\documentstyle[12pt,amssym,leqno]{article}
\newtheorem{theorem}{\sc Theorem}[section]
\newtheorem{lemma}{\sc Lemma}[section]
\newtheorem{definition}{\sc Definition}[section]
\newtheorem{corollary}{\sc Corollary}[section]
\newtheorem{proposition}{\sc Proposition}[section]
\newtheorem{remark}{\sc Remark}[section]
\newtheorem{example}{\sc Example}[section]
\begin{document}
\begin{titlepage}
\vspace*{1cm}
\begin{center}
{\huge\bf Harmonic endomorphism fields}
\end{center}
\bigskip
\bigskip
\begin{center}
{\large\sc Eduardo Garc\'{\i}a--R\'{\i}o\footnote{Supported by projects XUGA
20701B93 and DGICYT PB94 - 0633 - C02 - 01}, $\,$ Lieven Vanhecke}
\medskip
{\large\sc and M. Elena V\'azquez--Abal$^1$}
\end{center}
\vspace{1,5cm}
\begin{center}
{\large\bf Abstract}
\end{center}
An endomorphism field $\varphi$ on a pseudo--Riemannian manifold $(M,g)$
determines a map $\varphi:TM$ $\rightarrow$ $TM$. We derive a necessary and
sufficient condition for $\varphi$ to be a harmonic map when the tangent
bundle $TM$ is equipped with the complete lift metric. We give several
examples and applications.
\vspace{4cm}
\noindent{\normalsize\bf 1991 M.S. Classification.} 53C20, 53C25, 58E20.
\bigskip
\noindent{\normalsize\bf Key words and phrases.} Endomorphism fields, tangent
bundles, harmonic maps.
\end{titlepage}
\section{Introduction}
A $(1,1)$--tensor field $\varphi$ on a pseudo--Riemannian manifold $(M,g)$
determines a map $\varphi:TM$ $\rightarrow$ $TM$, where $TM$ denotes the
tangent
bundle of $M$. The main purpose of this paper is to determine a necessary and
sufficient condition for $\varphi:(TM,g^C)$ $\rightarrow$ $(TM,g^C)$ to be {\it
harmonic}, where $g^C$ is the (pseudo--Riemannian) complete lift metric as
introduced by Yano and Ishihara in \cite{18}. Our main result is that the
harmonicity condition is equivalent to $\nabla^*\varphi$ $=$ $0$, where
$\nabla^*$ is the formal adjoint of the Levi Civita connection $\nabla$ of
$(M,g)$.
In the remaining part we illustrate this result by means of several natural
examples of endomorphism fields. In particular we consider the Ricci operator
of $(M,g)$ and also the shape operator of a hypersurface. The harmonicity of
the corresponding map is equivalent to the constancy of the scalar curvature
or, when $(M,g)$ is Einsteinian, to the constancy of the mean curvature of the
hypersurface. From these results we derive characterizations of harmonic
manifolds and manifolds of constant sectional curvature by using geodesic
spheres or tubes about geodesics as hypersurfaces. Further we consider the
structure $J$ on an almost Hermitian manifold $(M,g,J)$ and we also treat the
case of almost product structures. We finish by looking at these two kind of
structures on the tangent bundle $TM$ of $(M,g)$. In this way we provide
examples of harmonic maps of some special pseudo--Riemannian manifolds.
We refer to \cite{17} where the above notion of a harmonic $\varphi$ is used to
define harmonic foliations.
\section{Harmonic endomorphism fields}
Let $(M,g)$ be a connected, smooth pseudo--Riemannian manifold, $\nabla$ its
Levi Civita connection and $R$ the corresponding Riemannian curvature tensor
defined by $R_{XY}$ $=$ $\nabla_{[X,Y]}$ $-$ $[\nabla_X,\nabla_Y]$ for smooth
vector fields $X$, $Y$ on $M$. $\rho$ and $Q$, respectively, denote the Ricci
tensor of type $(0,2)$ and $(1,1)$, respectively and $\tau$ denotes the scalar
curvature.
Next, let $\varphi:M$ $\rightarrow$ $N$ be a smooth map between two
pseudo--Riemannian manifolds with metric $g$ and $h$, respectively and let
$\varphi^{-1}(TN)$ be the pull--back bundle. The Levi Civita connections on
$TM$ and $TN$ induce a connection $\nabla$ in the bundle of one--forms on $M$
with values in $\varphi^{-1}(TN)$. Then $\alpha_\varphi$ $=$ $\nabla d\varphi$
is a symmetric bilinear form on $TM$ which is called the {\it second
fundamental form} of $\varphi$. The trace of $\alpha_\varphi$ with respect to
$g$ is called the {\it tension field} of $\varphi$, and denoted by
$\tau(\varphi)$. The map $\varphi$ is said to be harmonic if $\tau(\varphi)$
$=$ $0$. (See \cite{7}, \cite{8}, \cite{9} for more details and references.)
Now, let $U\subset M$ be a domain with coordinates $(x^1,\dots,x^m)$, $m$ $=$
dim$M$ and $V\subset N$ be a domain with coordinates $(z^1,\dots,z^n)$, $n$ $=$
dim$N$, such that $\varphi(U)\subset V$ and suppose that $\varphi$ is locally
represented by $z^\alpha$ $=$ $\varphi^\alpha(x^1,\dots,x^m)$, $\alpha$ $=$
$1,\dots,n$. Then we have
\begin{equation}\label{eq:1}
(\alpha_\varphi)^\gamma_{ij}=
\frac{\partial^2\varphi^\gamma}{\partial x^i\partial x^j}-
{}^M\Gamma^k_{ij}\frac{\partial\varphi^\gamma}{\partial x^k}+
{}^N\Gamma^\gamma_{\alpha\beta}(\varphi)
\frac{\partial\varphi^\alpha}{\partial x^i}
\frac{\partial\varphi^\beta}{\partial x^j}.
\end{equation}
\noindent{Here} ${}^M\Gamma^k_{ij}$ and ${}^N\Gamma^\gamma_{\alpha\beta}$
denote
the Christoffel symbols of $(M,g)$ and $(N,h)$, respectively. So, $\varphi$ is
harmonic if and only if
\begin{equation}\label{eq:2}
\tau(\varphi)^\gamma=g^{ij}(\alpha_\varphi)^\gamma_{ij}=0
\end{equation}
for $\gamma$ $=$ $1,\dots,n$.
Now, let $TM$ denote the tangent bundle of $M$. This $2m$--dimensional manifold
may be equipped with the pseudo--Riemannian complete lift metric $g^C$, of
signature $(m,m)$, defined by
\begin{equation}\label{eq:3}
\left\{\begin{array}{l}
g^C(X^H,Y^H)=g^C(X^V,Y^V)=0,\\
\noalign{\medskip}
g^C(X^H,Y^V)=g^C(X^V,Y^H)=g(X,Y)^V.
\end{array}\right.
\end{equation}
\noindent{Here}, the horizontal and vertical lifts of tangent vector fields
$X$,
$Y$ on $M$ refer to the decomposition of the tangent space $TM$ at every
point in
horizontal vectors with respect to $\nabla$ and canonical vertical vectors. For
vector fields $X$, $Y$ on $M$ the function $g(X,Y)^V$ on $TM$ is the pull--back
of $g(X,Y)$ under the projection $TM$ $\rightarrow$ $M$. For local coordinates
$(x^1,\dots,x^{2m})$ $=$ $(x^1,\dots,x^m;x^{\bar{1}},\dots,x^{\bar{m}})$, where
$\bar{i}$ $=$ $i+m$, $i$ $=$ $1,\dots,m$, we have the local expression
\begin{equation}\label{eq:4}
g^C=\left(\begin{array}{cc}
x^{\bar{k}}\frac{\partial g_{ij}}{\partial x^k} & g_{ij}\\
\noalign{\medskip}
g_{ij} & 0
\end{array}\right),
\end{equation}
\noindent{$i$}, $j$ $=$ $1,\dots,m$ with respect to $(\frac{\partial}{\partial
x^1},\dots, \frac{\partial}{\partial x^m},\frac{\partial}{\partial
x^{\bar{1}}},\dots, \frac{\partial}{\partial x^{\bar{m}}})$. We refer to
\cite{18} for more details.
Finally, let $\varphi$ be a tensor field of type $(1,1)$ on $M$. Then $\varphi$
determines a map $\varphi:TM$ $\rightarrow$ $TM$.
\bigskip
\noindent{\normalsize\sc Definition.}\begin{enumerate}
\item[(i)] The endomorphism field $\varphi$ (or $(1,1)$--tensor field
$\varphi$) on $(M,g)$ is said to be {\it harmonic} if the map
$\varphi:(TM,g^C)$
$\rightarrow$ $(TM,g^C)$ is a harmonic map.
\item[(ii)] The $(0,2)$--tensor field $\Phi$ on $(M,g)$ determined by
$\Phi(X,Y)$ $=$ $g(\varphi X,Y)$ for all tangent vectors $X$, $Y$ on $(M,g)$ is
called {\it harmonic} if $\varphi$ is harmonic on $(M,g)$.
\end{enumerate}
\bigskip
Following \cite[p. 34]{2}, we denote by $\nabla^*$ the formal adjoint of the
Levi Civita connection $\nabla$ on $(M,g)$. Then we have with respect to local
coordinates
\begin{equation}\label{eq:5}
(\nabla^*\varphi)^k=-g^{ij}(\nabla_i\varphi)_j^k.
\end{equation}
Now we are ready to state and prove the
\bigskip
\noindent{\normalsize\sc Main Theorem.}
{\it The endomorphism field $\varphi$ on $(M,g)$ is harmonic if and only if
$\nabla^*\varphi$ $=$ $0$.}
\bigskip
\noindent{\normalsize\sc Proof.}
In local coordinates,
$(x,y)$ $=$ $(x^1,\dots,x^m,$ $x^{\bar{1}},\dots,x^{\bar{m}})$,
the map $\varphi:TM$ $\rightarrow$ $TM$ is given by
$$
\varphi(x,y)=
(x^1,\dots,x^m,\varphi^1_k(x)x^{\bar{k}},\dots,\varphi^m_k(x)x^{\bar{k}}).
$$
Moreover, since at $(x,y)$ the Christoffel symbols
${}^{TM}\Gamma^\alpha_{\beta\gamma}$ of the Levi Civita connection
$\nabla^C$ of
$g^C$, where $\alpha$, $\beta$, $\gamma$ $=$ $1,\dots,2m$, are given by
$$
{}^{TM}\Gamma^k=\left(\begin{array}{cc}
\Gamma^k_{ij} & 0\\
\noalign{\medskip}
0 & 0
\end{array}\right),\quad {}^{TM}\Gamma^{\bar{k}}=\left(\begin{array}{cc}
x^{\bar{l}}\frac{\partial\Gamma^k_{ij}}{\partial x^l} & \Gamma^k_{ij}\\
\noalign{\medskip}
\Gamma^k_{ij} & 0
\end{array}\right)
$$
for $i$, $j$, $k$ $=$ $1,\dots,m$ \cite{18}, we have from (\ref{eq:1}),
(\ref{eq:2}), (\ref{eq:3}), (\ref{eq:4}) for the second fundamental form
$\nabla (d\varphi)$ and the tension field $\tau(\varphi)$:
$$
\begin{array}{rcl}
\nabla(d\varphi)^\gamma_{\alpha\beta}(x,y) & = & \displaystyle{
\frac{\partial^2\varphi^\gamma}{\partial x^\alpha\partial x^\beta}(x) -
{}^{TM}\Gamma^\delta_{\alpha\beta}(x,y)\frac{\partial\varphi^\gamma}{\partial
x^\delta}(x)}\\
\noalign{\medskip}
& &+\displaystyle{
{}^{TM}\Gamma^\gamma_{\lambda\mu}(\varphi(x,y))
\frac{\partial\varphi^\lambda}{\partial x^\alpha}(x)
\frac{\partial\varphi^\mu}{\partial x^\beta}(x),}\\
\noalign{\medskip}
\tau(\varphi)^\gamma(x,y) & = &
(g^C)^{\alpha\beta}(x,y)\nabla(d\varphi)^\gamma_{\alpha\beta}(x,y),
\end{array}
$$
where we have at $(x,y)$, in terms of the Christoffel symbols of $\nabla$:
$$
\begin{array}{rcl}
\nabla(d\varphi)^k_{ij} & = & 0,\\
\noalign{\medskip}
\nabla(d\varphi)^k_{\bar{i}j} & = & 0,\\
\noalign{\medskip}
\nabla(d\varphi)^k_{\bar{i}\bar{j}} & = & 0,\\
\noalign{\medskip}
\nabla(d\varphi)^{\bar{k}}_{ij} & = & \displaystyle{x^{\bar{l}}\left(
\frac{\partial^2\varphi^k_l}{\partial x^i\partial x^j} -
\Gamma^a_{ij}\frac{\partial\varphi^k_l}{\partial x^a} -
\frac{\partial\Gamma^a_{ij}}{\partial x^l}\varphi^k_a +
\frac{\partial\Gamma^k_{ij}}{\partial x^a}\varphi^a_l\right.}\\
\noalign{\medskip}
& & \displaystyle{\left. +
\Gamma^k_{ia}\frac{\partial\varphi^a_l}{\partial x^j} +
\Gamma^k_{aj}\frac{\partial\varphi^a_l}{\partial x^i}\right)},\\
\noalign{\medskip}
\nabla(d\varphi)^{\bar{k}}_{\bar{i}j} & = & (\nabla_j\varphi)^k_i,\\
\noalign{\medskip}
\nabla(d\varphi)^{\bar{k}}_{\bar{i}\bar{j}} & = & 0
\end{array}
$$
and hence, by means of (\ref{eq:5}):
$$
\tau(\varphi)^k=0,\quad \tau(\varphi)^{\bar{k}}=2g^{ij}(\nabla_i\varphi)^k_j
=-2(\nabla^*\varphi)^k.
$$
This yields the required result.$\hfill\Box$
\bigskip
\noindent{\normalsize\sc Remark}
It follows from the main result and \cite[p.34-35]{2} that when $\Phi$ is
skew--symmetric or symmetric, respectively, then $\varphi$ (or $\Phi$) is
harmonic if and only if $\Phi$ is coclosed (i.e., $\delta\Phi$ $=$ $0$) or has
vanishing divergence, respectively.
\bigskip
\section{Examples and applications}
In what follows we shall give several examples and applications of the notion
and result considered in Section 2. In this way we provide examples of harmonic
maps of some special pseudo--Riemannian manifolds.
We start by noting that in \cite{5} the authors also considered the notion of
harmonic symmetric $(0,2)$--tensors. They first considered the notion of a
harmonic Riemannian metric $g'$ on a Riemannian manifold $(M,g)$ and called
$g'$ a harmonic metric
with respect to $g$ if id$_M:(M,g)$ $\rightarrow$ $(M,g')$ is a harmonic map.
The analytic expression of this condition then led the authors to the
definition of a harmonic symmetric $(0,2)$--tensor $\Phi$. It turns out that
$\Phi$ is harmonic in the sense of \cite{5} if and only if
$$
\Phi'=\Phi - \frac{1}{2} (\mbox{\rm tr}\, \Phi) g
$$
is harmonic in the sense of Section 2. Hence, the examples given in \cite{5}
yield examples of harmonic endomorphism fields.
Before getting more results, we consider the Ricci tensor $\rho$ on a
Riemann\-ian
$(M,g)$ and note that
$$
(\mbox{\rm div}\,\rho)(X) =
-\sum_{i=1}^m(\nabla_{e_i}\rho)(e_i,X)=-\frac{1}{2}\nabla_X\tau,
$$
where $(e_1,\dots,e_m)$ is an arbitrary orthonormal basis of $T_pM$ at each
$p\in M$. So, we get at once.
\begin{proposition}\label{prop:1}
The Ricci endomorphism field on a Riemannian manifold is harmonic if and only
if the scalar curvature is constant. Moreover, the Einstein tensor $G$ $=$
$\rho$ $-$ $\frac{1}{2}$ $\tau$ $g$ is always harmonic.
\end{proposition}
Note that $\rho$ is always harmonic in the sense of \cite{5}. Our notion of
harmonicity for $\rho$ is thus more restrictive.
Using this result we may give another characterization of harmonic manifolds
$(M,g)$ with dim$M$ $>$ $2$. Indeed, in \cite{6} it is proved that a Riemannian
$(M^m,g)$, $m>2$, is a harmonic manifold if and only if every sufficiently
small geodesic sphere has constant scalar curvature, i.e., the scalar curvature
only depends on the radius of the sphere. Hence we have
\begin{proposition}\label{prop:2}
Let $(M^m,g)$, $m\geq 3$, be a Riemannian manifold. Then $(M,g)$ is a harmonic
space if and only if the Ricci endomorphism field of any sufficiently small
geodesic sphere is harmonic.
\end{proposition}
Instead of geodesic spheres one may also consider tubes of sufficiently small
radius about geodesics. Then, in \cite{10}, an $(M,g)$ is said to be scalar
curvature harmonic with respect to geodesics $\gamma$ if the scalar curvature
for all small tubes about all $\gamma$ only depends on the radius. It is proved
in \cite{10} that such an $(M,g)$ is a real space form. So, we obtain
\begin{proposition}\label{prop:3}
A Riemannian manifold $(M^m,g)$, $m\geq 3$, is a real space form if and only if
the Ricci endomorphism field of every small tube about all geodesics is
harmonic.
\end{proposition}
The result in Proposition \ref{prop:2} implies that all small geodesic spheres
in harmonic spaces provide examples of harmonic maps by means of the Ricci
tensor of type $(1,1)$. We refer to \cite{1} for the known examples of harmonic
spaces. In what follows we will show that the same holds when we consider the
shape operator of these geodesic spheres.
To prove this we turn to submanifold theory. Let $\overline{M}$ be an oriented
hypersurface in a Riemannian $(M,g)$. Let $\xi$ denote a unit normal vector
field on $\overline{M}$ and let $S$ be the shape operator of $\overline{M}$
defined by $\nabla_X\xi$ $=$ $-SX$ for $X$ tangent to $\overline{M}$. $S$ is
related to the second fundamental form $\sigma$ by $g(SX,Y)$ $=$
$g(\sigma(X,Y),\xi)$. Then the Codazzi equation reads
\begin{equation}\label{eq:7}
(R_{XY}Z)^\perp = (\nabla_Y\sigma)(X,Z) - (\nabla_X\sigma)(Y,Z)
\end{equation}
for $X$, $Y$, $Z$ tangent to $\overline{M}$. (See \cite{4} for more details.)
The mean curvature $h$ is given by $h$ $=$ tr$\, S$. So, from (\ref{eq:7}) we
have
\begin{lemma}\label{le:1}
Let $\overline{M}$ be an oriented hypersurface in $(M,g)$ with unit normal
vector field $\xi$. Then we have
$$
\rho(X,\xi) = (\mbox{\rm div}\, \sigma)(X) +Xh
$$
for any $X$ tangent to $\overline{M}$.
\end{lemma}
Using this lemma we get at once
\begin{proposition}\label{prop:4}
An oriented hypersurface $\overline{M}$ in an Einstein manifold $(M,g)$ has
constant mean curvature if and only if the shape operator is harmonic.
\end{proposition}
Since harmonic spaces, which are Einstein spaces, may be defined as $(M,g)$ all
of whose sufficiently small geodesic spheres have constant mean curvature (see,
for example, \cite{6}), we have
\begin{proposition}\label{prop:5}
An Einstein manifold $(M,g)$ is a harmonic space if and only if the shape
operator of each sufficiently small geodesic sphere is harmonic.
\end{proposition}
Using the notion of harmonicity with respect to geodesics as developed in
\cite{12} by means of the mean curvature of tubes about geodesics, we have in a
similar way as for Proposition \ref{prop:3}:
\begin{proposition}\label{prop:6}
An Einstein manifold $(M,g)$ is a real space form if and only if the shape
operator of every small tube about all geodesics is harmonic.
\end{proposition}
\bigskip
\noindent{\normalsize\sc Remark}
A $(0,2)$--tensor field $\Phi$ is a Killing tensor field if and only if for all
$X$ we have $(\nabla_X\Phi)(X,X)$ $=$ $0$. It follows that a symmetric Killing
tensor field of type $(0,2)$ is harmonic if and only if it has constant trace.
Since a manifold $(M,g)$ whose Ricci tensor $\rho$ is a Killing tensor has
automatically constant scalar curvature, we get that such $\rho$ is harmonic.
It follows from the general theory that any Riemannian space with
volume--preserving geodesic symmetries (up to sign), i.e., D'Atri spaces, has a
harmonic Ricci tensor. The same result holds for the $C$--spaces, i.e., spaces
such that the Jacobi operator field has constant eigenvalues along geodesics.
For both cases we refer to \cite{1} for more information and examples.
\bigskip
Now, we proceed the construction of examples by considering manifolds $(M,g)$
which are equipped with an additional structure given by an endomorphism field.
We start by looking at an almost Hermitian manifold $(M,g,J)$ and denote by
$\Omega$ its K\"ahler form defined by $\Omega(X,Y)$ $=$ $g(X,JY)$ for all
tangent $X$, $Y$. Then we have
$$
(\delta\Omega)(X)=-\sum_{i=1}^m(\nabla_{e_i}\Omega)(e_i,X)
=-\sum_{i=1}^m g(X,(\nabla_{e_i}J)e_i).
$$
\noindent{Since} an almost Hermitian manifold is said to be
semi--K\"ahlerian if
$\Omega$ is coclosed (see for example \cite{11}) we obtain easily
\begin{proposition}\label{prop:7}
Let $(M,g,J)$ be an almost Hermitian manifold. Then $J$ is harmonic if and only
if $(M,g,J)$ is semi--K\"ahlerian.
\end{proposition}
Next, let $P$ be an almost product metric structure on a Riemannian manifold
$(M,g)$, i.e., $P$ is a $(1,1)$--tensor field such that $P^2$ $=$ id. and
$g(PX,PY)$ $=$ $g(X,Y)$ for all tangent $X$, $Y$ (see for example \cite{19}).
Then the $(0,2)$--tensor $\varphi$ defined by $\varphi(X,Y)$ $=$ $g(X,PY)$
determines a pseudo--Riemannian metric on $(M,g)$. Note that conversely, any
pseudo--Riemannian metric on $(M,g)$ gives rise to an almost product metric
structure. Here we have
\begin{proposition}\label{prop:8}
The pseudo--Riemannian metric tensor $\varphi$ on an almost product metric
manifold $(M,g,P)$ is harmonic if and only if $P$ is harmonic.
\end{proposition}
Note that the eigenspaces of $P$ determine two complementary and orthogonal
distributions $D$, $D'$ on $M$. When these distributions are integrable and
determine foliations ${\cal D}$, ${\cal D'}$ with minimal leaves, then $P$
is certainly harmonic since $\nabla^*P$ $=$
$-(\alpha_{\cal{D}}+\alpha_{\cal{D'}})$,
where $\alpha_{\cal{D}}$ and $\alpha_{\cal{D'}}$ are the mean curvature
vectors of the leaves of ${\cal D}$ and ${\cal D'}$, respectively
\cite{16}. We refer to \cite{3} for examples of such manifolds.
We finish this short list of examples by considering the tangent bundle $TM$ of
a Riemannian manifold $(M,g)$. As is well-known, $TM$ may be equipped with a
Riemannian metric $g^S$, called the Sasaki metric, which is defined by
$$
g^S(X^H,Y^H)=g^S(X^V,Y^V)=g(X,Y)^V, \quad g^S(X^H,Y^V)=0
$$
for $X$, $Y$ tangent to $M$ \cite{14}, \cite{15}. Moreover, the endomorphism
field $J$ on $TM$ defined by
$$
JX^V=-X^H, \quad JX^H=X^V
$$
determines an almost Hermitian structure on $TM$ and the endomorphism field
$P$ determined by
$$
PX^V=X^H, \quad PX^H=X^V
$$
defines an almost product metric structure on $TM$. It follows that also $Q$
$=$ $PJ$ $=$ $-JP$ is an almost product metric structure.
A rather straightforward computation, which we omit here, using the
expressions for the Riemannian connections of $(TM,g^S)$ and $(TM,g^C)$ (see
\cite{13}, \cite{18}) then yields the following result:
\begin{proposition}\label{prop:9}
Let $(M,g)$ be a Riemannian manifold.
\begin{enumerate}
\item[(i)]
The endomorphism fields $J$ and $Q$
are harmonic on $(TM, g^S)$ and the endomorphism field $P$ is harmonic on
$(TM,g^S)$ if and only if $(M,g)$ is Ricci--flat.
\item[(ii)]
The endomorphism field $Q$ is harmonic on $(TM,g^C)$ and the endomorphisms
fields $J$ and $P$ are harmonic on $(TM,g^C)$ if and only if $(M,g)$ is
Ricci--flat.
\end{enumerate}
\end{proposition}
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%\newpage %\sc
E. Garc\'{\i}a--R\'{\i}o$^1$, M.E. V\'azquez--Abal$^2$
\bigskip
${}^1$ Departamento de An\'alise Matem\'atica,
${}^2$ Departamento de Xeometr\'{\i}a e Topolox\'{\i}a,
Facultade de Matem\'aticas,
Universidade de Santiago de Compostela,
15706 Santiago de Compostela, {\rm Spain}\\
\bigskip
\bigskip
L. Vanhecke
\bigskip
Department of Mathematics,
Katholieke Universiteit Leuven,
Celestijnenlaan 200 B, 3001 Leuven, {\rm Belgium}
\end{document}