%changes \documentclass[11pt,amsmath,amssym,latexsymb]{article} \usepackage{amssymb,latexsym} \usepackage{longtable,tabularx} \usepackage{makeidx} \usepackage{graphicx} \usepackage{psfrag} \begin {document} \def\newtext{}%\def\newtext{\bf} \def\finalmodifications{\bf} \def\hat{\widehat} \def\tilde{\widetilde} \def \bfo {\begin {eqnarray*} } \def \efo {\end {eqnarray*} } \def \ba {\begin {eqnarray*} } \def \ea {\end {eqnarray*} } \def \beq {\begin {eqnarray}} \def \eeq {\end {eqnarray}} \newtheorem {lemma}{Lemma}\newtheorem {proposition}{Proposition} \newtheorem {theorem}{Theorem} \newtheorem {definition}{Definition} \newtheorem {example}{Example} \newtheorem {corollary}{Corollary} \newtheorem {problem}{Problem} \newtheorem {ex}{Exercise} \def\picture #1 by #2 (#3){ \vsquare to #2{ \hrule width #1 height 0pt depth 0pt \hfill \special{picture #3}}} \def\scaledpicture #1 by #2 (#3 scaled #4){{ \dimen0=#1 \dimen1=#2 \divide\dimen0 by 1000 \multiply\dimen0 by #4 \divide\dimen1 by 1000 \multiply\dimen1 by #4 \picture \dimen0 by \dimen1 (#3 scaled #4)}} \def \cA {{\cal A}} \def \cB {{\cal B}} \def \cD {{\cal D}} \def \cC {{\cal C}} \def \cG {{\cal G}} \def \F {{\cal F}} \def \N {{\cal N}} \def \W {{\cal W}} \def \cM {{\cal M}} \def \cL {{\cal L}} \def \tX {{\widetilde X}} \def \Box {${\diamondsuit}$} \def \ssskip {\vskip -23pt \hskip 40pt {\bf.} } \def \sskip {\vskip -23pt \hskip 33pt {\bf.} } \def \stkip {\vskip -23pt \hskip 33pt {\bf*.}} \def \sstkip {\vskip -23pt \hskip 40pt {\bf*.}} \def \bigstar {$\oplus$} \def \Coordinate {{\cal X}} \def \coordinate {{X}} \def \B {{\cal B}} \def \cR {{\cal R}} \def \Z {{\bf {Z}}} \def \C {{\bf {C}}} \def \D {{\cal D}} \def \R {{\bf {R}}} \def \bv {v} \def \bw {w} \def \bp {p} \def \be {e} \def \bz {z} \def \bx {x} \def \by {y} \def \grad {\hbox{Grad\,}} \def \covector {{\bf p}} \def \Hs {H^{s+1}_0(\partial M\times [0,T])} \def \H2s {H^{s+1}_0(\partial M\times [0,T/2])} \def \smooth {C^\infty_0(\partial M\times [0,2r])} \def \supp {\hbox{supp }} \def \diam {\hbox{diam }} \def \dist {\hbox{dist}} \def\diag{\hbox{diag }} \def \det {\hbox{det}} \def\bra{\langle} \def\cet{\rangle} \def \e {\varepsilon} \def \g {\tau} \def \l {\sigma} \def \r {\rho} \def \o {\over} \def \i {\infty} \def \d {\delta} \def \G {\Gamma} \def \b {\beta} \def \a {\alpha} \def \t {\tau} \def\th{\theta} \def\k{\kappa} \def \tM {\tilde M} \def \tr {\tilde r} \def \tz {\tilde z} \def \wB {{\widehat B}} \def \wD {{\widehat D}} \def \wR {{\widehat R}} \def \tG {{\widetilde G}} \def \tC {{\widetilde C}} \def \tR {{\widetilde R}} \def \u {\underline } \def \RS {Schr\"od\-ing\-er\ } \def \wB {{\widehat B}} \def \wD {{\widehat D}} \def \wR {{\widehat R}} \def \tG {{\widetilde G}} \def \tC {{\widetilde C}} \def \tR {{\widetilde R}} \def \pat {\partial _t} \def \pa0 {\partial _0} \def \pan {\partial _n} \def \prp {\prime \prime} \def \re {\,\hbox{Re}\,} \def \im {\,\hbox{Im}\,} \def \Q {Q^\phi_{\e,\t}} \def \p {\partial} \def\star{$\oplus$} \def\g{{\bf g}} \def\de{\delta_{\varepsilon,\mu}} \def\dm{\delta_{\mu,\varepsilon}} \def\m{\mu} \def\e{\varepsilon} \def\HE{*_\e} \def\HM{*_\m} \def\d{d} \def\Om{\Omega}\def\om{\omega} \def\La{\Lambda} \def\tE{\tilde E} \def\tB{\tilde B} \def\M{{M}} \def \Co {{{C ^{\infty}}^{{\hskip -14pt {}^{\circ}}\hskip 10pt}}} \def\oF{{\overline\F}} \def\hf{{\hat f}} \def\hh{{\hat h}} \def\ud{{\underline \delta}} \def\oa{\aa} \def \hat {\widehat } \def \bar{\overline } % Macros introduced 2.3.2001 \def\N{{\cal N}} \def\cM{{\M}}\def\cN{{\N}} \def\tilde{\widetilde} \def\div{\,\hbox{Div}\,}\def\Div{\div} \def\bn{{\bf n}} \def\sign{\hbox{sign}} \def \Co {{{C ^{\infty}}^{{\hskip -14pt {}^{\circ}}\hskip 10pt}}} \def \oC {{{C ^{\infty}}^{{\hskip -14pt {}^{\circ}}\hskip 10pt}}} % Matti's macros \def\D{{\cal D}} \def\cQ{{\cal Q}} \def\lap{\breve} \def \mbeq {\begin {eqnarray}} \def \meeq {\end {eqnarray}} \def \U {{\cal U}} \def \HL {\Lambda} \def\L{{\cal L}} \def\la{\lambda} \def\oC{C_{00}} \def\cE{{\cal E}} \def\cH{{\cal H}} \def\cX{{\cal X}} \def\cF{{\cal F}} % end of Matti's macros \def \bfo {\begin {displaymath} } \def \efo {\end {displaymath} } \def \beq {\begin {eqnarray}} \def \eeq {\end {eqnarray}} \def \ba {\begin {eqnarray*}} \def \ea {\end {eqnarray*}} \def \cA {{\cal A}} \def \cB {{\cal B}} \def \cD {{\cal D}} \def \cC {{\cal C}} \def \cG {{\cal G}} \def \cW {{\cal W}} \def \F {{\cal F}} \def \sN {N} \def \cL {{\cal L}} \def \tX {{\widetilde X}} %\def \mbox {${\diamondsuit}$} \def \ssskip {} \def \sskip {} \def \stkip{\hspace*{-0.5em}$\bigstar$} \def \sstkip {\hspace*{-0.5em}$\bigstar$} %\def \ssskip {\vskip -23pt \hskip 40pt {\bf.} } %\def \sskip {\vskip -23pt \hskip 33pt {\bf.} } %\def \stkip {\vskip -23pt \hskip 33pt {\bf*.}} %\def \sstkip {\vskip -23pt \hskip 40pt {\bf*.}} %\def \bigstar {$\oplus$} \def \Coordinate {{\cal X}} \def \coordinate {{X}} \def \B {{\cal B}} \def \cR {{\cal R}} \def \Z {{\bf {Z}}} \def \C {{\bf {C}}} \def \R {{\bf {R}}} \def \bv { v} \def \bu {{ {u}}} \def \bw {{ {w}}} \def \bp {{ {p}}} \def \bq {{ {q}}} \def \be {{ {e}}} \def \bz {{ {z}}} \def \bx {{ {x}}} \def \by {{ {y}}} \def \grad {\,\hbox{Grad\,}} \def \Grad {\grad} \def \covector {{\bf p}} \def \Hs {H^{s+1}_0(\partial \M\times [0,T])} \def \H2s {H^{s+1}_0(\partial \M\times [0,T/2])} \def \smooth {C^\infty_0(\partial \M\times [0,2r])} \def \supp {\hbox{supp }} \def \diam {\hbox{diam }} \def \dist {\hbox{dist}} \def\diag{\hbox{diag }} \def \det {\hbox{det}} \def\bra{\langle} \def\cet{\rangle} \def \e {\varepsilon} \def \g {\tau} \def \l {\sigma} %\def \r {\rho} %%% Remove this to make \aa work properly ***** \def \o {\over} \def \i {\infty} \def \d {\delta} \def \G {\Gamma} \def \b {\beta} \def \a {\alpha} \def \t {\tau} \def\th{\theta} \def\k{\kappa} \def \Rm {\hbox{Riemannian manifold}} \def \tM {\tilde {\cal M}} \def \tr {\tilde r} \def \tz {\tilde z} \def \wB {{\widehat B}} \def \wD {{\widehat D}} \def \wR {{\widehat R}} \def \tG {{\widetilde G}} \def \tC {{\widetilde C}} \def \tR {{\widetilde R}} \def \u {\underline } \def \RS {(Riemann)-Schr\"odinger\ } \def \wB {{\widehat B}} \def \wD {{\widehat D}} \def \wR {{\widehat R}} \def \tG {{\widetilde G}} \def \tC {{\widetilde C}} \def \tR {{\widetilde R}} \def \pat {\partial _t} \def \pa0 {\partial _0} \def \pan {\partial _n} \def \prp {\prime \prime} \def \re {\Re} \def \im {\Im} \def \Q {Q^\phi_{\e,\t}} \def \p {\partial} \def \fR {R} \def \vrho {\varrho} \def \fg {{\tilde {g}}} \def \pM {\partial M} \def \bK {\bf K} %Matti's new MACROS \def \RS {(Riemann)-Schr\"odinger\ } \def \DR {Dirichlet-Robin\ } \def \RD {Robin-Dirichlet\ } \def \HL {\Lambda} \def \ball{ ^{\circ}} \def \tu {{\tilde u}} \def \fu {{\underline f}} \def \hu {{\underline h}} \def \U{{\cal U}} \def \Co {{{C ^{\infty}}^{{\hskip -14pt {}^{\circ}}\hskip 10pt}}} \def \vol {\hbox{vol}} \def \oD {one-dimensional} \def \mD {multidimensional} \def \BSD {\hbox {boundary spectral data}} \def \GH {\hbox {Gromov-Hausdorff}} \def \bB {{\bf B}} \def \bM {{\bf M}} \def \diam {\hbox {diam}} \def \inj {\hbox {inj}} %Matti's even newer MACROS \def \cT{{\cal T}} \def \tilde{\widetilde} \def \proofbox {$\Box$\medskip} \title{Stability and Reconstruction in Gel'fand Inverse Boundary Spectral Problem} \author{Atsushi Katsuda\footnote{Department of Mathematics, Okayama University, Tsushima-naka, Okayama, 700-8530, Japan }, Yaroslav Kurylev\footnote{ Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, UK}, and Matti Lassas\footnote{Rolf Nevanlinna Institute, University of Helsinki Helsinki, P.O.Box 4, FIN-00014, Finland } } \maketitle {\bf Abstract.} We consider stability and approximate reconstruction of Riemannian manifold when the finite number of eigenvalues of the Laplace-Beltrami operator and the boundary values of the corresponding eigenfunctions are given. The reconstruction can be done in stable way when manifold is a priori known to satisfy natural geometrical conditions related to curvature and other invariant quantities. \bigskip {\bf I. Introduction} In this paper we consider the questions of stability and approximate reconstruction in the inverse boundary spectral problem {\newtext which is also called the generalized Gelfand inverse problem \cite{Ge} } for Riemannian manifolds. To formulate the problem and the main results, we need to introduce some basic notations. We will denote by $(\M,g)$ an unknown, $m$-dimensional, compact connected Riemannian manifold with a (smooth) metric $g$ and non-empty boundary $\p \M$. Our goal is to find $(\M,g)$ from boundary data. The boundary $\p \M$ is itself an $(m-1)$-dimensional compact differentiable manifold. We note that we do not assume the knowledge of the metric $i^*(g)$ generated on $\p\M$ by the metric $g$ , where $i: \p \M \to \M$ is an embedding or even the corresponding area element $d S_g$. Because in the inverse problems the boundary $\p \M$ is usually known, we will consider a class $\bM = \bM_{\p \M}$ of compact, connected Riemannian manifolds which have the same, i.e. diffeomorphic, boundary, $\p \M$. Let $-\Delta _g$ be the Laplace operator on $(\M,g)$ with Neumann boundary condition. Denote by $\{\la_j, \, \varphi_j; \,j=1,2,\dots \}$ the complete set of eigenvalues and corresponding orthonormal eigenfunctions of $-\Delta _g,\,$ $0 =\la_1<\la_2,\dots,\, \varphi_1 = \vol ^{-1/2} (\M,g)$, where $\vol$ stands for the volume of $(\M,g)$. %The {\it boundary spectral data} is the collection %\bfo % \left \{ \la_j,\, \varphi_j|_{\p \M}; \, j =1, 2, \dots \right \}. %\efo Denote by $\bB $ the set of sequences $ \{\mu_j,\, \psi_j;\,$ $ j=1,2,\dots \}$ where $\mu _j \in \R$ and $\psi_j \in L^2(\p \M)$ and by $\D: \bM \to \bB $, the map \bfo \D\,(\M,g) = \left \{\la_j,\, \varphi _j|_{\p \M},\, j=1,2,\dots \right \}. \efo \begin{definition}\label{def:1} {\newtext The collection $ \left \{ \la_j,\, \varphi_j|_{\p \M}; \, j =1, 2, \dots \right \}$ is called {\it the boundary spectral data} of $(\M,g)$ and the map $\D$ - {\it the boundary spectral map}}. \end{definition} It was shown by Belishev-Kurylev \cite {BeKu} {\newtext who used the boundary control method \cite{Be} and the unique continuation result of Tataru \cite{Ta} } that \begin{theorem}\label{th:1} The map $\D: \bM \to \bB$ is injective. \end{theorem} {\newtext (For earlier uniqueness results in the Gelfand inverse problem for isotropic operators obtained by the boundary control method see, e.g. \cite{BeKu1}). The complex geometric optics method \cite{SU} was applied to this problem in \cite{NSU}, \cite{No}.) } In this paper, we will first analyse the question of stability of the inverse problem, i.e. the question of continuity of $\D ^{-1}$. Later, we will also describe a procedure of an approximate reconstruction of $(\M,g)$. {\newtext Alessandrini \cite{Al},\cite{Al2} considered an inverse boundary problem for a Schr\"od\-ing\-er operator, $-\Delta + q$ in a bounded domain of $\R^m$ and obtained a log-type stability estimate in the Gel'fand inverse problem (see also \cite{KuSt}). Alessandrini and Sylvester \cite{AlS} and Stefanov and Uhlmann \cite{StU} considered inverse boundary value problems for the wave equation which are closely related to the Gel'fand inverse problem. Let \begin{eqnarray*} u_{tt} + a(\bx,D)u = 0, \quad \hbox{in} \quad \Omega \times [0,T], \quad u|_{t<0} < 0, \quad u|_{\p \Omega \times [0,T]} = f, \end{eqnarray*} where $a(\bx,D) = -\Delta + q$ in \cite{AlS} and $a(\bx,D) = - \p_ia^{ij}\p_j$ in \cite{StU} and the inverse data is given as a non-stationary Dirichlet-Neumann map. For sufficiently large $T$ and, in \cite{StU} $a^{ij}$ close to $\delta^{ij}$, they showed H\"older-type stability. These results raise the question of an optimal stability estimates in the Gel'fand inverse problem and inverse boundary value problem for the wave equation. Although exact stability estimates lie outside the scope of this paper which is devoted to the analysis of geometric conditions upon $(\M,g)$ to provide continuity of $\D^{-1}$, we believe that H\"older-type estimates could be valid only under rather strong requirements upon $(\M,g)$, like strong geodesic property (see e.g. \cite{Cr}, \cite{Sh}). Further results on stability for different types of inverse boundary value problems could be found in \cite{Is}. For counterexamples see e.g. \cite{M}. } Clearly, to discuss the continuity of $\D ^{-1}$ it is necessary to define appropriate topologies in $\bM$ and $\bB $. However, it is well-known (see e.g. \cite {CoKr}) that inverse problems are, in general, ill-posed. Therefore, to gain stability we need to know {\it a priori} that the unknown object, which we are going to reconstruct, belongs to some compact class $\bK$ (see e.g. \cite {Is}). To obtain stability results, the topologies of $\bM$ and $\bB$ have to satisfy two properties. First, the map $$\D: \bK \to \bB$$ has to be continuous. Second, $\D$ has to be injective. When these two properties are satisfied and $\bK\subset \bM$ is a compact set then the continuity of the map $$\D^{-1}: \D (\bK) \to \bK$$ follows from basic results of general topology. In the first part of the paper we carry out the constructions of the appropriate topologies. We would like to note that a proper topology on $\bM$ is the Gromov-Hausdorff topology and the (pre)-compact subsets we actually use are the manifolds of bounded geometry (see e.g. \cite{BuBuI}, \cite {Gr}, \cite {Pe}, \cite{Sa}). The topology on $\bB$ is defined from the boundary spectral convergence. In the second part of the paper our attention will be focused on the problem of an approximate reconstruction of $(\M,g)$ when we do not know all $\BSD$. Rather, we know only the first eigenvalues, $\la_j < \delta^{-1}$ {\newtext with some small $\delta > 0$ } and boundary values of the corresponding eigenfunctions and, furthermore, we know them with some error. Namely, we have a finite collection of pairs \bfo \{\mu_j,\psi_j;\,j=1,\dots,{\it n}(\delta^{-1})\}, \quad \mu_j \in \R,\, \psi_j \in L^2(\p \M) \efo and know that they are close to $ \{ \la_j,\, \varphi_j|_{\p \M};\, j=1,\dots,{\it n}(\delta^{-1})\}$, where ${\it n}(\delta^{-1})$ is the number of eigenvalues that are smaller than $\delta^{-1}$. Our definition of topology on $\bB$ is adjusted to make the intuitive notion of closedness of $\{\mu_j,\psi_j\}$ to $\{ \la_j,\, \varphi_j|_{\p \M};\, j\leq {\it n}(\delta^{-1})\}$, rigorous. In the second part of the paper we describe the stable reconstruction procedure but omit the proof showing that this procedure is stable. These proof are to be published elsewhere. We note that to carry out the reconstruction procedure we will need to work under stronger requirements on $ (\M,g)$ than those that are needed for stability. Similarly, in the first part we will only formulate necessary results omitting all proofs. The proofs and detailed description of the reconstruction procedure will be given in a forthcoming paper. \medskip {\bf II. Main Results. Stability.} We start with a proper topology on $\bM$. \begin{definition}\label{def:1b} (Gromov-Hausdorff topology). Let $\e >0.$ The Riemannian manifolds $(\M^i,g^i) \in \bM,\,i=1,2$ are $\e$-close in the Gromov-Hausdorff topology, \bfo d_{GH}((\M^1,g^1),\,(\M^2,g^2)) \leq \e, \efo if there are $\e$-nets $\{\bx_j^i;\,j=1,\dots,J(\e) \} \subset \M^i,\, i=1,2,\,$ such that \bfo |d_{g_1}(\bx_j^1,\bx_k^1)-d_{g_2}(\bx_j^2,\bx_k^2)|\leq \e, \quad j,k=1,\dots,J(\e), \efo where $d_{g^i}(\cdot,\cdot)$ stands for the distance on $(\M^i,g^i)$. \end{definition} {\newtext We remind that $\{\bx_j;\,j=1,\dots,J(\e) \} \subset \M$ is an $\e$-net if for every $\bx\in \M$ there is $\bx_j$ such that $d_{g}(\bx_j,\bx)\leq \e$. } %{\bf Definition 1 ($C^{1,\alpha}$-topology on $\bM$).} Let $\e >0, %\alpha \in 90,1)$. %Riemannian manifolds $(\M^i,g^i) \in \bM,\,i=1,2$ %are $(\e,\alpha)$-close if there is %a diffeomorphism ${\cal X}: \M^1 \to \M^2$ such that %${\cal X} ^* g^2 $ and $g^1$ are %$\e$-close with respect to the $C^{1,\alpha}$-norm on $\M (=\M^i)$. \medskip \noindent {\bf Remark 1.} In the future we will identify diffeomorphic manifolds assuming implicitly that desired statements are valid after an automorphism of $\M$. \begin{definition}\label{def:2} (Riemannian manifolds of bounded geometry). For any $\Lambda,$ $D$, $i_0 >0$, ${\bM}(\Lambda, D, i_0 ) \subset \bM$ consists of Riemannian manifolds $(\M,g) \in \bM$ such that \begin{tabular}{ll} i) $\|\nabla Rm(\M,g)\|+\|Rm(\M,g)\| \leq \Lambda,$ & iii) $\diam (\M,g) \leq D$,\\ ii) $\|\nabla S(\M,g)\|+\|S(\M,g)\| \leq \Lambda, $ & iv) $\inj (\M,g) \geq i_0$.\\ \end{tabular} \end{definition} Here $Rm$ is {\newtext the Riemannian curvature tensor on $(\M,g)$ considered as a 4-linear form on $T\M$}, {$\nabla$ is the covariant derivative, and} $S$ is the second fundamental form with respect to the inner product, i.e. a quadratic form on $T \p \M$. Furthermore, $\diam$ is the diameter of $(\M,g)$ and $\inj$ is the injectivity radius of $(\M,g)$. Condition iv) consists of three different conditions: \smallskip \noindent $(\alpha)$ Let $\bx \in \p \M$. Denote, as usual, by $B_{\p \M}(\bx,r)$ the open metric ball (on $\p \M$) of the radius $r$ with center in $\bx$. Then, for any $r < i_0$, $B_{\p \M}(\bx,r)$ is a domain of normal coordinates on $\p \M$ centered at $\bx$. \smallskip \noindent $(\beta)$ Let $\bx \in \M$, $d(\bx,\p\M)\geq i_0$. Then, for any $r < i_0\,$, $B_{ \M}(\bx,r)$ is a domain of normal coordinates on $ \M$ centered at $\bx$. \smallskip \noindent ($\gamma$) Let $\bx \in \M$, $d(\bx,\p\M)0$. Collections $$ \{\mu_j^i,\, \psi_j^i ;\, j=1,2,\dots\}\in \bB, \quad i=1,2$$ are $\delta$-close if there are disjoint intervals $$I_p = (a_p,b_p) \subset (-\delta ^{-1} -\delta,\ \delta^{-1}+\delta), \quad p=1,\dots, P,$$ such that \smallskip \noindent i) $b_p-a_p <\delta,$ \smallskip \noindent ii) For any $\mu_j^i,\, i=1,2$ with $\mu_j^i < \delta^{-1}$ there is $p$ such that $\mu_j^i \in I_p$. \smallskip \noindent iii) For any $p$, the total number $n_p^i\, $ of $\mu_j^i$ inside $ I_p$ coincide, i.e. $$ n_p^1 = n_p^2 \,(=n_p),$$ \smallskip \noindent iv) For any $p$ there is a unitary matrix $U_p = [u_p^{kl}]_{k,l=1}^{n_p}$ such that $$ \|U_p \Psi_p^1 - \Psi_p^2\|_{L^2(\p \M)^{n_p}} \leq \delta. $$ Here, $\Psi_p^i$ is the vector-function $(\psi_{j(1)}^i \cdots \psi_{j(n_p)}^i)$, $j(1) < \cdots 0,$ the map $\D^{-1}$ exists and is continuous on $\bM (\Lambda, D,i_0)$. That is, there is $\delta >0$ such that if $(\M^i,g^i) \in {\bM}(\Lambda, D, i_0), \, i=1,2,$ and their \BSD, \ba \left \{ \la_j^1,\, \varphi_j^1|_{\p \M}; \, j =1, 2, \dots\right \}\quad\hbox{and}\quad \quad \left \{ \la_j^2,\, \varphi_j^2|_{\p \M}; \, j =1, 2, \dots\right \}, \ea are $\delta$-close then $\M^1$ and $\M^2$ are diffeomorphic. \noindent Moreover, for any $\alpha \in (0,1)$, $g^1 \to g^2$ in $C^{1,\alpha}$ -topology when $\delta \to 0$. \end{theorem} \medskip \noindent {\bf Remark 2.} {\newtext The convergence of the metrics in the $C^{1,\alpha}$-topology means that in a proper coordinate system on $\M =\M_i,\, i=1,2,$ the metric tensors $g_{ij}$ converge in the $C^{1,\alpha}$-sense, see e.g. \cite{Pe}.} % $\|g^1-g^2\|_{1,\alpha} \leq \e$, %where %\bfo %\|g^1-g^2 \|_{1,\alpha} = \max_{i,k =1,\dots,m}\, %\max_{{\cal U} ^j} \| g^1_{ik}-g^2_{ik}\|_{C^{1,\alpha}({\cal U} ^j)}, %\efo %and $\{ {\cal U} ^j, \, j =1,\dots, J \}$ is a finite covering of $\M$ by %coordinate charts. % %Clearly, thus defined $C^{1,\alpha}$-norm depends on %the choice of the covering, however, different coverings give rise to %equivalent %norms. \medskip \noindent {\bf Remark 3.} It follows from definition \ref{def:3} that to apply Theorem \ref{th:2} it is sufficient to know only the parts of the $\BSD$ of $(\M^1,g^1)$ and $(\M^2,g^2)$ that correspond to the eigenvalues $\la_j^i < \delta^{-1}$. \smallskip Theorem \ref{th:2} and the fact that in the class $ {\bM}(\Lambda, D, i_0)$ the Gromov-Hausdorff topology is equivalent to the Lipschitz topology (see (\cite {Kod}) implies the following corollary that describes the Lipschitz convergence of Riemannian manifolds. \begin{corollary}\label{cor:3} Let $(\M^i,g^i) \in {\bM}(\Lambda, D, i_0), \, i=1,2$. Then, for any $\e >0$ there is $\delta >0$ such that if the $\BSD$ of these manifolds are $\delta$-close then $\M^1$ and $\M^2$ are diffeomorphic and $$ (1+\e)^{-1} \leq \frac{d_{g^1}(\bx,\by)}{d_{g^2}(\bx,\by)} \leq (1+\e), \, \quad \bx,\by \in \M. $$ \end{corollary} \medskip \noindent {\bf Remark 4.} It should be noted that ${\bM}(\Lambda, D, i_0)$ is not closed in $\bM$ in the Gromov-Hausdorff topology. Therefore, to obtain the desired stability result we should first extend Theorem \ref{th:1} onto the closure $\overline {{\bM}(\Lambda, D, i_0)}$. The set $ {{\bM}(\Lambda, D, i_0)}$ consists of {some class of $C^{1,\alpha}$-smooth} Riemannian manifolds. Although the boundary control method that was instrumental to prove Theorem \ref{th:1} is, in general, not applicable to $C^{1,\alpha}$ Riemannian manifolds it was possible to show that $$ \D: \overline {{{\bM}(\Lambda, D, i_0)}} \to {{\bB}}$$ is injective (see section IV). \medskip In Theorem \ref{th:2}, the condition iii) in ${\bM}(\Lambda, D, i_0)$ is automatically satisfied from an estimate of $\lambda_2$ in \cite{Me}. The other conditions are natural for the stability of the inverse problem because, otherwise, the convergence of the $\BSD$ does not imply the convergence of the corresponding Riemannian manifolds. We illustrate this thesis with the following examples. \smallskip \noindent {\bf Example 1.} Let $(\M,g)$ be a two-dimensional manifold obtained by the following surgeries of the two-sphere $S^2$. Add a handle $H$ near the North pole and cut-off a small piece $B$ to make a boundary near the South pole. If the metric size of the handle tend to $0$ uniformly then, in the boundary spectral topology, the $\BSD$ of $(\M,g)$ tend to the $\BSD$ of the sphere with a hole but without a handle (see e.g. \cite{Tak}). This shows the necessities of i) and iv) in ${\bM}(\Lambda, D, i_0)$. %{\bf Example 2.} Similarly to Example 1, cut-off small %pieces $B_1$ and $B_2$ near %the North and South poles %respectively. If $B_2$ shrinks to a point, then, $H^1(M)$-capacity of %$B_2$ tends to $0$ and %thus, the $\BSD$ of $(\M,g)$ tend to the $\BSD$ of %$S^3 \setminus B_1$ in the boundary %spectral topology. This also shows the necessities of i) and iv) %in ${\bM}(\Lambda, D, i_0)$ in %Theorem \ref{th:2}. % Let $(\M,g)$ be a two-dimensional sphere with a handle %near the North pole. We cut-off a small piece near the South pole %to make it a manifold with boundary. Let the metric size of the handle % tend to $0$. Then, if we ``shrink'' the handle in %a proper manner \cite{Tak},the $\BSD$ of $(\M,g)$ tend to the $\BSD$ % % in the boundary spectral topology on $\p \cM$. %Thus, the conclusions % of the theorem is no more valid. \begin{figure}[htbp] \begin{center} \psfrag{1}{$\partial M_1$} \psfrag{2}{$\partial M_2$} \psfrag{3}{$\partial M_1$} \psfrag{4}{$\partial M_2$} \psfrag{5}{$A$} \includegraphics[width=12cm]{msri2.eps}% \label{pict2}%was 12 cm \end{center} \caption{Two representations of the same manifold: Cylinder in ${\bf R}^3$ pinched in the middle and its representation as an annulus in ${\bf R}^2$ with variable metric. When $\e \to 0$ the arclength of circles in the region $A$ tend to $0$ so that the manifold splits into two components. } \end{figure} \smallskip \noindent {\bf Example 2.} Let $(\M,g)$ be cylinder in ${\bf R}^3$ with the boundary made of two components, $\p \M = \p \M_1 \cap \p \M_2$ (see Fig.1). We pinch the cylinder in the middle so that the diameter tends to $0$. Therefore, the manifold eventually ``splits'' into two disjoint semispheres. Meanwhile, in the boundary spectral topology on $\p \M$, the $\BSD$ of $(\M,g)$ tend to the $\BSD$ of this disconnected manifold. Again, conclusion of the theorem is no more valid. This second example may be also described in terms of a perturbation of the metric in an annulus $A(1,4)= \{(r,\theta) : 1 \leq r \leq 4 \}\ ,$ where $(r,\theta)$) are the polar coordinates in ${\bf R}^2$. We define the length element by \bfo dl^2_{\e} = dr^2 + \frac{r^2}{1 +\e^{-1} \chi(r)} \,d\theta^2, \quad \chi(r) \in C^{\infty}_0, \efo where \bfo \chi = 0 \quad \hbox{for} \quad r<2 \quad \hbox{or} \quad r>3, \quad \chi >0 \quad \hbox{for} \quad r \in (2,3). \efo Then, for $\e \to 0$ each circle $r \in (2,3)$ shrinks to a point so that the internal part near $r=1$ of $A$ becomes separated from the external part near $r=4$. We would note that although this example does not precisely satisfies the definition of ${\bM}_{\p \M}$ because the limiting manifold is no longer connected. However, this definition can be extended onto the case of, probably disconnected Riemannian manifolds with the same boundary $\p \M$ which have no components without the boundary if we change the condition of boundedness of $\diam (\M)$ to the boundedness of $\vol (\M)$. \smallskip \noindent {\bf Example 3} Take $X$ and $Y$ to be isospectral manifolds without boundary of different topological types (see e.g. \cite {Sunada}) and $Z$ be a Riemannian manifold with boundary $\p Z$. Let $M_{\e}$ be a Riemannian manifold obtained by connecting $Z$ with $X$ by a thin tube of radius $\e ^{2}$ of length $\e$. Similarly, let $N_{\e}$ be a Riemannian manifold obtained by connecting $Z$ with $Y$ by a thin tube of radius $\e ^{2}$ of length $\e$. When $\e \to 0$, the boundary spectral data of $M_{\e}$ tend to the union of the boundary spectral data of $Z$ and $\{\mu_j, 0\}$, where $\mu_j$ is the spectrum of $X$. Clearly, the boundary spectral data of $N_{\e}$ tend to the same limit. Thus, for any $\delta > 0$, there exists $\e _0 > 0$ such that the boundary spectral data of $M_{\e}$ and $N_{\e}$ are $\delta$-close when $\e \leq \e _0$. However, $M_{\e}$ and $N_{\e}$ are of different topological types. %\begin{figure}[htbp] %\begin{center} %\psfrag{1}{$p_1$} %\psfrag{2}{$p_2$} %\psfrag{3}{$q_1$} %\psfrag{4}{$q_2$} %\psfrag{5}{$A$} %\includegraphics[width=12cm]{msri.eps} \label{pict2}%was 12 cm %\end{center} %\caption{Two representations of the same manifold. %The unit circle with a metric perturbed in the region $A$ near the %vertical diameter and the same domain in a metric close %to Euclidean. %Points $p_i, q_i,\, i=1,2$ correspond to the same points on the manifold. %When $\e \to 0$ then $d(p_1,p_2) \to 0$. %} %\end{figure} %{\newtext {\bf Example 3.} Let again $(\M,g)$ be a unit ball %$B(0,1) \subset {\bf R}^2$ with a perturbed metric, %\bfo %dl^2_{\e} = dx^2 +\frac 1 {1+\e^{-1}\chi(x/\e)}\,dy^2, %\efo %where now $\chi(s) = 1$ for $|s| \leq 1$ (see Fig. 2). When $\e \to 0$, %this corresponds to ``pulling %together'' opposite points $p_1 = (0,1)$ and $p_2=(0,-1)$ on the unit sphere %so that eventually the unit ball ``splits'' into two parts. %Clearly, the theorem is no more valid.} In these examples $\|Rm\| \to \infty$ and $\hbox{inj} \to 0$ when $\e \to 0$ which show the importance of conditions i) and iv) of the theorem. Similar examples can be given to illustrate the importance of condition ii). \medskip {\bf III. Main results. Approximate reconstruction.} Definition \ref{def:1b} of the Gromov-Hausdorff topology in $\bM$ can be easily extended to the set of all compact metric spaces $(X,d)$. Thus, two Riemannian manifolds $(\M^i,g^i)$ are $\e$-close in the Gromov-Hausdorff metric if they possess $\e$-nets $X^i$ which, being considered as finite metric spaces with distance inherited from $(\M^i,g^i)$, are $\e$-close in the Gromov-Hausdorff topology. Therefore, to approximately reconstruct the Riemannian manifold $(\M,g)$ we should construct a metric space $(Y,d), Y = \{\by_1,\dots,\by_J\}$ which is $\e$-close to $(\M,g)$, i.e. \bfo d_{GH}((\M,g),\,(Y,d)) \leq \e. \efo To construct a metric space approximation to $(\M,g)$ we will assume further regularity properties of the class of manifolds. %\medskip %\begin{definition}\label{def:3beta} For any $\beta \in (0,1),\,$ %${\bM}^{\beta}(\Lambda, D, i_0)$ consists of % %\noindent Riemannian manifolds that satisfy conditions %iii) and iv) of Definition 2 and also % %\smallskip %\noindent %$i^{\beta})\,\quad \|Rm(\M,g)\| _{\beta} \leq \Lambda,$ % %\smallskip %\noindent %$ii^{\beta})\,\quad \|S(\M,g)\| _{\beta} \leq \Lambda$. % %\noindent Here $\|Rm(\M,g)\|_{\beta}, \|S(\M,g)\| _{\beta}$ are the %$C^{0,\beta}$-norm of the Riemannian curvature tensor $R_{ijkl}$ and the %Second fundamental %form $S_{ij}$ % with respect to fixed coordinate charts on $M$. %(Note that this norm depends on the choice %of charts but equivalent each other as in Remark 3, thus %fixing charts is not essential.) %\end{definition} %\smallskip %{\bf Remark 6.} The Sasakian metric on $T\M$ is the metric inherited from %the metric $g$ on $\M$. % Namely, if $T_{\bx,\bu} (T \M)$ is %identified with $T_{\bx}\times T_{\bx}$ by considering %$(0,\bw) \in T_{\bx,\bu} (T \M)$ as velocity vector of path $t\mapsto %(x,\bu+t\bw)$, the Sasakian metric is defined by %\bfo %|(\bv,\bw)|^2 = |\bv|^2_g + |\bw|^2_g. %\efo \begin{theorem}\label{th:4.1} Let $\Lambda, D,i_0 >0$. Then for any $\e >0$ there is $\delta >0$ such that if $ \{\mu_j,\, \psi_j|_{\p M};\, j=1,2,\dots \} \in \bB \,$ is $\delta$-close to the $\BSD$ $ \{ \la_j,\, \varphi_j|_{\p \M}; \, j=1,2,\dots \}$ of a Riemannian manifold, $ (\M,g) \in \overline{{\bM}(\Lambda, D, i_0)} \, $, then there is a finite metric space $(Y,d)$ such that \bfo d_{GH}((\M,g),\,(Y,d)) \leq \e \efo Moreover, there is a constructive algorithm to find $(Y,d)$ from $ \{\mu_j^i,\, \psi_j^i;\,j=1,2,\dots \} \in \bB \,$. \end{theorem} The algorithm to construct $(Y,d)$ is described in section V. \medskip \noindent {\bf Remark 5.} By definition \ref{def:3} the $\delta$-closedness does not involve $ \{\mu_j^i,\, \psi_j^i \}$ with $|\mu _j| >\delta^{-1}$. Similarly, the construction of $(Y,d)$ does not use these data. \medskip {\bf IV. Geometric convergence and stability.} The proof of Theorem \ref{th:2} is based upon the following result. \begin{proposition}\label{th:5} For any $\Lambda, D,i_0 >0,\,$ the set ${\bM}(\Lambda, D, i_0)$ is pre-compact in the $\GH $ topology. Its closure, $\overline{{\bM}(\Lambda, D, i_0)}$ consists of differentiable manifolds with $C^{1,\alpha}$-smooth metric, where $\alpha \in (0,1)$ is arbitrary. \end{proposition} If a sequence of Riemannian manifolds $(\M^n,g^n)$ converges to $(\M,g)$ in the $\GH$ topology on $\overline{{\bM}(\Lambda, D, i_0)}$, there is $n_0 \geq 1$ such that \smallskip \noindent \smallskip \noindent { i) For $n \geq n_0$, there is a diffeormorphism $F_n:\M\to \M^n$, i.e. $\M^n$ and $\M$ are diffeomorphic. \smallskip \noindent ii) for any $\alpha \in (0,1)$, $F_n^*(g^n)$ converge to $g$ in the $C^{1,\alpha}$-topology. } \smallskip \noindent \noindent {\bf Remark 6.} { Proposition \ref{th:5} is obtained by applying the ideas developed by M. Anderson (\cite {And}, see also \cite {HH}) in the interior of the manifold and the fundamental equation of Riemannian geometry (see \cite{Pe}) near the boundary. The latter equations is also known as the Riccati equation for the second fundamental form along normal geodesics. This equation gives the desired regularity estimates for the metric tensor in the boundary normal coordinates.} \smallskip \noindent The geometric convergence described by Proposition \ref{th:5} yields the continuity of the direct problem. \smallskip \begin{proposition}\label{p:6} $ \D: \overline{{\bM}(\Lambda, D, i_0)} \to {{\bB}} $ is continuous. \end{proposition} \smallskip \noindent {\bf Remark 7.} {Proposition \ref{th:5} was proven by Kodani \cite {Kod} in the case of weaker regularity. Namely, Kodani showed that the manifolds satisfying \noindent \begin{tabular}{ll} i) $\|Rm(\M,g)\| \leq \Lambda,$ \hspace*{3cm} & iii) $\diam (\M,g) \leq D$,\\ ii) $\|S(\M,g)\| \leq \Lambda, $ & iv) $\inj (\M,g) \geq i_0$.\\ \end{tabular} \noindent are pre-compact in the Lipschitz topology rather than the $C^{1,\alpha}$-topology.} This result is sufficient for continuity of $\D$. This can be obtained by means of the perturbation theory of quadratic form (see e.g. \cite{Fu} in the case of manifolds without boundary). %Then the Lipschitz convergence %implies, in particular, %the convergence of the corresponding $\BSD$ in the locally uniform %topology %on $\p \M$. Therefore, the result of Kodani would have been sufficient to prove the continuity of $\D ^{-1} : \D(\overline {{\bM}(\Lambda, D, i_0)}) \to \overline {{\bM}(\Lambda, D, i_0)}$ if it were possible to show the injectivity of $\D$ on $\overline {{\bM}(\Lambda, D, i_0)}$. Unfortunately, the method used to prove Theorem \ref{th:1} fails in this case in its parts related to the approximate controllability result by Tataru \cite {Ta} and also to the injectivity of the boundary distance map (see the next section for the definition of this map). Proposition \ref{th:5}, on the contrary, guarantees a stronger regularity of the Riemannian manifolds in $\overline {{\bM}(\Lambda, D, i_0)}$. It was used to extend Theorem \ref{th:1} on the class \bfo \hat{{\bM}} = \bigcup _{\Lambda,D,i_0} \overline {{\bM}(\Lambda, D, i_0)}. \efo \begin{theorem}\label{th:1'} The map $ \D : \hat{{\bM}} \to {{\bB}}$ is injective. \end{theorem} \medskip {\bf V. Construction of a finite $\e$-net.} The procedure of constructing an approximation $(Y,d)$ consists of several steps which we will explain separately. We will also point out smoothness requirements which are necessary to carry out different steps of the procedure. \smallskip {\bf a. Construction of Fourier coefficients of waves.} For simplicity, we assume here that the metric on the boundary, $i^*(g),\, i:\p\M \to \M$ is given. Consider an initial-boundary value problem associated with the Neumann Laplace operator, \bfo u^f_{tt} - \Delta_gu^f = 0, \quad \hbox {in} \quad \M \times [0,D], \efo \bfo \p_{\nu}u^f|_{\p \M \times [0,D]} = f \in C_p^1([0,D], \,L^2(\p \M)), \efo \beq u^f|_{t=0} = u^f_t|_{t=0} =0, \label {7} \eeq where $C_p^1([0,D], \,L^2(\p \M))$ consists of piecewise $C^1$-functions of $t$. We {\newtext introduce the wave operator $W^t$ given by} \bfo W^t(f) = u^f(\cdot,t) \in C^1([0,D], \,L^2( \M)) \efo and, if we denote by $u^f_k(t)$ the $k$-th Fourier coefficient of $u^f$, \beq u^f_k(t) = (u^f(t),\,\varphi_k) = \int_0^t \,\int_{\p \M} s_k^t(\by,t')f(\by,t')\,dt'dS_g(\by), \label {8} \eeq where $dS_g$ is the boundary area element and \bfo s_k^{t}(\by,t') = \frac{\sin (\sqrt {\la_k}(t-t'))}{\sqrt {\la_k}}\, \varphi_k(\by). \efo This formula makes possible to find approximately the first Fourier coefficients of the waves $u^f$ if we know $ \{\mu_j, \psi_j;\, j=1,\dots,{\it n}(\delta^{-1})\}$. \smallskip {\bf b. Construction of domains of influence.} Let $\Gamma \subset \p \M$ be open. Consider the waves $u^f(\tau)$ with $f \in C_0^{\infty}(\Gamma \times [0,\tau]),\, \tau >0$. Clearly, \bfo \hbox {supp} [u^f(\tau)] \subset \M(\Gamma,\tau) = \{\bx \in \M: d_g(\bx, \Gamma) \leq \tau \}. \efo By the fundamental result of Tataru (\cite {Ta}, see also \cite {KaKuLa}, Ch.2.5), we have \smallskip \begin{theorem}\label{th:7} Assume that the metric tensor $g$ is $C^1$-smooth. For any $\tau >0$ and $\Gamma \subset \p \M$, \bfo \hbox {cl}_{L^2(\M)} \left \{u^f(\tau): f \in C_0^{\infty}(\Gamma \times [0,\tau]) \right \} = L^2(\M(\Gamma,\tau)), \efo where $L^2(\Omega) \subset L^2(\M)$ consists of functions with support in $\overline {\Omega}$. \end{theorem} Let $\eta >0$, $D>\diam(\M)$ as in part iii) of Definition \ref{def:3} and $ \Gamma_l,\quad l=1,\dots, L,$ be open subsets of $\p \M$ satisfying \bfo \diam (\Gamma_l) < \eta, \quad \Gamma_l \cap \Gamma_k = \emptyset, \quad \p \cM = \overline{\cup \Gamma_l}. \efo Let \beq \alpha = (\alpha_1,\cdots, \alpha_L),\, \quad \alpha_l \leq D/\eta, \label {8a} \eeq be a multi-index. Denote by $\Sigma_{\alpha} \subset \p \cM \times [0,D]$ the set \bfo \Sigma_{\alpha} = \bigcup _l \, (\Gamma_l \times [D-\alpha_l \eta,\,D]) \cap (\p \cM \times [0,D]) \efo and by $\M_{\alpha}\subset \M$ the subdomain \beq \M_{\alpha} = \bigcup_l \M(\Gamma_l,\,\alpha_l\eta). \label {3'} \eeq %\smallskip %{\bf Corollary 8} For any $\Sigma_{\alpha} $, %\bfo %\hbox {cl}_{L^2(\cM)} \left \{ W^Df: f \in C^{\infty}_0 (\Sigma_{\alpha}) %\right \} = L^2(\M_{\alpha}). %\efo By Theorem \ref{th:7}, for any $\sigma >0$ there is $f=f_{\alpha} \in C_0^{\infty}(\Sigma _{\alpha})$ such that \bfo \|u^{f}(D) - \chi_{\M_{\alpha}} \varphi_1 \|_{L^2(\M)} \leq \sigma/2, \efo where $\chi_{\Omega}$ is the characteristic function of a set $\Omega$. It then follows from the general results of control theory that there is a finite linear combination $\hat {f}$, \beq \hat {f}(\bx,t) = \sum_{j=1}^J a_j s_j^D(\bx,t) \chi_{\Sigma_{\alpha}}(\bx,t), \quad \bx \in \p\M, \label {9a} \eeq such that \beq \|u^{\hat {f}}(D) - \chi_{\M_{\alpha}} \varphi_1 \|_{L^2(\M)} \leq \sigma. \label {10} \eeq The coefficients $a_j$ of the function (\ref{9a}) which satisfies equation (\ref{10}) can be found using the $\BSD$ $\{ \la_k,\, \varphi_k|_{\p \M};\,k=1,\dots,K\},\,$ $ K>J$ by means of some variational procedure. %As we know $ \{\mu_k,\, \psi_k \}$ rather then %$\{ \la_k,\, \varphi_k|_{\p \M}\}$, this variational procedure %provides approximate coefficients $\tilde {a}_j$ which are close to %$a_j,\,j=1,\dots,J$ when $E>0$ is sufficiently large and $\delta >0$ -- %sufficiently small. Namely, %\beq %\|u^{\tilde {f}_{\Gamma,\tau}}(\tau) - \Xi_{\M(\Gamma,\tau)} \varphi_0 \| %\leq \sigma, %\label {11} %\eeq %where %\beq %\tilde {f}_{\Gamma,\tau} = \sum_{j=1}^J a_j s_j(\tau -t) \varphi_j %\Xi_{\Gamma}. %\label {12} %$\eeq To define this procedure rigorously, let $H_N(\Sigma_{\alpha}) \subset C_p^1([0,D], \,L^2(\p \M))$ be an $N$-dimensional subspace spanned by the functions $ \{s_n^D(\by,t)\,\chi_{\Sigma_{\alpha}} (\by,t);\,$ $n=1,\dots,N \}$. Let $A_{N,I,K}(f)$ be a functional on $H_N(\Sigma_{\alpha})$, \beq A_{N,I,K}(f) = \label {803} \eeq \bfo = \sup \left \{ \left \|(P_KW^Df - \chi_{\M_{\alpha}} \varphi_1,W^Dh) \right \|: \, h \in H_I(\Sigma_{\alpha}), \quad \|P_KW^Dh \| \leq 1 \right \}. \efo Here $P_N$ is the orthoprojector in $L^2(\M)$, \bfo P_Nu = \sum_{n\leq N} (u,\varphi_n)\,\varphi_n. \efo Clearly, due to (\ref {8}) the functional $A(f)$ can be evaluated in terms of $ \{ \la_j,\, \varphi_j|_{\p \M};\,$ $j=1,\dots,{\it n}(\delta^{-1})\}$ if ${\it n}(\delta^{-1}) \geq \max (N,I,K)$. As we have $\{\mu_j,\, \psi_j ;\,j=1,\dots, {\it n}(\delta^{-1})\}$ rather then $ \{ \la_j,\, \varphi_j|_{\p \M};\,j=1,\dots,{\it n}(\delta^{-1})\}$, the functional $A(f)$ can be evaluated only approximately. \smallskip \begin{theorem}\label{th:9} For any $\sigma >0$ there are parameters $C, N,I,K,\delta,\,$ $N \leq I \leq K \leq {\it n}(\delta^{-1}),\,$ such that if $ \left\{\mu_j,\, \psi_j \right \}$ is $\delta$-close to the boundary spectral data $\left \{ \la_j,\, \varphi_j|_{\p \M} \right \}$ of a Riemannian manifold $(\M,g)$, then a minimizer $f^* \in H_N$ of the functional $A_{N,I,K}$ such that \bfo \|f^*\|_{L^2(\p\M \times [0,D])} \leq C, \efo satisfies the equation \bfo \left \|W^Df^* - \chi_{\M_{\alpha}} \varphi _1 \right \| \leq \sigma. \efo \end{theorem} \medskip By Proposition \ref{p:6} the $\BSD$ depend continuously on the $\GH$ distance restricted on $\overline{{\bM}(\Lambda, D, i_0)}$. Consider \bfo u^f(\cdot,t) = u^f_{(\M,g)}(\cdot,t) \efo as a function of $(\M,g) \in \overline{{\bM}(\Lambda, D, i_0)}$. {\newtext By Proposition \ref{th:5}, the perturbation theory \cite{Ka} shows that the wave $u^f_{(\M,g)}(\cdot,t)$ depends continuously (in the $C^1([0,D],L^2(\M))$-topology) on the $\GH$ distance. As the set $\overline{{\bM}(\Lambda, D, i_0)}$ is compact, the parameters} $C,N,I,K$ and $\delta$ used in the above construction are uniform on $\overline{{\bM}(\Lambda, D, i_0)}$ and depend only on $\sigma$. \medskip {\bf c. Construction of the boundary distance map.} As \bfo \|\chi_{\M_{\alpha}} \varphi _1 \|^2 = \frac {\vol (\M_{\alpha})}{\vol (\M)}, \quad \vol(\M) = \varphi_1(z)^{-2},\ z\in \p M, \efo Proposition \ref{th:5} makes it possible to evaluate an approximate volume, $\vol ^a (\M_{\alpha})$. We will use this observation to construct a finite approximation to the set of the boundary distance functions associated with $(\M,g)$. Namely, for any $\bx \in \M$, let $r_{\bx} \in L^{\infty}(\p\M)$ be given by \bfo r_{\bx}(\bz) = d_g(\bx,\bz), \quad \hbox {for any} \quad \bz \in \p\M. \efo The boundary distance functions determine the boundary distance map $R: \,(\M,g) \to L^{\infty}(\p\M)$, \bfo R(\bx) = r_{\bx}. \efo It turns out that the map $R$ or, more precisely, its image $R(\M,g)$ is sufficient to reconstruct $(\M,g)$ (see \cite {Ku} for the procedure of the reconstruction of $(M,g)$ from $R(M)$). We will use an approximation $R^*$ to $R(\M,g)$ to construct the desired metric space $(Y,d)$ in Theorem \ref{th:4.1}. Note that the metric $d$ is different to the induced metric as a (metric) subspace of $L^{\infty}(\p\M)$ (cf. Step 1 in subsection {\bf d}. To find $R^*$ consider subdomains $\M_{\beta}^* \subset \M,$ \beq \M_{\beta}^* = \left \{\bx \in \M: \, d_g(\bx, \Gamma _l) \in ((\beta _l -2) \eta, (\beta _l +2) \eta),\quad l=1,\dots,L \right \}. \label {21} \eeq For any $\beta$ the subdomain $\M_{\beta}^*$ can be obtained as a finite number of unions, intersections and compliments of the subdomains $\M_{\alpha}$ of form (\ref{3'}). For any $\Omega, \Omega' \subset \M$, \bfo \vol (\Omega ^c) = \vol (\M) - \vol (\Omega), \efo where $\Omega ^c = \M \setminus \Omega$, and \bfo \vol (\Omega \cap \Omega') = \vol (\Omega) +\vol (\Omega') - \vol (\Omega \cup \Omega'). \efo Moreover, for any $\alpha, \beta$, \bfo \vol (\M_{\alpha} \cup \M_{\beta}) = \vol (\M_{\gamma}), \quad \hbox {where} \quad \gamma _l = \max (\alpha _l, \beta _l). \efo Therefore, it is possible to evaluate an approximate volume $\vol ^a (\M_{\alpha}^*)$, if we know $\{\mu_j,\, \psi_j\}$. Clearly, $| \vol ^a (\M_{\beta}^*) -\vol (\M_{\beta}^*) | < \sigma$ when $\{\mu_j,\, \psi_j \}$ is $\delta$-close to $\{ \la_j,\, \varphi_j|_{\p \M} \}$ with sufficiently small $\delta$. When $\vol ^a (\M_{\beta}^*) \geq \sigma$ we associate with this $\beta$ a function $r_{\beta} \in L^{\infty}(\p\M)$, \bfo r_{\beta}(\bz) = \beta _l \eta \quad \hbox {for} \, \quad \bz \in \Gamma _l. \efo Thus, if $\vol (\M_{\beta}^*) \geq 2\sigma$ so that $\vol ^a (\M_{\beta}^*) \geq \sigma$, there is an approximate function $r_{\beta}$ such that for any $\bx \in \M_{\beta}^*$, %By definition (\ref {21}), for any $\bx \in \M_{\alpha}^*,$ \bfo \|r_{\beta} - r_{\bx} \| \leq 2 \eta. \efo {\newtext In the construction of the approximate volume of a set $\M_{\beta}^* \neq \emptyset$ such that $\vol (\M_{\beta}^*) < 2\sigma$ it may happen that $\vol ^a (\M_{\beta}^*) < \sigma$. Thus, according to the procedure, $R^*$ do not contain $r_{\beta}$ corresponding to this multi-index $\beta$. Consider next a point $\bx \in \M_{\beta}^*$ for such $\beta$. By condition (\ref {8a}) there is an upper bound $N(\eta)$ for the number of all multi-indexes $\beta$. Moreover, due to condition iv) of definition \ref{def:2}, $\vol (B_{\M}(\bx,\eta)) \geq c \eta ^m$. Therefore, if $\sigma \leq C' \eta ^m N(\eta)^{-1}$, there is a multiindex $\gamma, \ \gamma \neq \beta$ such that \smallskip \noindent i. $\vol (\M_{\gamma}^*) \geq 2 \sigma$, so that $\vol ^a (\M_{\gamma}^*) \geq \sigma$, \smallskip \noindent ii. $d(\bx,\M_{\gamma}^*) \leq \eta$. \smallskip \noindent Thus \bfo \|r_{\gamma} - r_{\bx} \| \leq 4 \eta. \efo Summarizing the previous considerations we obtain the following result. } \begin{lemma}\label{l:10} For any $(\M,g) \in \overline {\bM}$ and any $\eta >0$ there is $\delta >0$ such that if $\{\mu_j,\, \psi_j\}$ is $\delta$-close to the $\BSD\,$ $\{ \la_j,\, \varphi_j|_{\p \M} \}$ then the set $R^*$ constructed from $\{\mu_j,\, \psi_j;\,j=1,\dots,{\it n}(\delta^{-1})\}$ is $\eta$-close to $R(\M,g)$, i.e. \bfo d_H(R^*,R(\M,g)) \leq \eta. \efo Here $d_H$ is the Hausdorff distance between subsets in the metric space $L^\infty(\p \M)$. Moreover, $\delta$ can be chosen uniformly on any $\overline{{\bM}(\Lambda, D, i_0)}$. \end{lemma} \medskip {\bf d. Construction of an $\e$-net} The set $R^*$ has a metric structure inherited from $L^{\infty}(\p\M)$, \bfo d_{\infty}(r_1,r_2) = \|r_1 -r_2\|_{L^{\infty}(\p\M)}. \efo If $(\M,g)$ is strongly geodesic (e.g. \cite{Cr}, \cite{Sh}), i.e. all geodesic inside $\M$ are minimal, then \beq d_g(\bx,\by) = \|r_{\bx} -r_{\by}\|_{L^{\infty}(\p\M)}, \label {23} \eeq for any $\bx,\by \in \M$. Thus, the metric space $(R^*,d_{\infty})$ provides a $2\eta$-net for $(\M,g)$, \bfo d_{GH}((R^*,d_{\infty}),\,(\M,g)) \leq 2\eta. \efo However, condition (\ref{23}) is not valid for general manifolds. %It should be replaced by a more general relation, %\bfo %c\,d_g(\bx,\by) \leq \|r_{\bx} -r_{\by}\| \leq d_g(\bx,\by), %\efo %where the constant $c$ is uniform on $\overline{{\bM}(\Lambda, D, i_0)}$. Therefore, $d_{\infty}$-metric is inappropriate to construct an approximation to $(\M,g)$ in the $\GH \,$ topology. In this section we will describe how to equip the finite metric space $R^*$ with another metric $d^*$ so that $(Y,d)=(R^*,d^*)$ becomes close to $(\M,g)$ in the $\GH$ metric. %{\newtext *GENERAL % COMMENTS*: %Below, the procedure is not very clear. I suggest that we %explain our procedure in the case when we know the whole set $R(\M)$ %and say then how the method can be modified to the case %where $R^*$ is given. I think that %we should clearly say following things: % %1. Clearly construct the boundary %collar. % %2. Say that we have a test to see when $r_1$ and $r_2$ are %near a geodesic determined by points $z$ and $z'$ in the boundary collar. % %3. Tell what is a good triangle and how to find it % %4. Say how to construct the small distances and go from them %to global distances. % %*END OF GENERAL COMMENTS*} {\newtext In our description we will assume that the whole $R(\M)$ rather then $R^*$ is known. We will show how to find an approximation to $d_g(\bx,\by)$ from $R(\M)$. This procedure is valid also in the case when we know only $R^*$. The corresponding result is given at the end of this section.} \smallskip \noindent {\bf 1.} Assume first that $\bx_1,\bx_2 \in \M ^{int}$ can be connected by a unique shortest geodesic, $\gamma (\bx_1,\bx_2)$. Assume in addition that this geodesic can be continued beyond $\bx_2$ to a boundary point $\bz \in \p\M$ so that $\gamma (\bx_1,\bz)$ is a shortest geodesic between $\bx_1$ and $\bz$. Then, \bfo d_g(\bx_1,\bx_2) = d_g(\bx_1,\bz) - d_g(\bx_2,\bz) = \|r_{\bx_1} -r_{\bx_2}\|, \efo can be found from $R(\M,g)$. It remains to identify those $r_1,r_2 \in R(M)$ which correspond to $\bx_1,\bx_2$ with this property. % As $R(\M,g)$ determines $i^*g$ on $\p\M$, {\newtext As $i^*g$ on $\p\M$ is known,} it is possible to extend $(\M,g)$ to a larger Riemannian manifold $(\tilde {\M}, \tilde g)$ which has a continuous, piecewise $C^1$ metric $ \tilde g$. We can then extend each function $r_{\bx} \in C(\p\M)$ to the function $\tilde r_{\bx} \in C(\tilde {\M} \setminus \M)$, $\,\tilde r_{\bx}(\by)= d_{\tilde{\M}}(x,y)$. %Neglecting %the case when the point $\bz$ is the cut point for $\bx_i, \,i=1,2,\,$ %along %$\gamma (\bx_i,\bz)$ {\newtext Take a point $\bz \in \p\M$ and consider minimal geodesics $\gamma (\bx_i,\bz),\, i=1,2$. Assume that $\gamma (\bx_i,\bz) \setminus \{\bz\} \subset \M ^{{int}}$, $\bz$ is not a cut point along any of $\gamma (\bx_i,\bz)$ and $\gamma (\bx_i,\bz)$ are transversal to $\p \M$. Then these geodesics can be extended as minimal geodesics beyond $\bz$ upto some points $\by_i \in \tilde {\M} \setminus \M$. Denote by $\gamma_i(\bz,\by_i) \subset \tilde {\M} \setminus \M$ these extensions. Then the geodesic $\gamma(\bx_1,\bx_2)$ connecting $\bx_1$ and $\bx_2$ extends to a minimal geodesic to $\bz \in \p \M$ if and only if $\gamma_1(\bz, \by_1) = \gamma_2(\bz, \by_2)$.} \begin{figure}[htbp] \begin{center} \psfrag{1}{$y_3$} \psfrag{2}{$x_2$} \psfrag{3}{$\hspace{-3mm}x_1$} \psfrag{4}{$\hspace{5mm}y_2$} \psfrag{5}{$\gamma_{31}$} \psfrag{6}{$\gamma_{32}$} \psfrag{7}{$\gamma_{12}$} \psfrag{8}{$\partial M$} \psfrag{9}{$\partial \tilde M$} \includegraphics[width=8cm]{C.eps} \label{pict3}%was 12 cm \end{center} \caption{Example of a good triangle used to construct $d(x_1,x_2)$. In this case $x_1=y_1$ and $x_2$ lies on the geodesic $\gamma(y_2,y_3)$.} \end{figure} \smallskip \noindent {\bf 2.} Let now $\bx_1,\bx_2 \in \M ^{{int}}$ lie on the sides $\gamma (\by_3,\by_1)$ and $\gamma (\by_3,\by_2)$ of a {\newtext good} geodesic triangle $\Delta (\by_1,\by_2,\by_3)$. {\newtext A geodesic triangle $\Delta$ is called good if } all three geodesics $\gamma (\by_i,\by_j)$ can be continued to $\p\M$ as the shortest geodesics (see Fig. 2). Then, by part {\bf 1} we can find $d_g(\by_i,\by_j) \,$ and also $d_g(\bx_i,\by_i)$. Let $\tilde{\Delta} (\tilde{\by}_1,\tilde{\by}_2,\tilde{\by}_3)$ be a triangle in $\R^2$ with $d_e(\tilde{\by}_i,\tilde{\by}_j) = d_g(\by_i,\by_j)$, where $d_e$ stands for the Euclidean distance. Let $\tilde{\bx}_i, i=1,2$ be the points on the sides $[\tilde{\by}_3,,\tilde{\by}_i]$ of $\tilde{\Delta}$ with $d_e(\tilde{\bx}_i,\tilde{\by}_i) = d_g(\bx_i,\by_i)$. Then, by the Alexandrov lemma, \bfo |d_g(\bx_1,\bx_2) - d_e(\tilde{\bx}_1,\tilde{\bx}_2)| \leq c \sigma^2, \efo where $\sigma = \max d_g(\by_i,\by_j)$ and $c$ is uniform on $\overline{{\bM}(\Lambda, D, i_0)}$. Therefore, if for some given $\bx_1,\bx_2$, we can find a proper geodesic triangle $\Delta (\by_1,\by_2,\by_3)$ of a small size, we can find $d_g(\bx_1,\bx_2)$ with an error of order $\sigma ^2$. \smallskip \noindent {\bf 3.} The existence of a good geodesic triangle for any sufficiently close $\bx_1,\bx_2 \in \M ^{{int}}$ is based on the following result. \begin{lemma}\label{l:11} Let $\Lambda,D,i_0 >0$ and $(\M,g) \in \overline{{\bM}(\Lambda, D, i_0)}$. Let $\bx \in \M$. Then there is a constant $\rho>0$ such that for any $(\M,g) \in \overline{{\bM}(\Lambda, D, i_0)}$ and $\bx \in (\M,g)$ with $d_g(\bx,\p\M) \geq C \rho$ there is a vector $\bu \in T_{\bx}\M\,$ $|\bu|=1$ with the following property: \smallskip \noindent Let $(\by,\bv) \in B_{\rho}(\bx,\bu),\,|\bv| =1$. Then the geodesic $\gamma _{\by}(t\bv),\, t\geq 0,\,$ can be extended as a shortest geodesic to the boundary. \end{lemma} Here $B_{\rho}(\bx,\bu)$ is the ball of radius $\rho$ with center at $(\bx,\bu)$ in the Sasakian metric on $T\M$. {\newtext Combining steps {\bf 1-3} we can find an approximation to $d_g(\bx_1,\bx_2)$ when $\bx_1,\bx_2$ are sufficiently close to each other and not too close to $\p\M$. (For the definition of the Sasakian metric, see e.g. \cite {Sa}.) \smallskip \noindent {\bf 4.} Let \bfo d_g(\bx_i,\p\M) = \min_{\bz \in \p\M} r_i(\bz) \leq \eta < i_0, \quad r_i =r_{x_i}. \efo There are unique $\bz_i \in \p\M$ with $d_g(\bx_i,\bz_i) = r_i(\bz_i)$. Denote \bfo \tilde{d} = \tilde{d}(\bx_1,\bx_2) = \left[d^2_{\p\M}(\bz_1,\bz_2) + |r_1(\bz_1) - r_2(\bz_2)|^2 \right]^{1/2}, \efo where $d_{\p\M}(\cdot,\cdot)$ is the distance along $\p\M$. Then, \bfo |d_g(\bx_1,\bx_2) - \tilde{d}(\bx_1,\bx_2)| \leq C(\eta \tilde{d} +\tilde{d}^2). \efo Therefore, $\tilde{d}$ is a good approximation to $d$ when $\bx_i$ are close to $\p\M$. \smallskip \noindent {\bf 5.} Steps {\bf 1-4} define approximate distance $ \tilde{d}(\bx_1,\bx_2)$ for sufficiently close $\bx_1,\,\bx_2$. Using a standard procedure we can extend $ \tilde{d}(\bx_1,\bx_2)$ onto arbitrary $\bx_i \in \M$. } \smallskip When we know the metric space $(R^*,d_{\infty})$ rather then $R(\M,g)$ we can carry out the above constructions approximately. This gives rise to the following result. \begin{lemma}\label{l:12} Let $Y \subset L^{\infty}(\p\M)$ be a set described in Lemma \ref{l:10}. It is possible to equip $Y$ with a metric structure $d(r_1,r_2)$ so that \bfo d_{GH}((Y,d),(\M,g)) \leq \e (\eta), \efo where $\e (\eta) \to 0\,$ when $\eta \to 0$. The function $\e(\eta)$ can be found uniformly for $\overline{{\bM}(\Lambda, D, i_0)}$. \end{lemma} \smallskip Combining steps {\bf a.-d.} we obtain a procedure to construct a finite $\e$-net for $(\M,g),\, \e = \e(\delta)$. \medskip {\bf VII. Concluding remarks.} {\bf Remark 8.} The above considerations lack quantitative estimate for the function $\e(\delta)$. Combining our construction with the Carleman estimates (see, e.g. \cite {Ta}, \cite {Ta1}, \cite {KaKuLa}) we will obtain quantitative estimates for $\e =\e(\delta)$. This will be described in a forthcoming paper. \smallskip {\bf Remark 9.} All considerations remain valid for the Dirichlet Laplacian. We need just to take into account that $\varphi_1 \neq 0$ in $\M^{{int}}$. \smallskip {\bf Remark 10.} It is possible to use the convergence of the heat kernels restricted to $\p\M$ instead of the boundary spectral convergence. Besides, it is possible to modify the procedure to find an approximation to the heat kernel associated with $-\Delta _g$ rather then $R^*$. This puts our results within the framework of the spectral convergence of Riemannian manifolds (see e.g. \cite {BeBeG}, \cite {KaKu}). This will be described in the forthcoming paper. \smallskip {\bf Remark 11.} Requirements of $C^1$-regularity of $Rm$ and $S$ are of technical nature. We intend to improve reconstruction procedure to be valid for weaker regularity assumptions for $Rm$ and $S$. Part of our results also briefly announced in \cite {Kat}. \medskip {\bf Acknowledgements} The authors would like to express their gratitude to Y. D. Burago, D. Y. Burago, M. Gromov, I. M. Gel'fand, A. Katchalov, T. Sakai, E. Somersalo, J. Takahashi for helpful discussions and friendly support. This work was partly supported by EPSRC (UK) grant EPSRC (UK) grants GR/M14463 and 36595, RiP Program, Oberwolfach (Germany), Grant-in Aid for Scientific Research 09640109 and 12640073 (Japan), IHES (France), Finnish Academy project 42013 and 172434, MSRI NSF grant DMS-9810361 (USA), and TEKES (Finland). We are grateful to all these organizations. \begin{thebibliography} {1} {\small \bibitem{Al} Alessandrini G. {\it Stable determination of conductivity by boundary measurements}, Appl. Anal., {\bf 27} (1988), 153--172. \vspace{-2mm} \bibitem{Al2} Alessandrini G., {\it Determining conductivity by boundary measurements, the stability issue}, in R. Spigler, Appl. and Ind. Math., Kluwer 1991. \bibitem{AlS} Alessandrini G., Sylvester J. {\it Stability for a multidimensional inverse spectral theorem.} Comm. Part. Diff. Eq. {\bf 15} (1990), 711--736. \vspace{-2mm} \bibitem{And} Anderson M.T. {\it Convergence and rigidity of manifolds under Ricci curvature bounds}, Invent. Math., {\bf 102} (1990), 429-445. \vspace{-2mm} \bibitem{Be} Belishev, M.I. {\it An approach to multidimensional inverse problems for the wave equation.} (Russian) Dokl. Akad. Nauk SSSR, {\bf 297} (1987), 524--527 \vspace{-2mm} \bibitem{BeKu1} Belishev, M.I., Kurylev, Y.V. {\it A nonstationary inverse problem for the multidimensional wave equation "in the large".} (Russian) Zap. Nauchn. Sem. LOMI, {\bf 165} (1987), 21--30. \vspace{-2mm} \bibitem{BeKu} Belishev M. and Kurylev Y. {\it To the reconstruction of a Riemannian manifold via its spectral data (BC-method)}, Comm. Part. Diff. Eq., {\bf 17} (1992), 767-804. \vspace{-2mm} \bibitem{BeBeG} Berard P., Besson G. and Gallot S. {\it Embedding Riemannian manifolds by their heat kernel}, Geom. Funct. Anal., {\bf 4} (1994), 373-398. \vspace{-2mm} \bibitem{BuBuI} Burago D., Burago Y. and Ivanov S. A Course in Metric Geometry. AMS, Providence (2001) \vspace{-2mm} \bibitem{Cr} Croke, C. {\it Rigidity and the distance between boundary points.} J. Diff. Geom., {\bf 33} (1991), 445--464. \vspace{-2mm} \bibitem{CoKr} Colton D. and Kress R. Inverse Acoustic and Electromagnetic Scattering Theory, Springer, Berlin (1998). \vspace{-2mm} \bibitem{Fu} Fukaya, K. {\it Collapsing of Riemannian manifolds and eigenvalues of Laplace operator.} Invent. Math. {\bf 87} (1987), 517--547. \vspace{-2mm} \bibitem{Ge} Gelfand I.M., {\it Some aspects of functional analysis and algebra.} 1957 Proceed. the Intern. Congr. Mathem., Amsterdam, 1954, {\bf 1}, 253--276. \vspace{-2mm} \bibitem{Gr} Gromov M. with appendices by Katz M., Pansu P. and Semmes S., Metric Structures for Riemannian and Non-Riemanian Spaces, based on `` Structures metriques pour les varietes riemanniennes'' ( LaFontaine J. and Pansu P.eds), English translation by M.Bates, Birkhauser (1999). \vspace{-2mm} \bibitem{HH} Hebey E. and Herzlich M. {\it Harmonic coordinates, harmonic radius and convergence of Riemannian manifolds}, Rend. di Matem., ser.VII, {\bf 17}, Roma (1997), 569-605. \vspace{-2mm} \bibitem{Is} Isakov V. Inverse Problems for Partial Differential Equations, Springer, New York (1998). \vspace{-2mm} \bibitem{KaKu} Kasue A. and Kumura H. {\it Spectral convergence of Riemannian manifolds}, Tohoku Math. J., {\bf 46} (1994), 147-179. \vspace{-2mm} \bibitem{KaKuLa} Katchalov A.P., Kurylev Y. and Lassas M. Inverse Boundary Spectral Problems, Chapman/CRC, Boca Raton (2001). \vspace{-2mm} \bibitem{Ka} Kato T. Perturbation Theory for Linear Operators, Springer, Berlin (1995). \vspace{-2mm} \bibitem{Kat} Katsuda A. BC-method and stability of Gel'fand inverse spectral problem, RIMS Kokyuroku {\bf 1208} (Spectral and Scattering Theory and Related Topics) (2001) 24-35. \vspace{-2mm} \bibitem{Kod} Kodani S. {\it Convergence theory for Riemannian manifolds with boundary}, Compos. Math., {\bf 75} (1990), 171-192. \vspace{-2mm} \bibitem{Ku} Kurylev Y. {\it Multidimensional Gel'fand inverse problem and boundary distance map}, in: Inv. Probl. related with Geom., Ibaraki Univ. (1997), 1-16. \vspace{-2mm} \bibitem{KuSt} Kurylev, Y.V. Starkov, A. {\it Directional moments in the acoustic inverse problem. } Inv. Probl. in Wave Propag. (Minneapolis, MN, 1995), 295--323, IMA Vol. Math. Appl., {\bf 90}. \vspace{-2mm} \bibitem{M} Mandache N. {\it Exponential instability in an inverse problem for Schrodinger equation}, to appear in Inverse Problems \vspace{-2mm} \bibitem{Me} Meyer, D. {\it Minoration de la premiere valeur propre non nulle du probleme de Neumann sur les varietes,} Ann. Inst. Fourier (Grenoble) {\bf 36} (1986), 113--125. \vspace{-2mm} \bibitem{NSU} Nachman, A. Sylvester, J., Uhlmann, G. {\it An $n$-dimensional Borg-Levinson theorem. } Comm. Math. Phys. {\bf 115} (1988), 595--605 \vspace{-2mm} \bibitem{No} Novikov, R.G. {\it A multidimensional inverse spectral problem for the equation $-\Delta\psi +(v(x)-Eu(x))\psi=0$.} (Russian) Funk. Anal. i Prilozhen. {\bf 22} (1988), 11--22. \vspace{-2mm} \bibitem{Pe} Petersen P. Riemannian Geometry, Springer, New York (1998). \vspace{-2mm} \bibitem{Sa} Sakai T. Riemannian Geometry. Trans. Math. Monogr., {\bf 149}, AMS (1995). \vspace{-2mm} \bibitem{Sh} Sharafutdinov, V. A. Integral Geometry of Tensor Fields. Inverse and Ill-posed Problems Series. VSP, Utrecht, 1994. \vspace{-2mm} \bibitem{StU} Stefanov, P., Uhlmann, G. {\it Stability estimates for the hyperbolic Dirichlet to Neumann map in anisotropic media.} J. Funct. Anal. {\bf 154} (1998), 330--358. \vspace{-2mm} \bibitem{Sunada} Sunada T. {\it Riemannian coverings and isospectral manifolds.} Ann. of Math. (2) {\bf 121} (1985), 169--186. \vspace{-2mm} \bibitem{SU} Sylvester J., Uhlmann G. {\it A global uniqueness theorem for an inverse boundary value problem.} Ann. of Math. (2) {\bf 125} (1987), 153--169. \vspace{-2mm} \bibitem{Tak} Takahashi J., {\it Collasping of connected sums and the eigenvalues of the Laplacian,} to appear in J. Geom. Phys. (2001). \vspace{-2mm} \bibitem{Ta} Tataru D. {\it Unique continuation for solutions to PDE's: between Hormander's theorem and Holmgren's theorem}, Comm. Part. Diff. Eq., {\bf 20} (1995), 855-884. \vspace{-2mm} \bibitem{Ta1} Tataru D. Carleman estimates, unique continuation and applications, preprint. \vspace{-2mm} } \end{thebibliography} \end {document}