
&=-\underbrace{\nabla_{[[X,\vartheta],\vartheta](p)}\xi'}_{=0\,
(\xi'\in {\mathfrak p})}+\nabla_{\xi(p)}[[X,\vartheta],\vartheta]\\
&\!\!\!\!\!\!\!\!\!\!\!
\text{(note the sign of the bracket in }{\mathfrak g} \text{)}\\
&=-\nabla_{R(X(p),v)v}\xi+[[[X,\vartheta],\vartheta],\xi](p)\qquad\,\,\quad\qquad\qquad\qquad\\
&=A_w(R_v(X(p)))+[[[X,\vartheta],\vartheta],\xi](p).
\end{align*}
Since $X$ and $\vartheta$ are Killing vector fields on $M$, we
have $[[X,\vartheta],\vartheta]\in
[[{\mathfrak h},{\mathfrak h}^\bot],{\mathfrak h}^\bot]\subset [{\mathfrak h}^\bot,{\mathfrak h}^\bot]\subset {\mathfrak h}$,
where for the last inclusion we used that $H$ is a symmetric
subgroup of $G$. Hence, Lemma \ref{Lem_H_Xi_Commute} implies that
$[[[X,\vartheta],\vartheta],\xi](p)=0$.
\end{proof}

For the singular orbits, we have the following corollary.
\begin{Cor}\label{Cor_CommuteSingular} Let $H\subset G$ be a symmetric
subgroup and $p\in M$ arbitrary. Then for all $v\in \nu_pHp$, the
tangent space $T_pHp$ is invariant under the curvature operator
$R_v$. If $\Sigma$ is a section of the $H$-action passing through
$p$ and $v,w\in T_p\Sigma$, then the restriction of $R_v$ to
$T_pHp$ commutes with the shape operator $A_w$ of $Hp$.
\end{Cor}
\begin{proof} Conjugate $H$ such that $p$ is the origin. Let $\vartheta\in
{\mathfrak h}^\bot\cap {\mathfrak m}$ be the Killing vector field induced by
transvections with $\vartheta(p)=v$. Then for any $X\in {\mathfrak h}$,
equation (\ref{eqn_CurvSymmSpace}) yields
$$R(X(p),\vartheta(p))\vartheta(p)=-[[X,\vartheta],\vartheta](p)\in
T_pHp$$ because ${\mathfrak h}$ is a symmetric subgroup.

Then the operators commute because the regular points in $\Sigma$
are dense in $\Sigma$.
\end{proof}

\begin{Cor}\label{Cor_AllCommute} Let $H\subset G$ be a symmetric
subgroup. Then $\{A_v,R_v\mid v\in T_p\Sigma\}$ is a commuting
family of endomorphisms of $T_pHp$, where $\Sigma$ is a section
passing through $p$.
\end{Cor}
\begin{proof} We first assume that $p$ is regular. Then the $R_v$ commute
because $\nu_pHp$ is abelian as one sees by combining
(\ref{eqn_CurvSymmSpace}) with the Jacobi identity. The Ricci
equation implies
\begin{align*}
\left<[A_v,A_w]x,y\right>&=\left<R(x,y)v,w\right>-\left<R^\bot(x,y)v,w\right>\\
&=\left<R(v,w)x,y\right>=0
\end{align*}
for all $x,y\in T_pHp$ and all $v,w\in \nu_pHp$ since the normal
bundle of $Hp$ is flat. Therefore, the shape operators of $Hp$
commute.

If $p$ is not regular, the claim then follows from the fact that
the regular points in $\Sigma$ are dense in $\Sigma$.
\end{proof}

\begin{Rem}
A submanifold $N$ of a Riemannian manifold $M$ is called {\it
curvature-adapted} if $T_pN$ is invariant under the curvature
operator $R_u$ and if the restriction of $R_u$ to $T_pN$ commutes
with the shape operator $A_u$ of $N$ for any $p\in N$ and all
$u\in \nu_p M$. Theorem \ref{Thm_Commute} and Corollary
\ref{Cor_CommuteSingular} immediately imply that all the orbits
of the $H$-action are curvature-adapted submanifolds of $M$.
\end{Rem}

\section{Adapted Root Space Decomposition}
\label{sec_RootSpaceDecomp}

The root space decomposition adapted to the $H$-action has
already been described in \cite{F-J}; we include it for the
convenience of the reader.

Let $H\subset G$ act on $M$ in a hyperpolar fashion; later we
will assume that it is a symmetric subgroup such that the
corresponding involution commutes with the one of ${\mathfrak k}$. Consider
the Cartan decomposition $${\mathfrak g}={\mathfrak k}\oplus {\mathfrak m}$$ and choose a
maximal abelian subalgebra ${\mathfrak t}\subset {\mathfrak h}^\bot \cap {\mathfrak m}$. Then
$\Sigma={\rm Exp}({\mathfrak t})\subset M$ is a section for the $H$-action on
$M$. Note that $\Sigma$ is a torus since we assume the metric to
be induced by the Killing form on ${\mathfrak g}$; see \cite{HPTT}, Theorem
2.3. Note that $${\mathfrak m}={\rm pr}_{\mathfrak m}{\mathfrak h}\oplus ({\mathfrak h}^\bot\cap {\mathfrak m})\cong
T_pHp\oplus \nu_pHp.$$ Let further ${\mathfrak a}$ be a maximal abelian
subalgebra of ${\mathfrak m}$ containing ${\mathfrak t}$. The set of restricted roots
of $M$ with respect to ${\mathfrak a}$ shall be denoted by $\Delta$ and a
choice of positive roots by $\Delta^+$.

Consider the corresponding root space decomposition
\begin{equation}
\label{eqn_RootSD_a} {\mathfrak k}={\mathfrak z}_{\mathfrak k}({\mathfrak a})\oplus \sum_{\alpha\in
\Delta^+} {\mathfrak k}_\alpha\quad \text{ and }\quad {\mathfrak m}={\mathfrak a}\oplus
\sum_{\alpha\in \Delta^+} {\mathfrak m}_\alpha,
\end{equation}
where the root spaces ${\mathfrak k}_\alpha$ and ${\mathfrak m}_\alpha$ are given by
(\ref{eqn_K_Alpha}) and (\ref{eqn_M_Alpha}).

In the following, the restrictions of the roots to ${\mathfrak t}$ will be
of greater importance than the roots themselves. We define
$$\Delta_{\mathfrak t}=\{\alpha|_{\mathfrak t}\mid \alpha\in \Delta\}\setminus\{0\}.$$ Note
that for two roots $\alpha,\alpha'\in \Delta^+$, it is possible
that $\alpha|_{\mathfrak t}=-\alpha'|_{\mathfrak t}$ (see the example in Section
\ref{sec_Example}). Therefore, we let $\Delta_{\mathfrak t}^+\subset
\Delta_{\mathfrak t}$ be the set of nonzero restrictions of elements in
$\Delta^+$ to ${\mathfrak t}$, but if this occurs, we include only one of
$\alpha|_{\mathfrak t}$ and $\alpha'|_{\mathfrak t}$ in $\Delta_{\mathfrak t}^+$. For any
$\beta\in \Delta^+_{\mathfrak t}$, we set
$${\mathfrak k}_\beta^{\mathfrak t}=\{X\in {\mathfrak k}\mid {\rm ad}_w^2(X)=-\beta(w)^2X \text{ for all }
w\in {\mathfrak t}\}$$ and
$${\mathfrak m}_\beta^{\mathfrak t}=\{X\in {\mathfrak m}\mid {\rm ad}_w^2(X)=-\beta(w)^2X \text{ for all }
w\in {\mathfrak t}\}.$$

\begin{Lemma}
\label{lem_RelationRootSDs} For any $\beta\in \Delta^+_{\mathfrak t}$, we
have
$${\mathfrak k}_\beta^{\mathfrak t}=\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta}
{\mathfrak k}_\alpha\quad \text{ and }\quad  {\mathfrak m}_\beta^{\mathfrak t}=\sum_{\alpha\in
\Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta} {\mathfrak m}_\alpha.$$ Furthermore,
$${\mathfrak z}_{\mathfrak m}({\mathfrak t})={\mathfrak a}\oplus \sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=0}
{\mathfrak m}_\alpha \quad \text{ and } \quad
{\mathfrak z}_{\mathfrak k}({\mathfrak t})={\mathfrak z}_{\mathfrak k}({\mathfrak a})\oplus \sum_{\alpha\in \Delta^+:\,
\alpha|_{\mathfrak t}=0} {\mathfrak k}_\alpha.$$
\end{Lemma}
\begin{proof}
We only prove the first equality. For all $\beta\in \Delta^+_{\mathfrak t}$
and all $\alpha\in \Delta^+$ such that $\alpha|_{\mathfrak t}=\pm\beta$, we
have ${\mathfrak k}_\alpha\subset {\mathfrak k}_\beta^{\mathfrak t}$ by definition. The desired
equality then follows from (\ref{eqn_RootSD_a}).
\end{proof}

Consequently we have the decompositions
\begin{equation}
\label{eqn_RootSD_t} {\mathfrak k}={\mathfrak z}_{\mathfrak k}({\mathfrak t})\oplus \sum_{\beta\in
\Delta^+_{\mathfrak t}}{\mathfrak k}_\beta^{\mathfrak t} \quad \text{ and }\quad
{\mathfrak m}={\mathfrak z}_{\mathfrak m}({\mathfrak t})\oplus\sum_{\beta\in \Delta^+_{\mathfrak t}}{\mathfrak m}_\beta^{\mathfrak t}.
\end{equation}

From now on until the end of Section \ref{sec_EV_ComInv}, we
assume that $H$ and $K$ are symmetric subgroups of $G$
corresponding to commuting involutions. Then the two Cartan
decompositions
\begin{equation}
{\mathfrak g}={\mathfrak k}\oplus {\mathfrak m} ={\mathfrak h}\oplus{\mathfrak p}
\end{equation}
are compatible in the sense that
\begin{equation}
\label{eqn_CartanRefined} {\mathfrak g}={\mathfrak k}\cap {\mathfrak p}\oplus {\mathfrak k}\cap {\mathfrak h}\oplus
{\mathfrak m}\cap {\mathfrak h} \oplus {\mathfrak m}\cap {\mathfrak p}.
\end{equation}

\begin{Lemma}
For all $\beta\in \Delta_{\mathfrak t}^+$, we have
\begin{equation}
\label{eqn_RootSpacesRefinedK} {\mathfrak k}_\beta^{\mathfrak t}={\mathfrak k}_\beta^{\mathfrak t} \cap
{\mathfrak p} \oplus {\mathfrak k}_\beta^{\mathfrak t}\cap {\mathfrak h}
\end{equation}

\begin{equation}
\label{eqn_RootSpacesRefinedM} {\mathfrak m}_\beta^{\mathfrak t}={\mathfrak m}_\beta^{\mathfrak t}\cap
{\mathfrak p} \oplus {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}.
\end{equation}
Furthermore ${\mathfrak z}_{\mathfrak k}({\mathfrak t})={\mathfrak z}_{\mathfrak k}({\mathfrak t})\cap {\mathfrak p}\oplus
{\mathfrak z}_{\mathfrak k}({\mathfrak t})\cap {\mathfrak h}$ and ${\mathfrak z}_{\mathfrak m}({\mathfrak t})={\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak p}\oplus
{\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}$.
\end{Lemma}

\begin{proof}
Let $X\in {\mathfrak k}_\beta^{\mathfrak t}$. According to (\ref{eqn_CartanRefined}),
we can decompose $X$ as $X=X_{\mathfrak p}+X_{\mathfrak h}$, where $X_{\mathfrak p}\in {\mathfrak k}\cap
{\mathfrak p}$ and $X_{\mathfrak h}\in {\mathfrak k}\cap {\mathfrak h}$. First of all we have
\begin{equation*}
-\beta(w)^2X_{\mathfrak p}-\beta(w)^2X_{\mathfrak h}={\rm ad}_w^2(X_{\mathfrak p})+{\rm ad}_w^2(X_{\mathfrak h}).
\end{equation*}
for all $w\in {\mathfrak t}$. Since
\begin{equation*}
{\rm ad}_w^2X_{\mathfrak p}\in [{\mathfrak t},[{\mathfrak t},{\mathfrak k}\cap {\mathfrak p}]]\subset [{\mathfrak m}\cap
{\mathfrak p},[{\mathfrak m}\cap {\mathfrak p},{\mathfrak k}\cap {\mathfrak p}]]\subset [{\mathfrak m}\cap {\mathfrak p},{\mathfrak m}\cap
{\mathfrak h}]\subset {\mathfrak k}\cap {\mathfrak p}
\end{equation*}
and
\begin{equation*}
{\rm ad}_w^2X_{\mathfrak h}\in [{\mathfrak t},[{\mathfrak t},{\mathfrak k}\cap {\mathfrak h}]]\subset [{\mathfrak m}\cap
{\mathfrak p},[{\mathfrak m}\cap {\mathfrak p},{\mathfrak k}\cap {\mathfrak h}]]\subset [{\mathfrak m}\cap {\mathfrak p},{\mathfrak m}\cap
{\mathfrak p}]\subset {\mathfrak k}\cap {\mathfrak h},
\end{equation*}
we conclude ${\rm ad}_w^2(X_{\mathfrak p})=-\beta(w)^2X_{\mathfrak p}$ and
${\rm ad}_w^2(X_{\mathfrak h})=-\beta(w)^2X_{\mathfrak h}$. Thus we have proven
(\ref{eqn_RootSpacesRefinedK}). The proof of the rest of the
lemma is similar.
\end{proof}

\begin{Rem}
Note that the equations (\ref{eqn_RootSpacesRefinedK}) and
(\ref{eqn_RootSpacesRefinedM}) do not have analogues for the root
spaces ${\mathfrak k}_\alpha$ and ${\mathfrak m}_\alpha$ since they do not
necessarily respect the decomposition (\ref{eqn_CartanRefined}).
\end{Rem}

We can now refine decomposition (\ref{eqn_RootSD_t}) as follows:
\begin{equation}
\label{eqn_RootSpacesRefined_k} {\mathfrak k}=({\mathfrak z}_{\mathfrak k}({\mathfrak t})\cap {\mathfrak h}\oplus
\sum_{\beta\in \Delta^+_{\mathfrak t}}{\mathfrak k}_\beta^{\mathfrak t}\cap
{\mathfrak h})\oplus({\mathfrak z}_{\mathfrak k}({\mathfrak t})\cap {\mathfrak p}\oplus \sum_{\beta\in
\Delta^+_{\mathfrak t}}{\mathfrak k}_\beta^{\mathfrak t}\cap {\mathfrak p})
\end{equation}
and
\begin{equation}
\label{eqn_RootSpacesRefined_m} {\mathfrak m}=({\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}\oplus
\sum_{\beta\in \Delta^+_{\mathfrak t}}{\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}) \oplus
({\mathfrak t}\oplus \sum_{\beta\in \Delta^+_{\mathfrak t}}{\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}).
\end{equation}

\section{Eigenvalues of the Shape Operator: Commuting Involutions}
\label{sec_EV_ComInv}

In this section, $H\subset G$ is a symmetric subgroup
corresponding to an involution commuting with the one of $K$.
Recall the refined Cartan decomposition (\ref{eqn_CartanRefined})
\begin{equation*}
{\mathfrak g}={\mathfrak k}\cap {\mathfrak p}\oplus {\mathfrak k}\cap {\mathfrak h}\oplus {\mathfrak m}\cap {\mathfrak h} \oplus
{\mathfrak m}\cap {\mathfrak p}
\end{equation*}
and that ${\mathfrak a}$ is a maximal abelian subspace of ${\mathfrak m}$ containing
a maximal abelian subspace ${\mathfrak t}$ of ${\mathfrak m}\cap {\mathfrak p}$. Then ${\mathfrak a}$ can
be written as ${\mathfrak a}={\mathfrak t}\oplus {\mathfrak t}'$, where ${\mathfrak t}'\subset {\mathfrak m}\cap
{\mathfrak h}$.

Let $w\in {\mathfrak t}$ and set $p={\rm Exp}(w)$ (which we do not assume to be
regular) and $\gamma(t)={\rm Exp}(tw)$. Our first goal is to express
the tangent space $T_pHp$ in terms of the restricted roots. Note
that the case $H=K$ in the following proposition is the content
of Proposition 3 of \cite{Verhoczki}.

\begin{Prop}
\label{prop_TangSpaceInRoots} The tangent space $T_pH p$ coincides
with the parallel displacement of $({\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h})\oplus
V_1\oplus V_2\subset {\mathfrak m}$ along $\gamma$, where
\begin{equation*}
V_1=\sum_{\beta\in \Delta^+_{\mathfrak t},\, \beta(w)\notin
\frac{\pi}{2}+\pi\mathbf{Z}}{\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h} \quad\text{ and }\quad
V_2=\sum_{\beta\in \Delta^+_{\mathfrak t},\, \beta(w)\notin
\pi\mathbf{Z}}{\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}.
\end{equation*}
\end{Prop}

\begin{proof}
Regarding the elements of ${\mathfrak h}$ as Killing fields on $M$, the
equations in (\ref{eqn_G_Killing}) yield
\begin{equation}
\label{eqn_H_as_KillingField} {\mathfrak m}\cap {\mathfrak h}=\{X\in {\mathfrak h}\mid (\nabla
X)_{eK}=0\},\quad {\mathfrak k} \cap {\mathfrak h}= \{X\in {\mathfrak h}\mid X(eK)=0\}.
\end{equation}
Of course $T_pHp=U_1+ U_2$, where $U_1= \{X(p)\mid X\in {\mathfrak h}\cap
{\mathfrak m}\}$ and $U_2=\{X(p)\mid X\in {\mathfrak h}\cap {\mathfrak k}\}$.

For any $\alpha\in \Delta^+$, let $\{X^\alpha_i\}_{i\in I_\alpha}$
be an orthonormal basis of ${\mathfrak m}_\alpha$; furthermore let
$\{X^0_i\}_{i\in I_0}$ be an orthonormal basis of ${\mathfrak t}'$. Let
$E^\alpha_i$ and $E^0_i$ be the parallel fields along $\gamma$
with $E^\alpha_i(0)=X^\alpha_i(eK)$ and $E^0_i(0)=X^0_i(eK)$,
respectively.

For $X\in {\mathfrak h}$ let $Y=X|_\gamma$ be the Jacobi field along
$\gamma$ obtained by restricting $X$ to $\gamma$. Since $Y(t)$ is
tangent to the orbit through $\gamma(t)$ for all $t$, it follows
from the description of Jacobi fields on symmetric spaces that

\begin{align}
\label{eqn_JacobiFieldGeneral} Y(t)=&\sum_{i\in I_0}
(a_i+b_it)E^0_i(t)+\nonumber \\
&\sum_{\alpha\in \Delta^+}\sum_{i\in I_\alpha}
(c_i\sin(\alpha(w)t)+d_i\cos(\alpha(w)t))E^\alpha_i(t)
\end{align}
for some constants $a_i,b_i,c_i,d_i$.

Consider first the case $X\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$ for some
$\beta\in \Delta^+_{\mathfrak t}$ and let $v=X(eK)$. According to Lemma
\ref{lem_RelationRootSDs}, we can write
\begin{equation*}
X=\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta}\sum_{i\in
I_\alpha} \lambda_{\alpha,i} X^\alpha_i
\end{equation*}
for some constants $\lambda_{\alpha,i}\in \mathbf{R}$. Since $X\in {\mathfrak m}$,
we have $Y'(0)=0$ because of (\ref{eqn_H_as_KillingField}), so we
get $b_i=c_i=0$. It follows that
\begin{align*}
Y(t)&=\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta}\sum_{i\in
I_\alpha}\lambda_{\alpha,i}\cos(\alpha(w)t)E^\alpha_i \\
&=\cos(\beta(w)t)\sum_{\alpha\in \Delta^+:\,
\alpha|_{\mathfrak t}=\pm\beta}\sum_{i\in I_\alpha}\lambda_{\alpha,i}E^\alpha_i(t)\\
&=\cos(\beta(w)t){}_\gamma ||_0^t v.
\end{align*}
We thus have shown that if $\beta(w)\notin \frac{\pi}{2}+\pi\mathbf{Z}$,
then the parallel transport of ${\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$ along
$\gamma$ is contained in $U_1$.

Let now $X\in {\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}$ and $v=X(eK)$. Lemma
\ref{lem_RelationRootSDs} yields
$$X=\sum_{i\in I_0} \mu_i X^0_i +
\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=0} \lambda_{\alpha,i}
X^\alpha_i$$ for some constants $\mu_i$ and $\lambda_{\alpha,i}$.
We obtain
\begin{align*}
Y(t)=\sum_{i\in I_0}\mu_i E^0_i(t)+\sum_{\alpha\in \Delta^+:\,
\alpha|_{\mathfrak t}=0}\sum_{i\in
I_\alpha}\lambda_{\alpha,i}\cos(\alpha(w)t)E^\alpha_i ={}_\gamma
||_0^t v,
\end{align*}
so the parallel transport of ${\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}$ along $\gamma$
is contained in $U_1$. It is now clear that $U_1$ is the direct
sum of the parallel transport of ${\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}\oplus V_1$.

It remains to describe $U_2$. For any $\alpha\in \Delta^+$, let
$\{Z_i^\alpha\}_{i\in I_\alpha}$ be the orthonormal basis of
${\mathfrak k}_\alpha$ which is related to $\{X_i^\alpha\}$ by
$[Z_i^\alpha,u]=\alpha(u)X^\alpha_i$ for all $u\in {\mathfrak a}$. For any
$\alpha\in \Delta^+$, we have
\begin{equation*}
[{\mathfrak k}_\alpha,u]=\begin{cases}{\mathfrak m}_\alpha & \text{if }\alpha(u)\neq
0 \\ 0 & \text{if }\alpha(u)=0.\end{cases}
\end{equation*}
Lemma \ref{lem_RelationRootSDs} now yields that an analogous
relation is true for the root spaces with respect to ${\mathfrak t}$: for
all $\beta\in \Delta^+_{\mathfrak t}$, we have

\begin{equation*}
[{\mathfrak k}^{\mathfrak t}_\beta,u]=\begin{cases}{\mathfrak m}^{\mathfrak t}_\beta & \text{if }\beta(u)\neq 0 \\
0 & \text{if }\beta(u)=0.\end{cases}
\end{equation*}
For $X\in {\mathfrak k}$ we have that
\begin{align*}
Y(t)&=X({\rm Exp}(tw))=\left.\frac{d}{ds}\right|_{s=0}\exp(sX)\exp(tw)K\\
&=\left.\frac{d}{ds}\right|_{s=0}\exp({\rm Ad}_{\exp sX} tw)K = (d\,
{\rm Exp})_{tw}(t[X,w]).
\end{align*}
Hence, those $X\in {\mathfrak h}$ which lie in ${\mathfrak z}_{\mathfrak k}({\mathfrak t})\cap {\mathfrak h}$ do
not contribute to $U_2$. We thus have
\begin{equation*}
U_2=(d\,{\rm Exp})_w\sum_{\beta\in \Delta^+_{\mathfrak t}}[{\mathfrak k}_\beta^{\mathfrak t}\cap
{\mathfrak h},w]=(d\,{\rm Exp})_w\sum_{\beta\in \Delta^+_{\mathfrak t},\, \beta(w)\neq 0}
{\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}.
\end{equation*}
Let $v\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}$, where $\beta(w)\neq 0$, and
write
\begin{equation*}
v=\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta}\sum_{i\in
I_\alpha}\lambda_{\alpha,i} X^\alpha_i.
\end{equation*}
for some constants $\lambda_{\alpha,i}$. Define $X\in
{\mathfrak k}_\beta^{\mathfrak t}\cap {\mathfrak h}$ to be
\begin{equation*}
X=\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta}\sum_{i\in
I_\alpha}\pm\lambda_{\alpha,i} Z^\alpha_i.
\end{equation*}
By definition, we have $[X,w]=\beta(w)v$.

Since $Y$ is the unique Jacobi field along $\gamma$ with $Y(0)=0$
and $Y'(0)=[X,w]=\beta(w)v$, we get
\begin{align*}
Y(t)&=\sum_{\alpha\in \Delta^+:\, \alpha|_{\mathfrak t}=\pm\beta}\sum_{i\in
I_\alpha}\pm\lambda_{\alpha,i}\sin(\alpha(w)t)E^\alpha_i(t)=\sin(\beta(w)t){}_\gamma
||_0^t v.
\end{align*}
It follows that $Y(1)$ vanishes if and only if $\beta(w)\notin
\pi\mathbf{Z}$. We have thus proven that $U_2$ is the parallel
displacement of $V_2$ along $\gamma$.
\end{proof}
\begin{Cor} \label{cor_RegularPointsRoots}
The point $p={\rm Exp}(w)\in \Sigma$ is a regular point of the
$H$-action if and only if
\begin{enumerate}
\item $\beta(w)\notin \frac{\pi}{2}+\pi\mathbf{Z}$ for all $\beta\in \Delta^+_{\mathfrak t}$
with ${\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}\neq \{0\}$ and
\item $\beta(w)\notin \pi\mathbf{Z}$ for all $\beta\in \Delta^+_{\mathfrak t}$ with
${\mathfrak m}_\beta\cap {\mathfrak p}\neq \{0\}$.
\end{enumerate}
\end{Cor}
Choose a vector $u\in{\mathfrak t}$, let $c(t)={\rm Exp}(w+tu)$ and
$u(p)=\overset{.}{c} (0)$. By Corollary \ref{Cor_AllCommute}, the
shape and curvature operators can be simultaneously diagonalized.
A concrete such diagonalization is given in the following theorem.
\begin{Thm} \label{thm_ShapeOp}
The decomposition of $T_pHp$ into parallel displacements of the
root spaces described in Proposition \ref{prop_TangSpaceInRoots}
is compatible with the decomposition into the eigenspaces of the
shape operator $A_{u(p)}$ of $Hp$. More precisely,
\begin{enumerate}
\item For $v\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$ with $\beta(w)\notin
\frac{\pi}{2}+\pi\mathbf{Z}$, we have
\begin{equation}\label{eqn_ShapeEV_H}
A_{u(p)}({}_\gamma||_0^1v)=\beta(u)\tan(\beta(w)){}_\gamma
||_0^1v.\end{equation}
\item For $v\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}$ with $\beta(w)\notin \pi\mathbf{Z}$, we
have \begin{equation}\label{eqn_ShapeEV_P}A_{u(p)}({}_\gamma
||_0^1v)=-\beta(u)\cot(\beta(w)){}_\gamma ||_0^1 v.\end{equation}
\item For $v\in {\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}$, we have
\begin{equation}\label{eqn_ShapeEV_T}A_{u(p)}({}_\gamma
||_0^1v)=0.\end{equation}
\end{enumerate}
\end{Thm}
\begin{proof}
For any $s\in [0,1]$, let $\gamma_s(t):={\rm Exp}(t(w+su))$. Note that
$\gamma_0=\gamma$.

First of all, let $X\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$, where
$\beta(w)\notin \frac{\pi}{2}+\pi\mathbf{Z}$, and set $v=X(eK)$. Let
$Y_s$ be the Jacobi field obtained by restriction of the Killing
field $X$ to $\gamma_s$. The initial values of $Y_s$ are
$Y_s(0)=v$ and $Y_s'(0)=0$; as in the proof of Proposition
\ref{prop_TangSpaceInRoots}, we get
\begin{equation}
Y_s(t)=\cos(\beta(w+su)t){}_{\gamma_s} ||_0^1 v,
\end{equation}
since $v$ is contained in a sum of root spaces corresponding to
roots whose restrictions to ${\mathfrak t}$ coincide. We are interested in
the $Hp$-Jacobi field $Y(t)=Y_t(1)$ along $c$. Its initial values
are $Y(0)=\cos(\beta(w)){}_{\gamma} ||_0^1 v$ and
\begin{align}
Y'(0)&=\left.\frac{d}{dt}\right|_{t=0}\cos(\beta(w+tu)){}_{\gamma_t}
||_0^1 v \nonumber \\
&= -\beta(u)\sin(\beta(w)){}_{\gamma} ||_0^1 v,
\end{align}
since $\left.\frac{\nabla}{dt}\right|_{t=0}{}_{\gamma_t}
||_0^1v=0$ (use Lemma 8.3.2 of \cite{BCO}, together with the fact
that the $\gamma_s$ lie in the flat section $\Sigma$). The fact
that $Y$ is an $Hp$-Jacobi field along $c$ now implies
\begin{align*}
Y'(0)+&A_{u(p)}Y(0)\\
&=-\beta(u)\sin(\beta(w)){}_\gamma
||_0^1v+\cos(\beta(w))A_{u(p)}({}_\gamma ||_0^1v)\in \nu_pHp,
\end{align*} so
we get \begin{equation} A_{u(p)}({}_\gamma ||_0^1
v)=\beta(u)\tan(\beta(w)){}_\gamma ||_0^1 v,\end{equation} which
is Equation (\ref{eqn_ShapeEV_H}).

In order to prove Equation (\ref{eqn_ShapeEV_P}), choose some
vector $v\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}$ with $\beta(w)\notin \pi\mathbf{Z}$.
Let $X\in {\mathfrak k}_\beta^{\mathfrak t}$ be such that $[H,X]=-\beta(H)v$ for all
$H\in {\mathfrak t}$. We have $X\in {\mathfrak h}$ since $\beta(w)\neq 0$. Now
continue exactly as above: let $Y_s$ be the Jacobi field obtained
by restriction of the Killing field $X$ to $\gamma_s$. Its initial
values are $Y_s(0)=0$ and $Y_s'(0)=[X,w+su]=\beta(w+su)v$, so we
get

\begin{equation}
Y_s(t)=\sin(\beta(w+su)t){}_{\gamma_s} ||_0^tv.
\end{equation}
The $Hp$-Jacobi field $Y(t)=Y_t(1)$ has initial values
$Y(0)=\sin(\beta(w)){}_\gamma ||_0^1v$ and
\begin{align}
Y'(0)&=\left.\frac{d}{dt}\right|_{t=0}\sin(\beta(w+tu)){}_{\gamma_t}
||_0^1v \nonumber\\
&=\alpha(u)\cos(\beta(w)){}_\gamma ||_0^1v, \end{align} so we
obtain
\begin{align*}
Y'(0)+&A_{u(p)}Y(0)\\
&=\beta(u)\cos(\beta(w)){}_\gamma
||_0^1v+\sin(\beta(w))A_{u(p)}({}_\gamma ||_0^1v)\in \nu_pHp;
\end{align*}
thus, Equation (\ref{eqn_ShapeEV_P}) follows.

Finally, let $X\in {\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}'$ and set $v=X(eK)$. Let
again $Y_s$ be the restriction of $X$ along $\gamma_s$ and
$Y(t)=Y_t(1)$. Then we see that $Y_s(t)={}_{\gamma_s} ||_0^1v$ and
hence $Y$ satisfies the initial conditions $Y(0)={}_\gamma ||_0^1
v$ and $Y'(0)=0$. Equation (\ref{eqn_ShapeEV_T}) follows
immediately.

\end{proof}

\section{Eigenvalues of the Shape Operator: General Case}
\label{sec_EV_General}

In this section we will determine, as far as possible, the
eigenvalues of the shape operators in the general case of an
arbitrary Hermann action. This is independent of the calculations
in section \ref{sec_EV_ComInv}.

The following proposition shows the dependence of the normal
direction. Let the origin be a regular point, denoted by $p$.
Note that $T_pHp\cong {\rm pr}_{\mathfrak m} {\mathfrak h}=\sum_\beta {\mathfrak m}_\beta^{\mathfrak t} \oplus
({\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\rm pr}_{\mathfrak m} {\mathfrak h})$.
\begin{Prop}\label{prop_ShapeOpGeneral}
\text{\emph{(1)}} There exists a refinement ${\mathfrak m}_\beta^{\mathfrak t}=\sum_i
V_{\beta,i}$ of the root spaces, together with constants
$c_{\beta,i}$, such that for $v\in \nu_pHp$ and all $x\in
V_{\beta,i}$, we have
$$A_vx=c_{\beta,i}\beta(v)x.$$
\text{\quad\emph{(2)}} For $x\in {\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\rm pr}_{\mathfrak m}{\mathfrak h}$, we
have
$$A_vx=0.$$
\end{Prop}
\begin{proof}
Let $x\in T_pHp$ be a common eigenvector of all curvature and
shape operators in normal directions.

Let the linear form $f:\nu_pHp\to \mathbf{R}$, depending on $x$, be
defined by $A_vx=f(v)x$. Choose $X\in {\mathfrak h}$ such that $X(p)=x$. We
write $X=X_{\mathfrak k}+X_{\mathfrak m}$ with $X_{\mathfrak k}\in {\mathfrak k}$ and $X_{\mathfrak m}\in {\mathfrak m}$.

For $v\in\nu_pHp$, we denote by $\xi'\in {\mathfrak m}$ the Killing vector
field induced by transvections with $\xi'(p)=v$ and by $\xi$ the
$H$-equivariant parallel normal vector field with $\xi(p)=v$.
Then we have
\begin{align*}
[\xi',X_{\mathfrak k}](p)&=-\nabla_vX_{\mathfrak k}+\nabla_{\underbrace{X_{\mathfrak k}(p)}_{=0}}\xi'=-\nabla_vX=-\nabla_x\xi-\underbrace{[\xi,X]}_{=0}\\
&=A_vx=f(v)x;
\end{align*}
since $[\xi',X_{\mathfrak k}]\in [{\mathfrak m},{\mathfrak k}]\subset {\mathfrak m}$, it follows that
\begin{equation}[\xi',X_{\mathfrak k}]=f(v)X_{\mathfrak m}.\label{Eqn_XkXpBracket}\end{equation}

Let us now first regard the case of $x\in {\mathfrak m}_\beta^{\mathfrak t}$ for some
root $\beta$. Since (\ref{Eqn_XkXpBracket}) is valid for all
$\xi'\in {\mathfrak m}$, we can write $X_{\mathfrak k}=X_{{\mathfrak k},0}+X_{{\mathfrak k},1}$ with
$X_{{\mathfrak k},0}\in {\mathfrak z}_{\mathfrak k}({\mathfrak m})$ and $X_{{\mathfrak k},1}\in {\mathfrak k}_\beta^{\mathfrak t}$; the
vector $Z\in {\mathfrak m}_\beta^{\mathfrak t}$ related to $X_{{\mathfrak k},1}$ is a multiple
of $X_{\mathfrak m}$, i.e.~there exists some constant $c$, independent of
$v$, with $f(v)=c\cdot \alpha(v)$.

If $x\in {\mathfrak m}$ is such that $[X_{\mathfrak m},{\mathfrak t}]=0$, it follows from
(\ref{Eqn_XkXpBracket}) that
$$f(v)\left<X_{\mathfrak m},X_{\mathfrak m}\right>=\left<[\xi',X_{\mathfrak k}],X_{\mathfrak m}\right>=-\left<X_{\mathfrak k},[\xi',X_{\mathfrak m}]\right>=0;$$
hence $A_vx=0.$
\end{proof}
\begin{Rem} The explicit description of the eigenvalues in the case of
commuting involutions was possible because there existed a point
$p\in M$ such that every $H$-Killing vector field could be
written as the sum of one vanishing at $p$ and one with
derivative vanishing at $p$, leading to eigenvalues containing
either a cotangent or a tangent.

In the general case such a point does not exist, but for each
Killing vector field we can choose a point where either it
vanishes or its derivative. For the following calculation we
choose to express the $c_{\beta,i}$ in terms of zeros of the
Killing vector fields themselves; hence, only the cotangent
occurs.
\end{Rem}

Now we can investigate how the $c_{\beta,i}$ and the eigenvalues
of the shape operators change when varying the orbit. In the
notation above, let $Y_w:=X\circ\gamma_w$, where
$\gamma_w(t)={\rm Exp}(tw)$ is the geodesic in direction $w$. For
$x=X(p)\in V_{\beta,i}\subset {\mathfrak m}_\beta^{\mathfrak t}$, we have
$$Y_w(t)=(-c_{\beta,i} \sin(\beta(w)t)+\cos(\beta(w)t)){}_{\gamma_w}||_0^t
x.$$ If $t_{\beta,i}$ is a fixed zero of some $Y_{w_0}$ with
$\beta(w_0)=1$, we can write $c_{\beta,i}=\cot(t_{\beta,i})$.
Regarding the $H\gamma_w(1)$-Jacobi field $Y(t):=Y_{w+tv}(1)$, we
can determine the shape operator of the orbit $H\gamma_w(1)$:
\begin{align*}
0&=Y'(0)+A_vY(0)\\
&=(-c_{\beta,i}\beta(v)\cos(\beta(w))-\beta(v)\sin(\beta(w))){}_{\gamma_w}||_0^1
x+A_v Y_w(1).
\end{align*}
Hence
\begin{align*}
A_v({}_{\gamma_w}||_0^1
x)&=\beta(v)\frac{c_{\beta,i}\cos(\beta(w))+\sin(\beta(w))}{-c_{\beta,i}\sin(\beta(w))+\cos(\beta(w))}{}_{\gamma_w}||_0^1x\\
&=\beta(v)\frac{\cos(t_{\beta,i})\cos(\beta(w))+\sin(t_{\beta,i})\sin(\beta(w))}{-\cos(t_{\beta,i})\sin(\beta(w))+\sin(t_{\beta,i})\cos(\beta(w))}{}_{\gamma_w}||_0^1x\\
&=\beta(v)\cot(t_{\beta,i}-\beta(w)){}_{\gamma_w}||_0^1x.
\end{align*}

\section{Applications}
\label{sec_Appl}
 Let $H\subset G$ be a symmetric subgroup such that the corresponding
involutions commute. Let $p\in M$ be regular and
$\Sigma={\rm Exp}({\mathfrak t})$ be the section through $p$. Denote the
generalized Weyl group of the action by $W$ and define a function
$\vartheta:{\mathfrak t}\to \mathbf{R}$ by

\begin{equation}
\vartheta(w)=\prod_{\beta\in
\Delta^+_{\mathfrak t}}|\sin(\beta(w))|^{p_\beta}|\cos(\beta(w))|^{h_\beta},
\end{equation}
where the exponents $p_\beta$ and $h_\beta$ are the relative root
multiplicities defined by $p_\beta=\dim {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}$ and
$h_\beta=\dim {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$. Since $\vartheta$ is
invariant under the reflections in the singular hyperplanes in
${\mathfrak t}$, it can be regarded as a function on $\Sigma$. We will
reprove the following theorem from \cite{F-J} using our
calculation of the shape operators. The proof is similar to the
arguments in \cite{CNV}, where the case $H=K$ is treated.

\begin{Thm}\label{thm_IntFormula}
For any integrable function $f$ on $M$, we have
\begin{equation}
\label{eqn_IntFormulaNotInvariant} \int_M
f(x)\,dx=\frac{1}{|W|\cdot\vartheta(p)}\int_\Sigma\left(\int_{H/H_p}f(hq)\,
d(hH_p)\right)\vartheta(q)\, dq,
\end{equation}
where the Riemannian measure on $H/H_p$ is chosen to be the one
induced by $H\cdot p\subset M$.

If $f$ is additionally $H$-invariant, we have

\begin{equation}
\label{eqn_IntFormulaInvariant} \int_M
f(x)\,dx=\frac{{\rm Vol}(M)}{\int_\Sigma \vartheta(q)\, dq}
\int_\Sigma f(q)\vartheta(q)\, dq.\end{equation}
\end{Thm}

\begin{Rem}
If $H$ is an arbitrary symmetric subgroup of $G$, the theorem is
true with $\vartheta$ defined as follows: Conjugate $H$ such that
the regular point $p$ is the origin $eK\in M$ and consider the
decomposition
$$T_pHp=({\mathfrak z}_{\mathfrak m}({\mathfrak h})\cap {\rm pr}_{\mathfrak m} {\mathfrak h})\oplus\sum_{\beta\in
\Delta^+_{\mathfrak t}}\sum_i V_{\beta,i}$$ of Proposition
\ref{prop_ShapeOpGeneral}. Choose nonzero $x_{\beta,i}\in
V_{\beta,i}$ and $X_{\beta,i}\in {\mathfrak h}$ with
$X_{\beta,i}(p)=x_{\beta,i}$. Then choose $w_{\beta,i}\in {\mathfrak t}$
with $\beta(w_{\beta,i})=1$ and let $t_{\beta,i}$ be a zero of
the Jacobi field $X_{\beta,i}\circ \gamma_{\beta,i}$, where
$\gamma_{\beta,i}$ is the geodesic in direction $w_{\beta,i}$.
Then define $\vartheta:{\mathfrak t}\to \mathbf{R}$ as
\begin{equation}
\vartheta(w)=\prod_{\beta\in \Delta_{\mathfrak t}^+}\prod_i
|\sin(\beta(w)-t_{\beta,i})|^{\dim V_{\beta,i}}.
\end{equation}
Since the proof is completely analogous using the calculations in
Section \ref{sec_EV_General} instead of Theorem
\ref{thm_ShapeOp}, we will prove the theorem only in the case of
commuting involutions.

We also remark that in the noncompact case, the theorem remains
true if $\vartheta$ is defined using hyperbolic functions.
\end{Rem}

\begin{Lemma} \label{lem_Fubini} {\bf (Generalized Cavalieri Principle)}
Let $M$ be a Riemannian manifold such that a subset $U\subset M$
of full measure can be written as $L\times N$, equipped with a
Riemannian metric of the form $g(q,r)=\left(\begin{matrix}h(q,r) &
0 \\ 0 & k(r)\end{matrix}\right)$, where $(q,r)\in L\times N$.
Then for any integrable function $f$ on $M$, we have
\begin{equation*} \int_M f(p)\, dM=\int_{L\times N} f(q,r)\,
d(L\times N)=\int_N\left( \int_{L_r} f(q,r)\, dL_r\right) dN.
\end{equation*}
\end{Lemma}
\begin{proof}
Applying Fubini's theorem in coordinates $\phi:V\times
W\overset{\sim}{\to} V'\times W'\subset U$ yields
\begin{align*}
\int_{V'\times W'}&f(q,r)\, dM=\int_{V\times W} f\circ
\phi(x,y)\sqrt{\det
(g_{ij}(x,y))}\, d(x,y)\\
&=\int_{W}\left(\int_V f\circ \phi(x,y)\sqrt{\det (h_{ij}(x,y))}\,
dx\right)\sqrt{\det (k_{ij}(y))}\, dy.\\
&= \int_{W'}\left(\int_{V'_r}f(q,r)\, dV'_r\right)dW'.
\end{align*}\vskip-20pt
\end{proof}

Applying Lemma \ref{lem_Fubini} to the regular set in $M$, we
obtain that for any integrable function $f$ on $M$,
\begin{align}
\int_M f(x)\, dx&=\int_{\bar{Q}} \left(\int_{H\cdot {\rm Exp}(w)}
f(x)\,
dx\right) \, dw \nonumber \\
&=\frac{1}{|W|}\int_\Sigma\left( \int_{Hq}f(x)\, dx\right)\, dq,
\label{eqn_IntSigmaOrbit}
\end{align}
where $\bar{Q}\subset {\mathfrak t}$ is a generalized Weyl chamber, and $W$
is the generalized Weyl group of the $H$-action on $M$.

\begin{Lemma} \label{lem_CompareVolumes}
The Riemannian densities $\mu$ and $\mu'$ of the $H$-orbits
through two regular points $p={\rm Exp}(w)$ and $q={\rm Exp}(w+tu)$ are
related by the formula
\begin{align*}
\mu'\circ \varphi=F_{p}(q)\cdot \mu,
\end{align*} where
\begin{align}
F_{p}(q)= &\prod_{\beta\in \Delta^+_{\mathfrak t}}
\left(\cos(\beta(tu))+\cot(\beta(w))\sin(\beta(tu))\right)^{p_\beta}
\nonumber \\ & \quad\,\times
\left(\cos(\beta(tu))-\tan(\beta(w))\sin(\beta(tu))\right)^{h_\beta}.\label{eqn_RelativeDensities}
\end{align}
Here the exponents are the relative root multiplicities
$h_\beta=\dim {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$ and $p_\beta=\dim
{\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}$.
\end{Lemma}
\begin{proof}
Let $\xi$ be the $H$-equivariant vector field on $Hp$ determined
by $\xi(p)=u(p)$. The polarity of the $H$-action implies that
$\xi$ is a parallel normal vector field (\cite{PalaisTerng},
Theorem 5.7.1) and the map $\varphi:={\rm Exp}_{Hp}\circ (t\xi)$ is a
diffeomorphism between $Hp$ and $Hq$. Since $\xi$ is
$H$-equivariant, so is $\varphi$, and consequently, for $X\in
{\mathfrak h}$, we have $d\varphi(X\cdot p)=X\cdot q$. In other words,
applying $d\varphi$ to $X\cdot p$ is the same as evaluating the
Jacobi field obtained by restriction of the Killing vector field
$X$ to the geodesic $c(t)={\rm Exp}_{Hp}(tu(p))$ at time $t$.

In order to compare the Riemannian densities of $Hp$ and $Hq$, we
need to calculate the determinant of $d\varphi_p$; the
argumentation above shows that this amounts to explicitly
calculate these Jacobi fields.

Let $X\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak h}$, $v= X(eK)$ and $v'={}_\gamma
||_0^1v\in T_pHp$. The initial values of the Jacobi field
$Y_v(t)=X\cdot c(t)$ along $c$ have been calculated in the proof
of Theorem \ref{thm_ShapeOp}. Therefore
\begin{equation}\label{eqn_NormalizedJacFieldH}
\frac{1}{\cos(\beta(w))}Y_v(t)=(\cos(\beta(tu))-\tan(\beta(w))\sin(\beta(tu))){}_c||_0^t
v'.\end{equation}

If $X\in {\mathfrak z}_{\mathfrak m}({\mathfrak t})\cap {\mathfrak h}$, $v=X(eK)$ and $v'={}_\gamma ||_0^1
v$, then \begin{equation}
\label{eqn_NormalizedJacFieldT}Y_v(t)={}_c ||_0^t v'\end{equation}
is the restriction of $X$ to $c$.

If $v\in {\mathfrak m}_\beta^{\mathfrak t}\cap {\mathfrak p}$, $v'={}_\gamma ||_0^1v$ and $X\in
{\mathfrak k}_\beta^{\mathfrak t}\cap {\mathfrak h}$ is such that $[H,X]=-\beta(H)v$ for all
$H\in {\mathfrak t}$, we analogously see that the Jacobi field $Y_v$,
defined by $Y_v(t)=X\cdot c(t)$, is given by
\begin{equation}
\label{eqn_NormalizedJacFieldP}
\frac{1}{\sin(\beta(w))}Y_v(t)=(\cos(\beta(tu))+\cot(\beta(w))\sin(\beta(tu))){}_c||_0^t
v'.\end{equation}

If $\mu$ and $\mu'$ are the Riemannian densities of $Hp$ and $Hq$,
respectively, we conclude from (\ref{eqn_NormalizedJacFieldH}),
(\ref{eqn_NormalizedJacFieldT}) and
(\ref{eqn_NormalizedJacFieldP}) that
\begin{align*}\mu'(\varphi(p))=\mu(p)\cdot &\prod_{\beta\in
\Delta^+_{\mathfrak t}}
\left(\cos(\beta(tu))+\cot(\beta(w))\sin(\beta(tu))\right)^{p_\beta}
\\ &\quad\,\times
\left(\cos(\beta(tu))-\tan(\beta(w))\sin(\beta(tu))\right)^{h_\beta},\end{align*}
since the Jacobi fields in question are multiples of parallel
vector fields and therefore stay orthogonal to each other.
\end{proof}
\begin{Cor}
\label{cor_CompareVolumes} With the same notation as in Lemma
\ref{lem_CompareVolumes}, the volumes of $Hp$ and $Hq$ are
related by \begin{equation*} {\rm Vol}(Hq)={\rm Vol}(Hp)\cdot
F_p(q).\end{equation*}
\end{Cor}

Lemma \ref{lem_CompareVolumes} now enables us to replace the inner
integral in Equation (\ref{eqn_IntSigmaOrbit}) by an integral over
a fixed regular orbit $Hp$, where $p\in\Sigma$. If we denote the
Riemannian density of the orbit $Hq$ by $\mu_q$, and
$\varphi_q:Hp\to Hq$ is the equivariant diffeomorphism introduced
in Lemma \ref{lem_CompareVolumes}, we have
\begin{align}
\int_{M} f(x)\,
dx&=\frac{1}{|W|}\int_{\Sigma}\left(\int_{Hq}f(x)\,
dx\right)\, dq \nonumber\\
&=\frac{1}{|W|}\int_{\Sigma}
\left(\int_{Hp}f(\varphi_q(x))\frac{\mu_q(\varphi_q(x))}{\mu_p(x)}\,
dx\right) \, dq\nonumber \\
&=\frac{1}{|W|}\int_{\Sigma} \left(\int_{H/H_p}f(hq)\,
dh\right)F_{p}(q)\, dq.\label{eqn_IntOverHp}
\end{align}

A short calculation using the addition theorems for the
trigonometric functions shows that
\begin{equation}\label{eqn_DensitiesTheta}
\frac{\mu_q\circ
\varphi_q}{\mu_p}=F_p(q)=\frac{\vartheta(q)}{\vartheta(p)}.
\end{equation}
Now equality (\ref{eqn_IntFormulaNotInvariant}) follows from
(\ref{eqn_IntOverHp}) and (\ref{eqn_DensitiesTheta}).

If we additionally assume $f$ to be $H$-invariant, we can argue
as in Theorem 3.5 of \cite{CNV}: on the one hand, we clearly have
\begin{equation*}
\int_M f(x)\, dx= \frac{1}{|W|}\int_\Sigma {\rm Vol}(Hq)f(q)\, dq.
\end{equation*}
On the other hand, Corollary \ref{cor_CompareVolumes} and Equation
(\ref{eqn_DensitiesTheta}) show that
\begin{equation*}
{\rm Vol}(Hq)=\frac{{\rm Vol}(Hp)}{\vartheta(p)}\cdot \vartheta(q),
\end{equation*}
so the quotient ${\rm Vol}(Hp)/\vartheta(p)$ does not depend on $p$,
and we may denote it by $V(M)$. Then we have
\begin{equation*}
{\rm Vol}(M)=\frac{1}{|W|}\int_\Sigma {\rm Vol}(Hq)\,
dq=\frac{V(M)}{|W|}\cdot \int_\Sigma \vartheta(q)\, dq.
\end{equation*}

Combining these three equalities, we have shown
(\ref{eqn_IntFormulaInvariant}). On the way, we have also proven
\begin{Prop} For any regular point $p$,
\begin{equation}
{\rm Vol}(Hp)={\rm Vol}(M)|W|\frac{\vartheta(p)}{\int_\Sigma
\vartheta(q)dq}.
\end{equation}
\end{Prop}

\section{Example: ${\rm U}(p+q)$ Acting on ${\rm SO}(2p+2q)/{\rm S}({\rm O}(2p)\times
{\rm O}(2q))$} \label{sec_Example}

In this section, we determine the function $\vartheta$ introduced
in Section \ref{sec_Appl} for the $H={\rm U}(p+q)$-action on the
symmetric space
$$M={\rm SO}(2p+2q)/{\rm S}({\rm O}(2p)\times {\rm O}(2q)).$$ Let us assume that
$p\le q$.

On the Lie algebra ${\mathfrak g}={\mathfrak{so}}(2p+2q)$, we consider the two
involutions $\sigma_1(X)=I_{2p,2q}XI_{2p,2q}$ and
$\sigma_2(X)=J_{p+q}XJ_{p+q}^{-1}$, where
$$\quad I_{2p,2q}=\left(\begin{matrix}I_p & 0 & 0 & 0 \\ 0 & I_p & 0 & 0
\\ 0 & 0 & -I_q & 0 \\ 0 & 0 & 0 & -I_q\end{matrix}\right) \quad \text{
and }    J_{p+q}= \left(\begin{matrix}0 & I_p & 0 & 0 \\ -I_p & 0
& 0 & 0
\\ 0 & 0 & 0 & I_q \\ 0 & 0 & -I_q & 0\end{matrix}\right). $$ Clearly,
$\sigma_1$ and $\sigma_2$ commute. The corresponding fixed point
algebras are ${\mathfrak k}={\mathfrak g}^{\sigma_1}={\mathfrak{so}}(2p)\oplus {\mathfrak{so}}(2q)$ and

\begin{equation*}
{\mathfrak h}={\mathfrak g}^{\sigma_2}=\left\{\left.\left( \begin{array}{cc|cc} a & -b
& c & -d \\ b & a & d & c \\ \hline -c^t & -d^t & e & -f \\
\smash{\underbrace{d^t}_{p}} & \smash{\underbrace{-c^t}_{p}} &
\smash{\underbrace{f}_{q}} & \smash{\underbrace{e}_{q}}
\end{array}\right)\right| \begin{matrix} a^t=-a & e^t=-e \\ b^t=b
& f^t=f \end{matrix} \right\}\cong{\mathfrak{u}}(p+q).
\end{equation*}\vskip12pt

Let the respective Cartan decompositions of ${\mathfrak g}$ be
${\mathfrak g}={\mathfrak k}\oplus {\mathfrak m}$ and ${\mathfrak g}={\mathfrak h}\oplus {\mathfrak p}$. We have
$${\mathfrak m}=\left\{\left(\begin{array}{c|c} 0 & a \\ \hline -a^t &
0\end{array}\right)\in {\mathfrak g}\right\}$$ and
\begin{equation*}
{\mathfrak p}=\left\{\left.\left(\begin{array}{cc|cc} u & v & w & x \\ v &
-u & x & -w \\ \hline -w^t & -x^t & y & z \\ -x^t & w^t & z & -y
\end{array}\right)\right| \begin{matrix}u^t=-u & v^t=-v \\ y^t=-y
& z^t=-z\end{matrix}\right\}.
\end{equation*}
The intersections ${\mathfrak m} \cap {\mathfrak p}$ and ${\mathfrak m} \cap {\mathfrak h}$ are given by
$${\mathfrak m}\cap {\mathfrak p}=\left\{\left(\begin{array}{cc|cc}0 & 0 & w & x \\ 0
& 0 & x & -w \\ \hline -w^t & -x^t & 0 & 0 \\ -x^t & w^t &  0 &
0\end{array}\right)\in {\mathfrak g}\right\}$$ and $${\mathfrak m}\cap
{\mathfrak h}=\left\{\left(\begin{array}{cc|cc} 0 & 0 & c & -d \\ 0 & 0 & d
& c \\ \hline -c^t & -d^t & 0 & 0 \\ d^t & -c^t &  0 &
0\end{array}\right)\in {\mathfrak g}\right\}.$$ For $1\le i\le p$ and $1\le
j\le q$, let $E_{i,j}$ be the $p\times q$-matrix having $1$ in the
$(i,j)$-th entry and zeros elsewhere, and define

$$Q_{i,j}=\left(\begin{array}{cc|cc} 0 & 0 & E_{i,j} & 0 \\ 0 & 0
& 0 & -E_{i,j}\\ \hline -E_{i,j}^t & 0 & 0 & 0 \\ 0 & E_{i,j}^t &
0 & 0\end{array}\right)\in {\mathfrak m}\cap {\mathfrak p},$$
$$
R_{i,j}=\left(\begin{array}{cc|cc} 0 & 0 & 0 & E_{i,j} \\ 0 & 0 &
E_{i,j} & 0\\ \hline 0 & -E_{i,j}^t & 0 & 0 \\ -E_{i,j}^t & 0 & 0
& 0\end{array}\right)\in {\mathfrak m}\cap {\mathfrak p},$$

$$F_{i,j}=\left(\begin{array}{cc|cc} 0 & 0 & E_{i,j} & 0 \\ 0 & 0 &
0 & E_{i,j}\\ \hline -E_{i,j}^t & 0 & 0 & 0 \\ 0 & -E_{i,j}^t & 0
& 0\end{array}\right)\in {\mathfrak m}\cap {\mathfrak h},$$
$$
G_{i,j}=\left(\begin{array}{cc|cc} 0 & 0 & 0 & -E_{i,j} \\ 0 & 0
& E_{i,j} & 0\\ \hline 0 & -E_{i,j}^t & 0 & 0 \\ E_{i,j}^t & 0 &
0 & 0\end{array}\right)\in {\mathfrak m}\cap {\mathfrak h}.$$

Then we define abelian subalgebras by $${\mathfrak t}=\sum_{i=1}^p \mathbf{R}
Q_{i,i}\subset {\mathfrak m}\cap {\mathfrak p},\quad {\mathfrak t}'=\sum_{i=1}^p \mathbf{R}
F_{i,i}\subset {\mathfrak m}\cap {\mathfrak h},\quad {\mathfrak a}={\mathfrak t}\oplus {\mathfrak t}'\subset {\mathfrak m}$$
and consider the root space decomposition of $({\mathfrak g},{\mathfrak k})$ with
respect to ${\mathfrak a}$: For $1\le i\le 2p$, let linear forms
$\lambda_i:{\mathfrak a}\to \mathbf{R}$ be defined by
$$
\lambda_i(Q_{j,j})=\begin{cases}\delta_{i,j} & \text{if }i\le p \\
-\delta_{i-p,j} & \text{if }i>p \end{cases},\quad
\lambda_i(F_{j,j})=\begin{cases}\delta_{i,j} & \text{if }i\le p \\
\delta_{i-p,j} & \text{if }i>p \end{cases}.
$$
The restricted roots are $\pm \lambda_i$ for $1\le i\le 2p$ and
$\pm (\lambda_i\pm \lambda_j)$ for $ 1\le i<j\le 2p$; we choose
as positive roots the $\lambda_i$ and the $\lambda_i\pm
\lambda_j$ for $i<j$. Denoting the restriction of $\lambda_i$ to
${\mathfrak t}$ by $\lambda_i^{\mathfrak t}$, we obtain the adapted root space
decomposition with respect to ${\mathfrak t}$
\begin{equation*}
{\mathfrak m}={\mathfrak z}_{\mathfrak m}({\mathfrak t})\oplus \sum_{1\le i\le
p}{\mathfrak m}_{\lambda_i^{\mathfrak t}}^{\mathfrak t}\oplus \underset{i<j\le 2p}{\sum_{1\le
i\le p}} {\mathfrak m}_{\lambda_i^{\mathfrak t}\pm \lambda_j^{\mathfrak t}}^{\mathfrak t}
\end{equation*}
where
\begin{align*}
{\mathfrak m}_{\lambda_i^{\mathfrak t}}^{\mathfrak t}= \underbrace{\sum_{k=p+1}^q \mathbf{R}
F_{i,k}\oplus \mathbf{R} G_{i,k}}_{\subset {\mathfrak m}\cap{\mathfrak h}} \oplus
\underbrace{\sum_{k=p+1}^q \mathbf{R} Q_{i,k}\oplus \mathbf{R} R_{i,k}}_{\subset
{\mathfrak m}\cap {\mathfrak p}}
\end{align*}
for $1\le i\le p$,
\begin{equation*}
{\mathfrak m}_{\lambda_i^{\mathfrak t}\pm \lambda_j^{\mathfrak t}}^{\mathfrak t}= \underbrace{\mathbf{R}
(F_{i,j}\pm F_{j,i})}_{\subset {\mathfrak m}\cap {\mathfrak h}} \oplus \underbrace{\mathbf{R}
(Q_{i,j}\pm Q_{j,i})}_{\subset {\mathfrak m}\cap {\mathfrak p}}
\end{equation*}
for $1\le i<j\le p$ and
\begin{equation*}
{\mathfrak m}_{\lambda_i^{\mathfrak t}\pm \lambda_j}^{\mathfrak t}=\underbrace{\mathbf{R} (G_{i,j-p}\pm
G_{j-p,i})}_{\subset {\mathfrak m}\cap {\mathfrak h}} \oplus \underbrace{\mathbf{R}
(R_{i,j-p}\pm R_{j-p,i})}_{\subset {\mathfrak m}\cap {\mathfrak p}}
\end{equation*}
for $1\le i\le p$ and $p< j\le 2p$. Therefore,
\begin{align*}
\vartheta&=\prod_{i=1}^p\prod_{j=1}^{2p}
|\sin(\lambda_i^{\mathfrak t}+\lambda_j^{\mathfrak t})||\sin(\lambda_i^{\mathfrak t}-\lambda_j^{\mathfrak t})||\cos(\lambda_i^{\mathfrak t}+\lambda_j^{\mathfrak t})||\cos(\lambda_i^{\mathfrak t}-\lambda_j^{\mathfrak t})|
\\
& \quad\times \prod_{i=1}^p |\sin (\lambda_i^{\mathfrak t})|^{2(q-p)} |\cos
(\lambda_i^{\mathfrak t})
|^{2(q-p)} \\
& = \prod_{i,j=1}^p
|\sin(\lambda_i^{\mathfrak t}+\lambda_j^{\mathfrak t})|^2|\sin(\lambda_i^{\mathfrak t}-\lambda_j^{\mathfrak t})|^2|\cos(\lambda_i^{\mathfrak t}+\lambda_j^{\mathfrak t})|^2|\cos(\lambda_i^{\mathfrak t}-\lambda_j^{\mathfrak t})|^2
\\
& \quad\times \prod_{i=1}^p |\sin (\lambda_i^{\mathfrak t})|^{2(q-p)} |\cos
(\lambda_i^{\mathfrak t}) |^{2(q-p)},
\end{align*}
where we have used that $\lambda_{i+p}^{\mathfrak t}=-\lambda_{i}^{\mathfrak t}$ for
$1\le i\le p$.

redpajama_set_name: RedPajamaArXiv


Noel Keywood covers the bases.
Jon Myles loves the smooth sound of Quadral's Chromium Style 8.
Tannoys you can afford, reviewed by Jon Myles.
Martin Pipe is overwhelmed by the sound of Chord's Hugo TT DAC.
Noel Keywood checks out a �ber-budget hi-res portable.
A preamp that does it all, Noel Keywood finds.
Big power from Naim, now with extra refinement, Jon Myles finds.
Jon Myles listens to a new Bluetooth loudspeaker.
Jon Myles travels to Hastings to meet Harry and Lucy.
Martin Pipe explains all about the micro cassette.
A new affordable turntable/arm combination from Pro-Ject impresses Martin Pipe.
Paul Rigby extols the virtue of Audio Origami's latest uni-pivot tonearm.
The upgraded phonostage from Parasound is a potent performer, says Paul Rigby.
107 The Beatles Are Coming!
Paul Rigby reviews a new book on The Beatles' time in America.
Jon Myles connects up to top In-ear 'phones.
Marvin Gaye & Tammi Terrell 'United', by Paul Rigby.


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ACCESSWIRE: Business
PRESS RELEASE: Paid content from ACCESSWIRE
The Top 10 Business Podcasts to Listen to in 2020
NEW YORK, NY / ACCESSWIRE / January 13, 2020 / Whether you are looking for motivation, teachings, or practical business tools, there is a podcast for your needs. Today, there are over 500,000 active podcasts online. This year, VIP Media Group, selected the top 10 business podcasts to listen to in 2020.
-Ace Weekly
Ace Weekly is a personal development podcast focused on the business and entrepreneurship space. The host of the podcast is Andrew Evans, a young millennial entrepreneur based in Miami. Andrew invites peak performers and guests from around the world to share with his listeners the skills, habits, and the mentality that is needed to succeed in the modern world. The current landscape of sales, entrepreneurship, and achievement is changing radically. Ace Weekly teaches people how to dominate their industry (regardless of whether or not they are a business owner or "entrepreneur") and find fulfillment in the process.
-The Ten Minute Podcast
The 10 Minute Entrepreneur is a podcast for entrepreneurs and small business owners, hosted by serial entrepreneur Sean Castrina. He started his first company when he was 23. In just over 20 years, he has started over 15 successful companies in various industries such as retail, marketing and real estate, and has a portfolio that continues to grow. In this podcast, Sean shares quick hitting tips to help equip you for success. Both tactical and practical, Sean's advice will help anyone thrive.
-The Garyvee Audio Experience
The Garyvee Audio Experience is hosted by entrepreneur, CEO, investor, vlogger, and public speaker Gary Vaynerchuk. Gary is the chairman of VaynerX, a modern-day media and communications holding company, and the active CEO of VaynerMedia, a full-service advertising agency servicing Fortune 100 clients across the company's 4 locations. In his podcast, Gary shares keynote speeches on marketing and business, segments from his DAILYVEE video series, interviews and fireside chats he has given, as well as new and current thoughts he records for this audio experience.
-Cardone Zone Podcast
Cardone Zone Podcast is hosted by Grant Cardone, one of the top sales trainers, real estate investors and bestselling authors in the world. Grant is the CEO of CardoneCapital, he is also an international speaker, entrepreneur, and creator of 21 best-selling business programs. He also owns and operates seven privately held companies and a $1.5B portfolio of multifamily properties. In this show, Grant covers his main areas of interest with content on topics such as Real Estate Investing Made Simple, as well as power player interviews with superstar entrepreneurs, authors, experts, coaches, and business leaders.
- Start Today Podcast with Chris Cavallini
The Start Today Podcast with Chris Cavallini is a show about taking responsibility over your life and making the conscious decision to raise your standards. In every episode, Chris Cavallini, an entrepreneur, philanthropist, United States Navy Veteran, speaker, and social media influencer, takes you on a journey of personal development, mental toughness and self-mastery by teaching the mindset and practical action steps required to reach the next level.
- Investing in Real Estate with Clayton Morris
Investing in Real Estate is a podcast hosted by Clayton Morris, Morris Invest founder who has mastered how to acquire rental real estate, build net worth and accelerate financial freedom. This podcast shares expert interviews, case studies with normal everyday investors, and Clayton's own methods for achieving passive income. If you want to learn how to acquire turnkey rentals, discounted properties, passive income and true legacy wealth this podcast is for you.
- RISE podcast
The RISE podcast is for anyone who wants to acquire tangible and tactical tools that can be useful for their life and business. This show is hosted by New York Times Best-Selling Author Rachel Hollis, who is interested in arming people with the tools to make positive and lasting change. In RISE, listeners will find a series of bold conversations with other business powerhouses and personal development leaders that provides the listener with real-life valuable takeaways.
- Real AF with Andy Frisella
In Real AF, host Andy Frisella and his guests debate, discuss and laugh about trending topics and hot-button issues. Andy Frisella is a highly successful entrepreneur and speaker who leads the #100to0 movement, for people of integrity and intensity who want to win big at business and life. That is why this podcast is for anyone interested in growing and scaling a business, but also about modern hottakes and trending debates.
-The Tai Lopez Show
The Tai Lopez Show is a podcast that showcases the best business education straight from the world's top entrepreneurs. In this podcast, investor, entrepreneur and author Tai Lopez also reviews the best books in health, wealth, love and happiness that will help you achieve your maximum potential and live the best life possible.
-The Tim Ferriss Show
The Tim Ferriss Show deconstructs world-class performers from eclectic areas (investing, chess, pro sports, etc.), digging deep to find the tools, tactics, and tricks that listeners can use. The host, Tim Ferriss, is a self-experimenter and bestselling author, best known for The 4-Hour Workweek, which has been translated into 40+ languages. Newsweek calls him "the world's best human guinea pig," and The New York Times calls him "a cross between Jack Welch and a Buddhist monk."
Paula Henderson
About VIP Media Group
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text:
Storm sweep Mystics, win third WNBA title
She's overcome a knee injury that forced her to sit out the 2013 season and survived a four-year roster reconstruction that resulted in two years of missing the postseason followed by a pair of first-round playoff exits.
The Seattle Storm hold a commanding 2-0 lead in the five-game series, so it's do or die for the Washington Mystics in game three of the WNBA Finals Wednesday night.
In addition to Stewie's impressive stats, forward Natasha Howard hit a career-high 29 points and 14 rebounds. Seattle won 26 games during the regular season - 11 more than the 2017 campaign - entered the playoffs as the No. 1 seed, and swept the finals.
Seattle may have lost its SuperSonics over a decade ago but the city has now earned itself a championship courtesy of the WNBA's Storm.
"Stewie was just wonderful", Storm coach Dan Hughes said.
Skylar Diggins-Smith, Wings: Now a three-time All-WNBA honoree, she averaged a career-high 6.2 assists (third in the WNBA) to go with 17.9 points while playing a league-high 34.1 minutes per game. Her running mate for Seattle's first two titles was Lauren Jackson. It was really imperative that Jewell [Loyd] kind of become part of our flow, and I thought she did a great, great job of giving us that flow at that point.
Bird, also a member of a Seattle's championship teams in 2004 and 2010, was certainly appreciative of the title - and the growth of the Storm's younger players.
And now it is Bird and Stewart and Loyd and Howard and more, a new championship cast is Seattle.
Kristi Toliver and Elena Delle Donne sit on the bench during the final seconds of Game 3 while the Seattle Storm become WNBA champions
"There is probably no comparison to be honest".
The Mystics, on the other hand, didn't have a single player score in double figures, and only had two points off the bench. Our team went through a rebuild and yes, I made a decision to stay. "You knew that this was game two and they were going to come out like a better D.C. team than we saw in game one".
The coach sensed something brewing early in his first year with the franchise.
Ok, maybe not to that extreme, but I do think shots will fall. "All year you could just see the escalation".
Elena Delle Donne scored 23 points, and Kristi Toliver had 22 for the Mystics, who were appearing in the WNBA Finals for the first time in franchise history. "So we ran into each other in the lounge, and obviously we knew each other before that, but not that well. This is a great arena, could be electric in here if we fill it up".
"We understand that the difference here is we have a chance to win a championship and not just advance to the next round", Bird said. The Mystics were relegated to Smith Center on the campus of George Washington for their home Semifinals games, but scheduling issues there pushed Wednesday night's WNBA Finals game all the way out to Fairfax.
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Florence dumps heavy rain on Carolinas, kills 7
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Apple A12 iPhone XS AnTuTu benchmark score is mind-blowing, crosses 360K!
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GM recalls more than 1 million pickups and SUVs
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Thimbleweed Park's Wireframe Writing Betrays Its Hefty Puzzles
By Stuart Arias | April 13, 2017 | 12:00pm
Games Reviews Thimbleweed Park
For the first two hours or so, Thimbleweed Park, the latest game by ex-LucasArts adventure game vets Ron Gilbert and Gary Winnick, convinces players that they are in for a standard murder mystery. It introduces its first two main characters, Agent Ray and Reyes, through snappy banter that feels reminiscent of Mulder and Scully from The X-Files. Ray comments on how the dead body before them is starting to "pixelate", which Reyes jots down in his journal like an eager student. It was a cute moment, despite the grim situation at hand. I didn't mind the game proclaiming "We're in a video game!" so blatantly and so early into the game. There's nothing wrong with a welcoming wink and nod, as long as it leads to something more.
Seven hours into the game, I find myself in control of five different characters, with access to a variety of locations. The two FBI agents have been relegated to walking inventories, accompanied by an über-geek game designer, a foul-mouthed circus clown, and a ghost. What distinguishes these other characters from their duller counterparts are unique moments of characterization early in the game: Dolores, the game designer, torn between her dreams and her rich uncle's wishes; Ransome the Clown, whose vitriolic standup leads to his mystical downfall; and Franklin, an inventor whose ambitions and needs are suppressed by his meekness, both in life and death. These moments offer promising glimpses into something deeper than just a murder mystery, hinting at themes about family dysfunction, corporate intrusion, and the boundaries of comedy. However, sadly, all of this is put to the wayside once the ruse is up and the game's intentions become clear: Thimbleweed Park is more invested in the logical efficiency of its puzzle design and incessant self-indulgent humor than developing any kind of originality and depth in its storytelling.
The former isn't in and of itself a bad thing. The puzzles in Thimbleweed Park have a pace and structure that effectively correct the meandering, nonsensical design decisions of adventure games of yore. They are all very absurd in nature, but make sense within the universe the game exists in. The puzzles shine most brightly in the first three chapters, during the sections you play as the FBI agents. Their singular united goal of solving a murder case breaks down the task into a series of smaller puzzles, each one rewarding for the journey, not for the punchlines carried through. Exploring the town and talking to its peculiar residents at first feels fresh, genuinely evoking the best of its adventure game ancestry. Fresh enough to make the first signs of pronounced eye-rolling meta humor ("This cut-scene is starting to get long and it's only going to get longer!") and pop culture regurgitation seem inoffensive.
Once the gameplay shifts towards working on checklists rather than focused objectives, the mediocrity begins to show itself more clearly. It turns into a formula of passing items between characters who never properly meet or talk to each other, working as one mutually disconnected team. Puzzles grow wider but not deeper, by having the player constantly traverse multiple places for items and clues while abandoning any sense of direction. Environments start to lose their visual and tonal appeal the more they become vessels for nostalgic pandering. Each story revelation becomes a new chance to throw more boiler plate genre clichés into the pot, as if relying on TV murder mystery iconography wasn't enough. Despite its visual crudeness by today's standards, Gilbert's seminal Maniac Mansion evoked B-movie sci-fi with a vivacious playfulness that revealed a genuine love for the subgenre. In Thimbleweed Park, sci-fi and horror are dismantled into various alluring pieces, each one used to desperately keep the player's attention.
This reliance on basic geek absurdities slowly evolves into a spiteful blend of self-deprecating and self-aggrandizing humor. One minute, the game sarcastically proclaims, "Who the hell wants to be a game designer?", the next, it proudly explains to the player how it's totally NOT like those old, ineffective adventure games. These detours into "the plight of the game designer" aren't so much as looking into the funhouse mirror, but more like gleefully smashing your head into it. All of this is compounded during the final act, where puzzles become easier and very linear, shifting the player's attention to the fourth wall that will be demolished before them. That early game meta-humor is revealed to have been foreshadowing for the game's actual narrative intentions. It wholeheartedly gives into trendy video game navel gazing, the kind that has prevailed in the medium ever since Bioshock made telling players they're playing a game sound like an original revelation.
After the game's credits finish rolling, we are presented with a Commodore 64 loading screen. To many, this might instill a sense of nostalgia for their youth growing up in this era of video game history. To me, it recalled a small moment early in the game, where Dolores expresses her love for her own C64. Her elated observations display an impassioned investment within the writing. The self-deprecation is, mostly, dropped, and through the tech jargon, we see the same kind of vibrant eagerness that Maniac Mansion had for its own subjects. All these moments, made most effective by the charming voice acting and animations, left me frustrated. Not because they were bad, but because they represented unfulfilled potential.
It wasn't long ago that Tim Schafer and his team at Double Fine released Broken Age, a game that was, admittedly, one half-terrible adventure game design, but managed to be two parts a powerful tale about family expectations, duty, and heroism, told with Schafer's signature sense of humanistic humor and warmth. In his own journey of reliving his old LucasArt days, Schafer made a game that not only was funny, but intimate in both its design and writing. In Thimbleweed Park, Gilbert & Co. seem afraid to let their passions truly shine with earnest depth, focusing only on the most superficial aspects of them. In a way, the game succeeds in feeling like relic, though not for the right reasons. It's caught in a time where its references and plot twists might have been compelling and essential, but now sound like the ramblings of someone trying to catch up and stay relevant within the zeitgeist. For some, none of this matters, as long as the puzzles achieve acceptable levels of fun and logic. And I admit, they game does reach those levels, many times in fact. But if we hope to see this often-maligned genre grow out of the worst parts of its history, we should also set a higher standard for the kind of stories it can tell. Or in Thimbleweed Park's case, how it tells them and with what amount of conviction.
Thimbleweed Park was developed and published by Terrible Toybox. Our review is based on the PC version. It is also available for PC, Xbox One and PlayStation 4.
Stuart Arias is a freelance writer based in Chicago. You can find his daily ramblings on music and video games @Vida_Honrada. For longer ramblings, check out his blog.
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This body is not me; I am not caught in this body, I am life without boundaries, I have never been born and I have never died. Over there the wide ocean and the sky with many galaxies. All manifests from the basis of consciousness. Since beginningless time I have always been free. Birth and death are only a door through which we go in and out. Birth and death are only a game of hide-and-seek. So smile to me and take my hand and wave good-bye. Tomorrow we shall meet again or even before. We shall always be meeting again at the true source, Always meeting again on the myriad paths of life.
Edmonton, Royal Alberta Museum and Blackfoot Provincial Park ~ April 2015.
Hedy – Thank you for sharing this… today will be different. Your views have resonated deeply; the Cockburn reference is a perfect trailing flourish. I will return.
alive and calm….love that…and yes feel good…so thank you Markus for saying…makes me smile 😀 sending you joy!
These photos definitely resonates those incredible words ~ the feelings of life, "We shall always be meeting again at the true source" ~ timeless and true.
i appreciate your words Randall and perhaps it's true…i don't know, you know 🙂 …compose a happy day!
yes it's Alberta! 😀 and Gina thank you for your kind words…and taking time to write…please know i appreciate it…and my humble thanks to you 😀 compose a happy day Gina!
Thanks kindly Alois ☺compose a happy day!
How did you catch these animals in hibernation??


The pathway of a moving point usually made by depositing material on a contrasting surface.
the essence of a subject captured in the quickest and most economical way.
The imaginary centerline of a volume; the centerline is assumed to run in a volume's longest dimension is called the long axis, the centerline running at a right angle to the long axis is called a short axis.
A series of short, parallel strokes used to build up tone.
Consecutive sets of strokes is added over the first hatching lines at different angles to build even darker tones.
The condition of an absent line, but implied edge, that describes the presence of an important contour needed to complete a form. Used to depict a strong light falling on a smooth, unbroken surface, overlapping forms of the same or similar value or the turning away of a form.
Lines drawn across the surface of a form to depict the surface topography of that form.
Literally, beautiful writing. The beauty of line becomes a major aesthetic aim.
Shapes and volumes such as circles, spheres, squares, cubes, cylinders, triangles and cones seldom seen in nature as perfect forms, serve as forms that underlie natural forms or form the basis for actual human-made objects.
Shapes and surface structure of forms in nature.
Any three dimensional form, regardless of identity, structure of size.
The effect and degree of bulk, density and weight of three dimensional volume in space or effect of a two dimensional shape consisting of value, texture or color on the picture plane.
The quality of space (2D) or object (3D) in whole or part dependent on its outline.
In drawing serves as the subject of the drawing. In the figure ground relationship it is usually the figure and the opposite defined by negative shapes (space).
In drawing serves as the shapes or spaces around the subject or positive shapes (spaces). Must be equally considered as visually important as the positive shapes.
In drawing to treat the surface of the picture plane so as to create the illusion of depth, either shallow or deep, using representational or non-representational subject matter (as a window).
In drawing to treat the drawing surface in such a way to affirm its two dimensional nature (as an object).
A system (one, two and three point) by which the dimensions and spatial relationships of forms are represented on a flat surface to create an accurate depiction of form and space in the picture plane.
Literally "fool the eye", subject is drawn so that the eye is deceived into perceiving the formss and spaces are real.
The use of blurred or softened contours and dimmed or dulled colors (tending towards the blue) to suggest space by imitating the visual effect of atmosphere interveing between the observer and objects at a distance.
Referring to the subject of the drawing or the positive shapes (forms, volumes).
In drawing referring to the spaces around the subject also called negative space.
Referring to the designed interplay of positive shapes and forms with negative shapes and forms so as to create a visually interesting and balanced relationships between the two.
Defines a form positioned in such a way (usually forced perspective) so that it's shape is shortened in the direction of depth (line of sight).
One of the methods used to create the illusion of depth in the picture plane. Accomplished by positioning one form or mass in front fo another.
Using our intuitive understanding that obejects decrease in size as they recede into the distance, an artist may create the illusion depth in the picture plane by systematically reducing the size of similar forms in a composition.
The point of intersection where the converging lines of perspective meet the eye level/horizon line.
Always at your eye level and may be different than the horizon of the terrestrial plane.
The gradation of tone from light to dark, form white though grey to black.
The intrinsic value of an object regardless of the amount or quality of light on it.
In modeling a form or volume, the gradual blending of light to dark to create a three dimensional illusion.
A means of creating a smoky haziness that softens outlines, literally translated from Italian "in the manner of smoke".
Relating to light and dark, the absence of color, as opposed to chromatic (relating to color).
The design of contrasting light and dark masses in a drawing or painting.
A feeling of equality in weight, attention, or attraction of the various visual elements within the picture plane as a means of accomplishing organic unity.
A form of balance attained when the visula units on either side of a vertical axis are not identical but are placed in position within the pictorial field so as to create a "felt" equilibrium of the total from concept (eastern).
A form of balance achieved by the use of identical compositional units on either side of a vertical axis within the picture plane. (western).
Oppostion, unlikeness, arrangement so as to intensify differences.
The principle of visual organization which suggests that certain elements should assume more importance than others in the same composition. It contributes to organic unity by emphasizing the fact that there is one main feature and that other elements are subordinate to it.
The arbitrary organization or inventive arrangement of all of the visual elements accoring to principles which will develop an organic unity in the total work of art.
The wole or total effec of a work of art which results from the combination of all of its component parts.
The unity of all of the visual elelments of a composition achieved by the repetition of the same characteristics or those which are similar in nature.
The use of the same visual element a number of times in the same composition. It may accomplish dominance of one visual idea, a feeling of harmonious relationship, or an obviously planned pattern.
A continuance, a flow, or a feeling of a movement achieved by repetition of regulated visual units; the use of measured accents.
The obvious emphasis on certain visual form relationships and certain directional movements within the picture plane. Also refers to teh repetition of elements or the combination of elements in a readily recognized systematic organization.
The surface feel of an object or the representation of surface character. Texture is the actual and "visual feel" of surface areas as they are arranged and altered by man or nature.
A surface which stimulates a tactile response when actually touched.
A representation of an actual texture created by careful copying of the light and dark pattern characterstic of its surface.
Two dimensional patterns sometimes derived from actual textures, frequently varied to fit pictorial needs, and often freely created without reference to any item.