Outdated Archival Version

These pages are not updated anymore. They reflect the state of 20 August 2005. For the current production of this journal, please refer to http://www.math.psu.edu/era/.

%_ **************************************************************************
%_ * The TeX source for AMS journal articles is the publishers TeX code     *
%_ * which may contain special commands defined for the AMS production      *
%_ * environment.  Therefore, it may not be possible to process these files *
%_ * through TeX without errors.  To display a typeset version of a journal *
%_ * article easily, we suggest that you either view the HTML version or    *
%_ * retrieve the article in DVI, PostScript, or PDF format.                *
%_ **************************************************************************
% Author Package file for use with AMS-LaTeX 1.2
\controldates{18-MAR-1999,18-MAR-1999,18-MAR-1999,18-MAR-1999}
 
\documentclass{era-l}
\sloppy

\newtheorem{prop}{Proposition}
\newtheorem{lemma}{Lemma}
\newtheorem{thm}{Theorem}
\newtheorem{cor}{Corollary}

\theoremstyle{remark}
\newtheorem{remark}{Remark}
\newtheorem*{note}{Note}

\theoremstyle{definition}
\newtheorem{ex}{Example}
\newtheorem*{example}{Example}



\newcommand{\dr}{\partial}
\newcommand{\R}{\mathbf{R}}
\newcommand{\Z}{\mathbf{Z}}
\newcommand{\II}{I\hspace{-0.1cm}I}
\newcommand{\III}{I\hspace{-0.1cm}I\hspace{-0.1cm}I}
\newcommand{\tr}{\operatorname{tr}}
\newcommand{\rk}{\mbox{rk}}
\newcommand{\ric}{\mbox{ric}}
\newcommand{\deltab}{\overline{\delta}}
\newcommand{\Db}{\overline{D}}
\newcommand{\Kb}{\overline{K}}
\newcommand{\cM}{\mathcal{M}}
\newcommand{\So}{\overline{S}}
\newcommand{\vt}{\tilde{v}}
\newcommand{\cV}{\mathcal{V}}
\newcommand{\ricb}{\overline{\ric}}
\newcommand{\psym}{\mbox{Psym}}

\newcommand{\kn}{\bigcirc\hspace{-0.37cm}\wedge \;}

%\newcommand{\vs}{\vspace{0.3cm}}

\begin{document}
\title{The Schl{\"a}fli formula in Einstein manifolds with boundary}

%first author

\author{Igor Rivin}
\address{Department of Mathematics, 
University of Manchester,
Oxford Road, Manchester M13 9PL, G.B.}
\email{irivin@ma.man.ac.uk}

%second author

\author{Jean-Marc Schlenker}
\address{Topologie et Dynamique (URA 1169 CNRS), 
B{\^a}t. 425, 
Uni\-ver\-sit{\'e} de Paris-Sud, 
91405 Orsay Cedex, France}
\email{jean-marc.schlenker@math.u-psud.fr}



\issueinfo{5}{03}{}{1999}
\dateposted{March 22, 1999}
\pagespan{18}{23}
\PII{S 1079-6762(99)00057-8}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}

\subjclass{Primary 53C21; Secondary 53C25}
\date{July 31, 1998}

\commby{Walter Neumann}

\keywords{Vanishing theorems; null spaces}

\begin{abstract} We give a smooth analogue of the classical Schl{\"a}fli
formula, relating the variation of the volume bounded by a hypersurface moving
in a general Einstein  manifold and the integral of the variation of the mean
curvature. We extend it to variations of the metric in a Riemannian Einstein
manifold with boundary, and apply it to Einstein cone-manifolds, to isometric
deformations of Euclidean hypersurfaces, and to the rigidity of Ricci-flat
manifolds with umbilic boundaries. 
\\ \\ 
\noindent {\sc R\'esum\'e.} On donne
un analogue r{\'e}gulier de la formule classique de Schl{\"a}fli, reliant la
variation du volume born{\'e} par une hypersurface se d{\'e}pla{\c c}ant dans
une vari{\'e}t{\'e} d'Einstein {\`a}   l'int{\'e}grale de la variation de la
courbure moyenne. Puis nous l'{\'e}tendons aux variations de la m{\'e}trique
{\`a} l'int{\'e}rieur d'une vari{\'e}t{\'e} d'Einstein  riemannienne {\`a}
bord. On l'applique aux cone-vari{\'e}t{\'e}s d'Einstein, aux d{\'e}formations
isom{\'e}triques d'hypersurfaces de $E^n$, et {\`a} la  rigidit{\'e} des
vari{\'e}t{\'e}s Ricci-plates {\`a} bord ombilique.   \end{abstract}

\maketitle

%\vs

Let $M$ be a Riemannian $(m+1)$-dimensional space-form of constant
curvature $K$, and 
$(P_t)_{t\in [0,1]}$ a one-parameter family of polyhedra in $M$ bounding
compact domains, all having the same combinatorics. Call $V_t$ the
volume bounded by $P_t$, $\theta_{i,t}$ and $W_{i,t}$ the dihedral angle
and the $(m-1)$-volume of the codimension 2 face $i$ of $P_t$.
The classical Schl{\"a}fli formula (see \cite{milnor-schlafli}
or \cite{geo2}) is
\begin{equation} \label{for-schlafli}
\sum_i W_{i,t} \frac{d \theta _{i,t}}{dt} = m K \frac{dV_t}{dt}.
\end{equation}

This formula has been extended and used on several occasions recently; see
for instance \cite{hodgson}, \cite{bonahon}. 

We give a smooth version of this formula, for 1-parameter families of
hypersurfaces in (Riemannian of Lorentzian) Einstein manifolds. Then we
extend it to variations of an Einstein metric inside a manifold
with boundary (a much more general process in dimension above
3). Finally, we give three applications: to the variation of the volume
of Einstein cone-manifolds, to isometric deformations of hypersurfaces
in  the Euclidean space, and to the rigidity of Ricci-flat manifolds
with umbilic boundaries. The reader can find the details in
\cite{rivinschlenk}. 

Throughout this paper, $M$ is an Einstein manifold of dimension
$m+1\geq 3$, and $D$ is its Levi-Civita connection. 
When dealing with a hypersurface $\Sigma$ (resp. with the boundary $\dr
M$), we call $I$ the {\em induced metric}, also called the {\em first
fundamental form}, of the corresponding immersion in $M$. $\Db$ is
the Levi-Civita connection of $I$, $B$ the shape operator, and  
$\II$, $\III$ are the second and third fundamental forms of $\Sigma$;
$B$ and $\II$ are defined with respect to an oriented unit normal vector
to $\Sigma$ (resp. to the exterior unit normal to $\dr M$).

The trace $H$ of $B$ is called the ``mean curvature'' (some definitions
differ by a factor $m$) and the ``higher mean curvatures''
$H_k, k\geq 1$, are the higher symmetric functions of the principal
curvatures of $\Sigma$ (resp. $\dr M$). For instance, $H_2 = (H^2 -
\tr(B^2))/2$. 
$dV, dA$ are the volume elements in $M$ and on $\Sigma$ (resp. $\dr
M$) respectively.

We denote by $\delta$ the divergence acting on symmetric tensors, and
by $\delta^*$ its formal adjoint. We will 
often implicitly identify (through the metric)
vector fields 
and 1-forms, as well as quadratic forms and linear morphisms. 

\section{Deformation of hypersurfaces}

Here is an analogue of the Schl{\"a}fli formula for
deformations of (smooth) hypersurfaces in a {\em fixed} Einstein
manifold $M$, which can 
be Riemannian or Lorentzian (the other pseudo-Riemannian cases can be
treated in the same way).

\begin{thm} \label{def-hyp-h}
Let $\Sigma$ be a smooth oriented hypersurface in a
(Riemannian) Einstein $(m+1)$-manifold $M$ with
scalar curvature $S$, and $v$ a section of the restriction of $TM$ to
$\Sigma$. $v$ defines a deformation of $\Sigma$ in $M$, which induces 
variations $V', H'$ and $I'$ of the volume bounded by $\Sigma$, mean
curvature, and induced metric on $\Sigma$.
Then: 
\begin{equation} \label{schlafli-h}
\frac{S}{m+1} V' = \int_{\Sigma} \left(H' + \frac{1}{2} \langle
I',\II\rangle \right) dA.
\end{equation}
\end{thm}

The unit normal used to define $H$ and $\II$ should be toward the
outside of the volume bounded by $\Sigma$.
Actually $\Sigma$ does not need to bound a finite volume domain for this
formula to hold. Otherwise, $V^\prime$ is just the derivative of the
(signed) volume 
contained between $\Sigma$ and $\Sigma_t$, for small $t$. This volume is
then oriented by the unit normal to $\Sigma_t$ used to define $H$ and $\II$,
which should be toward the exterior.

The proof can be given separately for normal and for tangent deformations
of $\Sigma$. For tangent deformations, $V'=0$, and the relation between
$I'$ and $H'$ comes from the equation: $\delta \II = -dH$, which holds
because $M$ is Einstein. For normal deformations, $H'$ is obtained using
the trace of $\II'$, which is related to the Laplacian of the amplitude
of the deformation and to the Ricci curvature of $M$ on the normal to
$\Sigma$. 

The ``classical'' Schl{\"a}fli formula (\ref{for-schlafli}) for
polyhedra in space-forms follows from Theorem \ref{def-hyp-h}. Namely,
one can apply Theorem \ref{def-hyp-h} 
to the set of points at distance $r$ on the outside of a convex
polyhedron and let $r\rightarrow 0$.

In the Lorentzian case, the only difference is 
that now $g(n,n)=-1$, so the volume variation has a minus sign in the
formula. 
Applying the Lorentzian formula to the set of points at distance
$\epsilon$ from a 
polyhedron in $S^n_1$, one
obtains the Schl{\"a}fli formula for de Sitter polyhedra as in
\cite{suarez}. 

\section{Einstein manifolds with boundary}

Here $(M, \dr M)$ is a compact manifold with boundary with
an Einstein metric $g$ of scalar curvature $S$. We will
prove the same formula as in the previous section, but in a much more
general setting: instead of moving a hypersurface in an Einstein
manifold, we will be changing the metric (among Einstein metrics of
a given scalar curvature) inside this  manifold with boundary. Although
the two 
operations are equivalent in dimension at most 3, moving the inside
metric is much more general in higher dimensions. 
On the other hand, our proof only works for Riemannian Einstein
manifolds. It is not obvious whether it can be extended to the
pseudo-Riemannian setting.



If $g$ is an Einstein metric, we say that a 2-tensor $h$ is an
``Einstein variation'' of $g$ if the associated variation of the metric
induces a variation of the Ricci tensor which is proportional to $h$,
so that $g+\epsilon h$ remains, to the first order, an Einstein
manifold with constant scalar curvature.

\begin{thm} \label{schlafli-einstein}
Let $h$ be a smooth Einstein variation of $g$. Then:
\begin{equation} \label{schlafli-e}
\frac{S}{m+1} V' = \int_{\dr M} H' + \frac{1}{2} \langle
h_{|\dr M},\II\rangle dA.
\end{equation}
\end{thm}


As always when studying deformations of Riemannian metrics, the proof
needs 
put some kind of restriction to remove the geometrically trivial
deformations, which
only correspond to the action of vector fields on the metric. We 
prevent those deformations in the same way as e.g. in \cite{graham-lee},
\cite{deturck} 
or \cite{biquard}, that is, we only consider metric variations $h$ such
that $2\delta h + d\tr h = 0$. The following proposition, which is
proved by a fairly simple variational argument, shows that we
do not forget any metric variation when doing this.

\begin{prop}
Let $h'$ be a smooth variation of $g$. Suppose that either $S\leq 0$, or
$M$ is strictly convex. There exists another smooth
variation $h$ of $g$ such that $2 \delta h + d\tr(h)=0$ and 
$h=h'+\delta^* v_0$, where $v_0$ is a vector field vanishing on $\dr M$.
\end{prop}

The variation $h$ of $g$ obtained in Proposition 1 satisfies (because
the metric remains Einstein) a simple, elliptic equation. Taking its
trace shows that the trace of $h$ also satisfies an elliptic equation
which, when integrated over $M$, expresses $V'$ as some integral over
$\dr M$. A careful examination of this boundary term leads to  
Theorem \ref{schlafli-einstein}. 


\section{Applications}

A first application can be found by looking at ``singular
objects'', just as to go from the smooth Theorem \ref{def-hyp-h} to the
classical Schl{\"a}fli formula. There are no polyhedra in general Einstein
manifolds, but we can check what happens when we deform Einstein
manifolds with cone singularities. It should be pointed out that some of
the most interesting modern uses of the classical Schl{\"a}fli formula
concern hyperbolic 3-dimensional cone-manifolds. 

Let $M$ be a compact $(m+1)$-manifold, and $N$ a compact
codimension 2 submanifold of $M$. Suppose $(g_t)$ is a 1-parameter
family of Einstein metrics with fixed scalar curvature $S\leq 0$ on
$M\setminus N$, with a conical singularity 
on $N$ in the sense that, in normal coordinates around $N$, $g_t$ has an
expansion like
\begin{equation*}
g_t = h_t + dr^2 + r^2 d\theta^2 + o(r^2),
\end{equation*}
where $h_t$ is the metric induced on $N$ by $g_t$, and $\theta\in
\R/\alpha_t \Z$ for some $\alpha_t\in \R$. Call $V_t$ the volume of
$(M\setminus N, g_t)$, and $W_t$ the volume of $(N, h_t)$. Then 

\begin{cor}
$V_t$
varies as:
$\frac{S}{m+1} \frac{d V_t}{dt} = W_t \frac{d \alpha_t}{dt}$.
\end{cor}

%{\bf Note:} 
\begin{note}
The same result holds when $N$ has several connected
components, each with a different value of $\alpha_t$. $N$ could also be
replaced by a stratified submanifold. 
\end{note}
%\vs

%{\bf Example:}
\begin{example} Take $m+1=3$ in the previous example. We find the
Schl{\"a}fli formula for the variation of the volume of a hyperbolic
cone-manifold \cite{hodgson}.
\end{example}
%\vs


When the $W_{i,t}$ are constant, the left-hand side of
(\ref{for-schlafli}) is a 
polyhedral analogue of  the variation of the mean curvature integral
of a hypersurface. 
When $K=0$, the right-hand side is $0$. This shows that the ``mean
curvature'' of a 1-parameter family of Euclidean polyhedra with
constant induced metric is constant. 
This was used in
\cite{almgren-rivin} to prove, using geometric measure theory methods,
the following

\begin{thm} \label{eucl}
In $\R^{m+1}$, the integral of
the mean curvature remains constant in an isometric deformation of a
hypersurface. 
\end{thm}

This theorem follows immediately from our Theorem 1.

On the other hand, the
integral mean curvature is \textbf{not} determined by the metric on $\dr
M$: this is already visible in $\R^3$. Namely,
some metrics on $S^2$ admit two isometric embeddings in $\R^3$: the
classical example is that a (topological) sphere in $\R^3$ which is
tangent to a plane along a circle can be ``flipped'' so as to obtain
another embedding with the same induced metric \cite{spivak}. Those two
embeddings do not in general have the same integral mean curvature---and
thus we have a complicated way of seeing that the two flipped 
surfaces cannot be bent one into the other.

Formula (\ref{schlafli-e}) is even simpler for variations which vanish
on $\dr M$: 
 
\begin{thm} \label{cor-e}
If $h$ is a smooth Einstein variation of $g$ which does not change the
induced metric on $\dr M$, then 
\begin{equation*}
\int_{\dr M} H' dV= \frac{S}{m+1} V'.
\end{equation*}
\end{thm}

In particular, for $S=0$, this implies that the integral of the mean
curvature of the boundary is constant under an Einstein variation which
does not change the induced metric on $\dr M$; this is a direct
generalization of Theorem \ref{eucl}.

The analogue of Theorem \ref{eucl} is also true, but in a pointwise
sense, for the higher mean curvatures: 

\begin{thm} \label{higher}
In $\R^{m+1}$, the integral of
$H_k$ $(k\geq 2)$ remains constant in an isometric deformation of a
hypersurface. 
\end{thm}

This comes from the following (probably classical) description
of the possible 
isometric deformations of a hypersurface for $m+1\geq 4$:

\begin{remark} \label{deform}
Let $(\Sigma_t)_{t\in [0,1]}$ be a 1-parameter family of hypersurfaces
in a space-form, such
that the induced metric $I_t$ is constant to the first order at
$t=0$. Then, at each point, one of the following is true:
\begin{itemize}
\item $\II_0=0$;
\item $\rk(\II_0)\leq 2$, and $\II_0'$ vanishes on the kernel
of $\II_0$;
\item $\II_0'=0$;
\end{itemize}  
where $\II_t$ is the second fundamental form of $\Sigma_t$, and $\II'_t$
its variation.
\end{remark}

Theorem \ref{higher} clearly follows, because $H_k'$ is zero
for $k\geq 3$ in each case, and the Gauss formula gives the proof for
$k=2$. 


Denote by $\Sigma^\epsilon_t$ the parallel surface at distance 
$\epsilon$ from $\Sigma_t$. It is well known (see, e.g., Santalo's book
\cite{santalo}) that the area of $\Sigma^\epsilon$ is a polynomial in
$\epsilon$ where the coefficent of $\epsilon^k$ is (essentially) the
$k$-th mean curvature of $\Sigma$. The two Theorems $\ref{eucl}$ and
$\ref{higher}$ can then be combined as stating that:

\begin{thm}\label{nbhd}
The area of $\Sigma_t^\epsilon$ stays constant when $\Sigma_t$ is a
bending of $\Sigma_0.$
\end{thm}

%\vs

Finally, we use the Schl{\"a}fli formula above to prove a rigidity
result for Ricci-flat manifolds with umbilic boundary; it is a
generalization of the classical result (see \cite{spivak}) that the
round sphere is rigid in 
$\R^3$, that is, it cannot be deformed smoothly without changing its induced
metric. 

\begin{lemma} \label{rigidity}
Suppose $(M, \dr M)$ is a compact $(m+1)$-manifold with boundary, and
$(h_t)_{t\in [0,1]}$ is a nontrivial $1$-parameter family of Ricci-flat
metrics on $M$ inducing 
the same metric on $\dr M$, and such that $\dr M$ is umbilic and convex (or concave)
for $h_0$. Then $\dr M$ has at least two connected components, and $(h_t)$
corresponds to the displacement of some connected component(s) of $\dr
M$ under the flow of some Killing field(s) of $M$.
\end{lemma}

This kind of rigidity result could be used in the future to prove that,
given a Ricci-flat manifold $M$ with umbilic boundary and induced metric
$g_0$ on the boundary, any metric close to $g_0$ on $\dr M$ can be
realized as induced on $\dr M$ by some Ricci-flat metric on $M$. In this
setting, rigidity corresponds to the local injectivity of an operator
sending the metrics on $M$ to the metrics on $\dr M$. In dimension 3,
this would be a part of the classical result (see \cite{N}) that metrics with
curvature $K>0$ on $S^2$ can be realized as induced by immersions into
$\R^3$. This circle of ideas is illustrated in \cite{ecb}.

The first point is to understand what an umbilic hypersurface in an
Einstein manifold is. By definition, if $N$ is a Riemannian manifold and
$S$ is a hypersurface, then $S$ is umbilic if, at each point $s\in S$, $\II$
is proportional to $I$, with a proportionality constant $\lambda(s)$
depending on $s$. Now

\begin{remark}
If $N$ is Einstein, then $\lambda$ is constant on each connected
component of $S$.
\end{remark}




The next step is an inequality concerning the integral of the mean
curvature squared.

\begin{prop} \label{umbilic}
Let $(M, g)$ be an Einstein manifold with boundary, with scalar
curvature $S$. Call $\So$ the scalar curvature of $(\dr M, g_{|\dr M})$.
Then the mean curvature $H=\tr(\II)$ of $\dr M$ satisfies
\begin{equation*}\frac{\So}{m-1} - \frac{S}{m+1} \leq \frac{H^2}{m}
\end{equation*}
with equality if and only if $\dr M$ is umbilic.
\end{prop}

Proposition \ref{umbilic} and Theorem \ref{cor-e} lead to the
proof of Lemma \ref{rigidity}. The key point is that by Proposition
\ref{umbilic}, $H^2$ is 
pointwise minimal when the boundary is umbilic, while Theorem
\ref{cor-e} shows that the integral of $H$ is constant. The
boundary therefore has to remain umbilic in an Einstein variation which
vanishes on the boundary.



\section*{Acknowledgements} The authors are happy to acknowledge their
debt to Fred Almgren, without whom this paper would not have been
possible. I.~Rivin would like to thank the Institut Henri Poincar{\'e}
and the Institut des Hautes {\'E}tudes Scientifiques for their
hospitality at crucial moments.

\bibliographystyle{alpha}

\begin{thebibliography}{{Hod}93}

\bibitem[AR97]{almgren-rivin}
F.~O. Almgren and I.~Rivin.
\newblock The mean curvature integral is invariant under bending.
\newblock Manuscript, 1997.

\bibitem[BG93]{berger-gostiaux}
M.~Berger and B.~Gostiaux.
\newblock {\em G{\'e}om{\'e}trie diff{\'e}rentielle: vari{\'e}t{\'e}s, courbes et
  surfaces}.
\newblock Presses universitaires de France, 1993.
\MR{93j:53001} 
\bibitem[Biq97]{biquard}
O.~Biquard.
\newblock M{\'e}triques d'{Einstein} asymptotiquement sym{\'e}triques.
\newblock Pr{\'e}\-pu\-blication no. 97-24, Ecole Polytechnique, 1997.

\bibitem[Ble96]{bleecker}
David~D. Bleecker.
\newblock Volume increasing isometric deformation of convex polyhedra.
\newblock {\em Journal of Differential Geometry}, 43:505--526, 1996.
\MR{97g:52035}
\bibitem[Bon]{bonahon}
F.~Bonahon.
\newblock A {Schl{\"a}fli}-type formula for convex cores of hyperbolic
  3-manifolds.
\newblock {\em Journal of Diff. Geometry}.
\newblock To appear.

\bibitem[Con77]{connelly}
R.~Connelly.
\newblock {A counterexample to the rigidity conjecture for polyhedra}.
\newblock {\em {Inst. Haut. Etud. Sci., Publ. Math.}}, 47:333--338, 1977.
\MR{58:7642}
\bibitem[DeT81]{deturck}
D.~M. DeTurck.
\newblock Existence of metrics with prescribed {Ricci} curvature: local theory.
\newblock {\em Invent. Math.}, 65:179--207, 1981.
 \MR{83b:53019}
\bibitem[GL91]{graham-lee}
C.~R. Graham and J.~M. Lee.
\newblock Einstein metrics with prescribed conformal infinity on the ball.
\newblock {\em Adv. Math.}, 87:186--225, 1991.
\MR{92i:53041} 
\bibitem[Her79]{herglotz}
G.~Herglotz.
\newblock {\em {Gesammelte Schriften. Herausgegeben im Auftrage der Akademie
  der Wissenschaften in Goettingen von Hans Schwerdtfeger.}}
\newblock Vandenhoeck \& Ruprecht, 1979.
\MR{80h:01040}
\bibitem[{Hod}86]{hodgson}
{C. Hodgson}.
\newblock PhD Thesis, Princeton University, 1986.

\bibitem[Lie00]{liebmann}
H.~Liebmann.
\newblock Ueber die {Verbiegung} der geschlossenen {Fl{\"a}schen} positiver
  {Kr{\"u}mmung}.
\newblock {\em Math. Annalen}, 53:81--112, 1900.

\bibitem[Mil94]{milnor-schlafli}
J.~Milnor.
\newblock The {Schl{\"a}fli} differential equality.
\newblock In {\em Collected papers, vol. 1}. Publish or Perish, 1994.
\MR{95c:01043}
\bibitem[Nir53]{N}
L.~Nirenberg.
\newblock The {Weyl} and {Minkowski} problems in differential geometry in the
  large.
\newblock {\em Comm. Pure Appl. Math}, 6:337--394, 1953.
\MR{15:347b}
\bibitem[Pog73]{Po}
A.~V. Pogorelov.
\newblock {\em Extrinsic Geometry of Convex Surfaces}.
\newblock American Mathematical Society, 1973.
\newblock Translations of Mathematical Monographs. Vol. 35.
\MR{49:11439}
\bibitem[RS98]{rivinschlenk}
I.~Rivin and J-M.~Schlenker.
\newblock{Schl{\"a}fli formula for Einstein manifolds}.
\newblock IHES Preprint, 1998.

\bibitem[San76]{santalo}
L.~Santal{\'o}.
\newblock {\em Integral geometry and geometric measure theory}.
\newblock Addison-Wesley, Reading, MA, 1976.
\MR{55:6340}
\bibitem[Sch98]{ecb}
J.-M. Schlenker.
\newblock Einstein manifolds with convex boundaries.
\newblock Pr{\'e}\-publi\-ca\-tion no. 98-12, Universit{\'e} de Paris-Sud, 1998.

\bibitem[SP97]{suarez}
E.~Suarez-Peir{\'o}.
\newblock A {Schl{\"a}fli} formula for simplices in semi-Riemannian
  hyperquadrics, {Gauss-Bonnet} formulas for simplices in the {de Sitter}
  sphere and the dual volume of a hyperbolic simplex.
\newblock Manuscript, 1997.

\bibitem[Spi75]{spivak}
M.~Spivak.
\newblock {\em A comprehensive introduction to geometry, Vols. I--V}.
\newblock Publish or perish, 1970-1975.
\MR{42:2369}; \MR{42:6726}; \MR{51:8962}; \MR{52:15254a}; \MR{52:15254b}
\bibitem[Vin93]{geo2}
D. V. Alekseevskij, E. B. Vinberg, and A. S. Solodovnikov.
\newblock {\em Geometry of Spaces of Constant Curvature},
  Encyclopaedia of Math. Sci., 29, 
\newblock Springer, Berlin, 1993.
\MR{95b:53042}
\end{thebibliography}



\end{document}
<\PRE>