# Steklov isospectrality of conformal metrics

Benjamin Florentin

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Benjamin Florentin. Steklov isospectrality of conformal metrics. 2025. hal-04912528v2

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# STEKLOV ISOSPECTRALITY OF CONFORMAL METRICS

## BENJAMIN FLORENTIN

ABSTRACT. The Steklov spectrum of a smooth compact Riemannian manifold $(M, g)$ with boundary is the set of eigenvalues counted with multiplicities of its Dirichlet-to-Neumann map. (DN map)

This article is devoted to the Steklov spectral inverse problem of recovering the metric $g$, up to natural gauge invariance, from its Steklov spectrum. Positive results are established in dimension $n \ge 3$ for conformal metrics under the assumption that the geodesic flow on the boundary is Anosov with simple length spectrum. The paper combines wave trace formula techniques with the injectivity of the geodesic X-ray transform for functions on closed Anosov manifolds. It is shown that knowledge of the Steklov spectrum determines the jet at the boundary of the underlying Riemannian metric within its conformal class. In this particular context, this parallels the well-known results of the Calderón problem, where we are given the entire Dirichlet-to-Neumann map instead. As a simple corollary, assuming real-analyticity of the conformal factor, Steklov isospectral metrics must coincide.

Using similar arguments, we are also able to prove under the same assumption of hyperbolicity of the geodesic flow on the boundary, that generically any smooth potential $q$ can be recovered from the Steklov spectrum, in the sense that its jet at the boundary is determined by the spectrum of the DN map for the Schrödinger operator with potential $q$. Consequently, in this case, two analytic Steklov isospectral potentials must be equal.

## CONTENTS

<table><tr><td>I. Introduction</td><td>2</td></tr><tr><td>I.1. Inverse problems for the Steklov spectrum</td><td>2</td></tr><tr><td>I.2. Main results</td><td>5</td></tr><tr><td>I.3. Plan of the paper</td><td>8</td></tr><tr><td>Acknowledgements</td><td>8</td></tr><tr><td>II. Preliminaries</td><td>9</td></tr><tr><td>II.1. X-ray transform and Livšic theorem</td><td>10</td></tr><tr><td>II.2. Spectral invariants for the DN map</td><td>11</td></tr><tr><td>III. Recovery of the normal derivative at the boundary</td><td>14</td></tr><tr><td>IV. Boundary determination of the conformal factor</td><td>20</td></tr><tr><td>V. Further results and remarks</td><td>29</td></tr><tr><td>Appendix A. A few results on the principal order</td><td>34</td></tr><tr><td>Appendix B. A suitable expression of the DN map</td><td>38</td></tr><tr><td>Appendix C. Cohomology of a sphere bundle</td><td>40</td></tr><tr><td>References</td><td>41</td></tr></table>
# I. INTRODUCTION

## I.1. Inverse problems for the Steklov spectrum.
Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 2$ with smooth boundary $\partial M$. Recall that the Steklov eigenvalue problem consists in finding scalars $\sigma$ for which the system

$$ egin{cases} -\Delta_g u = 0 & \text{in } M \\ \partial_\nu u = \sigma u & \text{on } \partial M \end{cases} $$

admits a non-trivial solution in $H^1(M)$; where $\Delta_g$ is the Laplace-Beltrami operator on $(M, g)$ and $\partial_\nu$ denotes the Riemannian outward normal derivative along the boundary.

This famous problem, introduced at the beginning of the 20th century by V.A. Steklov (see [KKK+14] for background), has been particularly studied over the last two decades and is now one of the most active research topics in the field of spectral geometry. We refer the reader to the excellent surveys [GP17] and [CGGS24] for details.

Equivalently, one can consider the eigenvalues of the Dirichlet-to-Neumann map (DN map) $\Lambda_g$ defined by

$$ \Lambda_g : \left. \begin{array}{c} H^{1/2}(\partial M) \\ f \end{array} \right| \begin{array}{c} \longrightarrow \\ \longmapsto \end{array} \begin{array}{c} H^{-1/2}(\partial M) \\ \partial_\nu u|_{\partial M} \end{array}, $$

where $u$ is the unique solution in $H^1(M)$ of the Dirichlet problem

$$ \begin{cases} -\Delta_g u = 0 \\ u|_{\partial M} = f \end{cases}. $$

The DN map is a positive self-adjoint elliptic and classical (in the sense of (1)) pseudodifferential operator of order 1 on the closed manifold $\partial M$. (see [Tay23])
In particular, its spectrum is discrete and is given by a sequence of eigenvalues

$$ 0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \dots \to \infty. $$

The multiset

$$ Sp(\Lambda_g) := (\sigma_k)_{k \ge 0} $$

defines the so-called Steklov spectrum of the Riemannian manifold $(M, g)$.

In this context, we are interested in probably the most famous question in spectral geometry, namely “Can you hear the shape of a drum ?”, originally formulated by Kac in [Kac66]. In other words, does the Steklov spectrum determine the metric up to a natural gauge invariance ? This problem is called the Steklov inverse problem, we refer once again to the surveys [GP17] and [CGGS24].

It is well known that the answer is generally negative; in fact, there are many non-isometric Steklov isospectral (with the same Steklov spectrum) manifolds; the constructions are mainly adaptations of two general methods originally introduced to construct Laplace isospectral manifolds : Sunada’s technique and the torus action technique. We refer to [GHW21] and references therein for details.
In this paper, we will study isospectral sets of conformal metrics, that is, metrics of the form $h = cg$ where $c: M \to \mathbb{R}$ is a smooth positive function in $M$. In that case, the Steklov spectral inverse problem boils down to showing that the map

$$h \mapsto Sp(\Lambda_h)$$

is injective on the space of metrics conformal to $g$ on $M$. Of course the problem is difficult since this map is highly non-linear and it is also possible to find many counterexamples, in particular by adapting Sunada's method cited above, and more specifically [BPY89, Theorem 1]. The aim of this paper is to establish 'positive' results, i.e. to exhibit a certain class of metrics for which it is possible to recover the geometry from the Steklov spectrum. The natural approach is to look for spectral invariants, i.e geometric quantities that can be determined by the spectrum. These are usually provided by considering functions constructed from eigenvalues; the first example is given by the well-known Weyl's asymptotic on the spectral counting function, which shows that the volume of the boundary can be recovered from the Steklov spectrum. (see **Proposition II.4**)

More generally, a complete set of spectral invariants can be defined by considering 'special' functions, in particular, the trace of the heat kernel (for the Steklov spectrum, see for example [PS15]) or the trace of the wave operator, recalled in Section II.2. We refer the reader to the excellent survey [Zel04] for further discussions on this topic. On a closed Riemannian manifold, the spectral invariants arising from the trace of the wave operator associated with any self-adjoint positive elliptic pseudodifferential operator are given by the trace formula of Duistermaat and Guillemin [DG75, Theorem 4.5], providing a careful analysis of its singularities as a distribution in time. In the case of the Laplace-Beltrami operator, this formula was first established independently by Y.Colin de Verdière [CdV73a], [CdV73b] and then by Chazarain [Cha74].

Note that the exploitation of these trace formulas requires important assumptions about the dynamics of the Hamiltonian flow associated with the principal symbol of the operator under consideration, e.g. geodesic flows in the case of the Laplacian and the DN map. For geodesic flows, a particularly appropriate framework is provided by the negative curvature assumption or, more generally, by the Anosov property. (see Section II)

One of the most important examples of application is given by Guillemin and Kazhdan [GK80a], who were able to show the absence of non-trivial Laplace isospectral deformations (i.e one-parameter families ($g_s$) of smooth metrics preserving the Laplacian spectrum) on surfaces of negative curvature, and then in any dimension in [GK80b]. (see also [CS98])

Recently, Paternain, Salo and Uhlmann [PSU14] were able to extend these results to the case of Anosov surfaces. The restriction of this spectral inverse problem to isospectral deformations is actually a linearization and leads more precisely to a linear tomography problem. The latter asks whether it is possible to recover a tensor from its integrals along closed geodesics, which corresponds to an injectivity property of the so-called X-ray transform. (see Section II.1)

Inspired by this approach, this paper also considers isospectral deformations for the Steklov spectrum:
**Definition I.1.** A deformation of g is a smooth one-parameter family of (smooth) Riemannian metrics $(g_s)_{0\le s\le\epsilon}$ on M such that $g_0 = g$.

It is called Steklov isospectral if the spectrum of $\Lambda_{g_s}$ is preserved :

$$ \forall s \in [0, \epsilon] , \quad Sp(\Lambda_{g_s}) = Sp(\Lambda_g) . $$

It is easy to establish, under natural assumptions, that isospectral deformations of conformal metrics are trivial in the sense that each metric automatically coincides at the boundary. (see **Proposition A.1**)

But, on the other hand, the trace formula techniques do not allow us to prove such a result for the non-linear problem. (i.e. considering two conformal metrics)

However, we point out that this can be shown under more restrictive assumptions on the geometry, exploiting the volume of the boundary as a spectral invariant. But this is actually not very satisfactory as very little spectral information is used, and then consequently the geometric assumptions required are relatively strong. We refer the reader to Appendix A.

More generally, one cannot expect to obtain positive results on the general inverse Steklov problem without sufficiently strong assumptions, especially since, by comparison, we start with even less information than in the famous anisotropic Calderón inverse problem, which consists in recovering the metric g from knowledge of the DN map $\Lambda_g$ and is still far from being fully understood despite a vast literature on the subject. The reader may wish to consult, for example, the survey [DKN18] and all references therein. Regarding this problem, a standard step is to show a boundary determination result, namely that the DN map determines, in boundary normal coordinates, all the normal derivatives of the metric tensor g at the boundary at any order $j \ge 0$. (and thus the full Taylor series of g at the boundary in these coordinates) We would like to mention in particular the important papers [LU89], [LU01], [LTU03], [DSFKSU09] and the recent [Cek20], [LLS22].

In this work, we establish similar boundary determination results for the Steklov spectrum. In particular, combining the boundary determination mentioned above, the use of wave trace invariants and the injectivity of the X-ray transform, we show that the Steklov spectrum determines the jet of the metric at any point $p \in \partial M$ in boundary normal coordinates. Note that we are working here in dimension $n \ge 3$, the case of dimension 2 is specific due to the well-known conformal invariance (see **Remark B.2**) ; we will simply observe that the Steklov inverse spectral problem considered in this paper then reduces to showing that two conformal and isospectral metrics coincide at the boundary which, in the non-linear case, is a priori as difficult (and not true in general) as in higher dimensions. This point is actually the most troublesome one in the boundary determination, since wave invariants provide no useful information and then, as mentioned above, we are only able to obtain results under much more restrictive geometric assumptions by using more global spectral invariants, in this case the volume of the boundary. (see Appendix A)

An important idea in this paper is to exploit the subprincipal symbol of the DN map (see the beginning of Section IV and the references therein for a reminder of this notion), which unlike the case of the Laplace-Beltrami operator, is non-zero. This makes it possible to obtain more information from the principal wave invariants given by **Theorem II.6**, under the additional assumption that the length spectrum
of $\partial M$ is simple; recall that a metric is said to have simple length spectrum if all its closed geodesics have different lengths. It is well-known that this assumption is satisfied by generic Riemannian metrics even in the non-Anosov case. (cf. [Abr70], [Kli82, Lemma 4.4.3] or [Ano83])

To the author's knowledge, this is the first example of a spectral inverse problem to exploit this observation on the subprincipal symbol. A similar yet interesting example, involving the recovery of a potential $q$ from the spectrum of the associated DN map $\Lambda_{g,q}$, will be discussed in part V.

It should also be noted that the literature establishing positive results for the Steklov spectral inverse problem is extremely limited, and this paper therefore also serves as a contribution to this lack.

To conclude, note that another version of the wave trace formula in the context of isospectral connections is used in a similar way in the recent [CL22] to show an in-jectivity result for the spectrum map (see [CL22, Theorem 1.1]) solving in particular positively, on closed negatively curved manifolds with simple length spectrum, the inverse spectral problem for the magnetic Laplacean, namely recovering the mag-netic potential from the spectrum. On the Steklov spectral inverse problem, let us also mention the work of [JS14] on compact and simply connected surfaces with boundary, [Gen20] and [DKN21] within a conformal class of certain warped product manifolds or the very recent [CS24] for the magnetic Dirichlet-to-Neumann map on surfaces. Finally, for the Steklov spectrum in the context of conformal metrics, [Jam14] shows that on any compact manifold of dimension $n \ge 3$ with boundary, it is possible to prescribe any finite part of the Steklov spectrum within a given conformal class.

Throughout the article, for any Riemannian manifold $(M, g)$ with boundary, we will note $g^\circ$ the induced metric on the closed manifold $\partial M$.

## I.2. Main results.
This section gathers all the main spectral rigidity results obtained in the paper. Turning first to the non-linear problem in a conformal class, we need to require equality of the metrics at the boundary, information that we are unable to obtain directly via the wave trace invariants and which is actually not verified in general. The following result is essentially established in Section IV and more particularly in **Theorem IV.6**.

**Theorem I.2.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$ with boundary. Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum. Let $c: M \to \mathbb{R}_+^*$ be a smooth function satisfying

$$c|_{\partial M} \equiv 1.$$

If the Steklov spectrum is preserved

$$Sp(\Lambda_{cg}) = Sp(\Lambda_g),$$

then the Taylor series of $g$ and $cg$ at each point $p \in \partial M$ are equal in $g$-boundary normal coordinates and the two operators coincide modulo a smoothing operator :

$$\Lambda_{cg} \equiv \Lambda_g.$$
The proof relies on a recursive procedure using an adapted version of the aforementioned Duistermaat-Guillemin trace formula and the injectivity of the X-ray transform. Initialization, which requires the use of Livšic theory, is therefore dealt with separately in Section III and established more precisely in **Theorem III.5**. As an easy consequence, adding the analyticity of the conformal factor, we deduce at the end of Section IV:

**Corollary I.3.** Let $M$ be a compact connected real-analytic manifold of dimension $n \ge 3$ with boundary and $g$ a smooth Riemannian metric on $M$.

Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum.

Let $c : M \to \mathbb{R}_*^+$ be an analytic function s.t

$$c|_{\partial M} \equiv 1.$$

If the Steklov spectrum is preserved

$$Sp(\Lambda_{cg}) = Sp(\Lambda_g),$$

then

$$c \equiv 1.$$

As already mentioned, note that it is possible to remove this condition of equality of metrics at the boundary by adding some relatively strong geometric assumptions, due to the lack of usable spectral invariants. This point is covered in Appendix A.

For the linear problem, i.e., considering isospectral deformations, the major difference is that it is easy to establish the following result by differentiating with respect to the parameter $s$:

**Theorem I.4.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$ with boundary.

Let $(g_s)_{0 \le s \le \varepsilon}$ be a Steklov isospectral deformation of the metric $g$ s.t there exists a family of smooth functions $(c_s : M \to \mathbb{R}_*^+)$$_{0 \le s \le \varepsilon}$ satisfying for any $s \in [0, \varepsilon]$,

$$g_s = c_s g$$

and $(\partial M, g_s^\circ)$ is Anosov with simple length spectrum.

Then for all $s \in [0, \varepsilon]$,

$$\begin{cases} c_s|_{\partial M} \equiv 1 \\ \partial_{\nu} c_s|_{\partial M} \equiv 0 \end{cases}.$$

This result is a consequence of **Proposition A.1** and **Theorem III.2**.

It should also be mentioned that, for simplicity of writing, we assume that the Anosov property is conserved along the deformation, but it is in fact sufficient for it to be verified only by $g^\circ$. This simply follows from the structural stability of Anosov flows. (see the beginning of Section II)

Similarly, if we already assume that the metrics coincide at the boundary, only the initial metric $g^\circ$ needs to verify the simple length spectrum assumption.

Finally, of course, everything else works the same way at lower order just as in the non-linear problem, so we get:
**Corollary 1.5.** Let $M$ be a smooth compact manifold of dimension $n \ge 3$ with boundary and $g$ a smooth Riemannian metric on $M$.

Let $(g_s)_{0 \le s \le \epsilon}$ be a Steklov isospectral deformation of the metric $g$ s.t there exists a family of smooth functions $(c_s : M \to \mathbb{R}_+^*)_{0 \le s \le \epsilon}$ satisfying for any $s \in [0, \epsilon]$,

$$g_s = c_s g.$$

and $(\partial M, g_s^\circ)$ is Anosov with simple length spectrum.

Then for all $s \in [0, \epsilon]$, the Taylor series of $g$ and $g_s$ at each point $p \in \partial M$ are equal in $g$-boundary normal coordinates and the two operators coincide modulo a smoothing operator :

$$\Lambda_{g_s} \equiv \Lambda_g.$$

Consequently, if $M$ is additionally connected, real-analytic and $c_s$ is analytic for every $s \in [0, \epsilon]$, then

$$c_s \equiv 1.$$

For the sake of completeness, let us also mention that a simple adaptation of the method established in Section IV allows us to obtain an analogue in the case of the Steklov inverse problem with potential, i.e. regarding the DN map $\Lambda_{g,q}$:

**Theorem 1.6.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$ with boundary. Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum. Let $q_1$ and $q_2$ be two smooth potentials on $M$.

If the Steklov spectrum is preserved

$$Sp(\Lambda_{g,q_1}) = Sp(\Lambda_{g,q_2}),$$

then the Taylor series of $q_1$ and $q_2$ at each point $p \in \partial M$ are equal in $g$-boundary normal coordinates and the two operators coincide modulo a smoothing operator :

$$\Lambda_{g,q_1} \equiv \Lambda_{g,q_2}.$$

Of course, we also get the analogue of **Corollary 1.3** by a simple argument of analytic continuation on functions as in the case of the conformal factor at the end of Section IV:

**Corollary 1.7.** Let $M$ be a compact connected real-analytic manifold of dimension $n \ge 3$ with boundary and $g$ a smooth Riemannian metric on $M$. Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum.

Let $q_1$ and $q_2$ be two analytic potentials on $M$.

If the Steklov spectrum is preserved

$$Sp(\Lambda_{g,q_1}) = Sp(\Lambda_{g,q_2}),$$

then

$$q_1 = q_2.$$
This spectral inverse problem with potential is discussed in Section V.

To conclude, as mentioned in the previous section, recall that the geometric assumption of simple length spectrum required to exploit the wave trace invariants (Theorem II.6 and Theorem IV.2) are generic in the space of Anosov metrics which makes all the above results, and in particular the non-linear ones, more consistent. In addition, note that all these results are, to the author's knowledge, the first positive results for the Steklov spectral inverse problem in a general framework.

**Remark I.8.** The following points are worth mentioning here :

* In all these results there is no need to assume boundary connectedness, since we rely on wave trace formula techniques on closed manifolds, and these techniques do not require connectedness of the underlying manifold.

* The Anosov assumption is made only on the boundary; the entire manifold does not need to be Anosov, it can be anything.

* The class of manifolds considered here seems rather special, although natural for the method, so it is important to note that one can easily construct such manifolds and there are plenty of them.

As a simple example, we know from a classic theorem of R.Thom, that any closed manifold of dimension 3 can be realized as the boundary of a 4-manifold, since its Stiefel-Whitney numbers are all zero. So in this case, all we have to do is take any closed Anosov 3-manifold and consider a manifold whose boundary it is.

I.3. **Plan of the paper.** Section II introduces the fundamental tools used throughout the paper, mainly the injectivity of the X-ray transform on functions in II.1 and spectral invariants for the Steklov spectrum in II.2.

Section III deals with the recovery of the normal derivative of the conformal factor at the boundary, exploiting principal wave trace invariants combined with Livšic theory. Concerning the linear problem, **Theorem I.4** is completely established by **Theorem III.2** and **Proposition A.1**. In Section IV, we present the recursive procedure that allows us to show more generally that one can recover, in boundary normal coordinates, all the higher-order normal derivatives of the conformal factor at the boundary. This completes the proof of **Theorem I.2**.

To conclude, in Section V, we discuss other inverse problems on the Steklov spectrum where the techniques used in this paper could be applied and show in particular **Theorem I.6** on the spectral problem for the DN map $\Lambda_{g,q}$.

Appendix A gives some statements that establish equality at the boundary of two isospectral conformal metrics, the only point that cannot be dealt with by trace formula techniques. The last two appendices contain proofs of useful results that are more or less well known, but not necessarily explicitly written in the literature. Appendix B provides an appropriate formula for calculating the subprincipal symbol in **Proposition III.1** and Appendix C recalls an algebraic result on the first cohomology group of a unit fiber bundle, used in the proof of **Theorem III.4**.

**Acknowledgements.** First and foremost, I would like to thank David dos Santos Ferreira for the many discussions that helped me enormously in writing this paper. I am also grateful to Samuel Tapie for enlightening me on certain notions of hyperbolic dynamics. Finally, I would like to thank Mikko Salo for his pertinent comments and advice on the organization of the document.
Throughout this article, we denote by $(N, h)$ a smooth closed Riemannian manifold and $Vol_h(N)$ its Riemannian volume with respect to the metric $h$. In applications, we will systematically take $(N, h) = (\partial M, g^\circ)$.
We will adopt the classical notations

$$Xu := \mathcal{L}_X u = \frac{d}{dt}\Big|_{t=0} \phi_t^* u$$

for the Lie derivative of the function $u$ along the vector field $X$, where of course $\phi_t^* u$ denotes the pullback of $u$ by the flow $\phi_t$ generated by $X$.

Finally, we will denote by $\Psi^k(N)$ the set of pseudodifferential operators of order $k$ on the closed manifold $N$.

## II. PRELIMINARIES

Recall that a flow $(\phi_t)$ generated by a smooth vector field $X$ on a closed manifold $M$ is Anosov if there is a continuous splitting

$$TM = \mathbb{R}X \oplus E_s \oplus E_u,$$

of the tangent bundle $TM$ into three flow-invariant subbundles, respectively, tangential to the flow direction, exponentially contracting and expanding. Note that two Anosov flows that are $C^1$-close enough are topologically conjugate: This is known as the *structural stability*. In particular, the Anosov property is robust to $C^1$-perturbations, it is a $C^1$-open condition. We refer, for example, to [FH19, Section 5.1] and to the fundamental [DILMM86] for more details.

A particularly important special case, which will be considered exclusively in this paper, is that where $M$ is the unit tangent bundle (resp. the unit cotangent bundle)

$$SN := \{(x, v) \in TN ; h_x(v, v) = 1\}$$

$$(\text{resp. } S^*N := \{(x, \xi) \in T^*N ; h_x^{-1}(\xi, \xi) = 1\})$$

of the closed Riemannian manifold $(N, h)$ and $(\phi_t)$ the associated geodesic flow. We say that $(N, h)$, or simply $h$, is Anosov if its geodesic flow is Anosov.

As already mentioned, any closed Riemannian manifold with negative curvature is Anosov. (see [FH19, Theorem 5.2.4], [KH97, Section 17.6] or [Kni02] for a proof)
Such flows are transitive<fnref n="1" />, their periodic orbits are dense, and knowledge of a function along these periodic integral curves allows us to reconstruct it up to natural obstructions. This is the purpose of the standard Livšic theory, and in particular of **Theorem II.1** below.

More recent elements of the so-called non-Abelian Livšic theory are used in the proof of **Proposition III.3**. Finally, as mentioned in the Introduction, the Anosov property provides a general framework particularly suited to obtaining spectral rigidity results by using trace formulas. (see Section II.2)

<footnote n="1">¹They admit a dense orbit.</footnote>
II.1. **X-ray transform and Livšic theorem.** We begin by recalling a famous theorem for Anosov flows originally due to Livšic [Liv71, Theorem 1] in Hölder regularity $C^\alpha$, $\alpha \in (0,1)$. We also refer to [GK80a, Appendix] and to [KH97, Theorem 19.2.4].

Here we need the smooth regularity version proved in [DILMM86, Theorem 2.1] :

**Theorem II.1** (Livšic [Liv71], De La Llave-Marco-Moriyon [DILMM86]). Let $\mathcal{M}$ be a closed Riemannian manifold, $f: \mathcal{M} \to \mathbb{R}$ a smooth function and $(\phi_s)$ a smooth transitive Anosov flow on $\mathcal{M}$ generated by the vector field $X$.

The following propositions are equivalent :

(1) For every periodic orbit $\gamma$ of the flow,

$$ \int_{\gamma} f = 0. $$

(2) There is $u \in C^\infty(\mathcal{M}, \mathbb{R})$ such that

$$ f = Xu. $$

This important result characterizes the kernel of a linear integration operator along periodic orbits, called X-ray transform. In the following, we recall the definition of this operator for geodesic flows.

Let us denote by $\mathcal{C}$ the set of free homotopy classes of $N$. The latter is in 1-1 correspondence with the set of conjugacy classes of the fundamental group $\pi_1(N)$ of $N$ and consequently $\mathcal{C}$ is countable. In addition, if $N$ is Anosov, it is well-known that there exists a unique closed geodesic for $h$ in each free homotopy class. (see [Kli74], [Kni02] or, more generally, [CIPP00, Theorem 29])

Thus in this case, the set $\mathcal{G}$ of periodic orbits of the geodesic flow $(\phi_t)$ can be identified with $\mathcal{C}$. The X-ray transform on $SN$ is the operator defined by :

$$ \mathcal{I}^h: \begin{array}{c|c} C^0(SN) & \ell^\infty(\mathcal{C}) \\ f & (\mathcal{I}^h f([\gamma]))_{[\gamma] \in \mathcal{C}} \end{array} $$

and

$$ \mathcal{I}^h f([\gamma]) := \frac{1}{\ell_h(\gamma_h)} \int_0^{\ell_h(\gamma_h)} f(\phi_s(x, v)) ds, $$

where $\gamma_h : [0, \ell_h(\gamma_h)] \to N$ denotes any parametrization by arclength of the unique closed geodesic in the free homotopy class $[\gamma] \in \mathcal{C}$, $\ell_h$ the length functional for $h$ and $(x, v) \in SN$ is any point on the closed curve $t \mapsto (\gamma_h(t), \dot{\gamma}_h(t))$.

**Remark II.2.** The map

$$ \mathcal{L}_h: \begin{array}{c|c} \mathcal{C} & (0, \infty) \\ [\gamma] & \ell_h(\gamma_h) \end{array} $$

is the well-known marked length spectrum of $(N, h)$.
We usually consider this operator acting on symmetric tensor fields on N :
Let $S^m T^* N$ be the bundle of symmetric tensors of order $m \in \mathbb{N}$ on $N$, we refer the
reader to [Sha94, Section 3] for definitions.

Denoting by $C^\infty(N, S^m T^*N)$ the set of smooth sections of $S^m T^*N \to N$, we can
see any symmetric tensor field $f$ of order $m$ on $N$ as a function on $SN$ by considering
the lift $\pi_m^* : C^\infty(N, S^m T^*N) \to C^\infty(SN)$ defined by

$$
\pi_m^* f : (x, v) \mapsto f_x(v, \dots, v),
$$

with the convention that $S^0 T^* N := N \times \mathbb{R}$ is the trivial bundle, so that $\pi_0^*$ is simply
the pullback by the projection $\pi : SN \to N$.

Then with the same notations as above, the X-ray transform on symmetric tensors
of order *m* is the operator

$$
\mathcal{I}_m^h := \mathcal{I}^h \circ \pi_m^* \left| \begin{array}{l} C^\infty(N, S^m T^* N) \\ f \end{array} \right. \longrightarrow \begin{array}{l} \ell^\infty(\mathcal{C}) \\ (\mathcal{I}^h \pi_m^* f([\gamma]))_{[\gamma] \in \mathcal{C}} \end{array},
$$

where

$$
\mathcal{I}^h \pi_m^* f([\gamma]) := \frac{1}{\ell_h(\gamma_h)} \int_0^{\ell_h(\gamma_h)} f_{\gamma_h(t)}(\dot{\gamma}_h(t), \dots, \dot{\gamma}_h(t)) \, dt.
$$

As mentioned in the Introduction, the problem considered here will mainly involve
$\mathcal{I}_0^h$, the X-ray transform on functions which is injective in the case of Anosov man-
ifolds :

**Theorem II.3 ([DS03]).** Suppose $h$ is Anosov. Then for any $f \in C^\infty(N)$,

$$
\mathcal{I}_0^h f = 0 \Rightarrow f = 0.
$$

The conclusion of **Theorem II.3** is a simple special case of a more general notion of injectivity on symmetric tensors of order *m*, called s-injectivity. The literature on this subject is quite extensive; we refer, for example, to [DS03], [CS98] and the more recent [PSU14] and [Gui17]. Its study in the context of general Anosov manifolds is still largely open, at least in dimension *n* ≥ 3, even if major progress has recently been made by M. Cekić and T. Lefeuvre in [CL25].

II.2. Spectral invariants for the DN map. As in the Introduction, let us consider $(M, g)$ a smooth compact Riemannian manifold of dimension $n \ge 2$ with boundary $(N, h) := (\partial M, g^\circ)$. The aim of this section is to introduce the spectral invariants used throughout this paper. The first and probably the best known, which will be used in Appendix A, follows from the asymptotic of the spectral counting function of the DN map ; namely applying standard results of [Hör85b, Chapter 29] we have the following distribution of Steklov eigenvalues :

**Proposition II.4 (Weyl's law).**

$$
\#(\sigma_k \le \sigma) = \frac{\mathrm{Vol}(\mathbb{B}^{n-1})}{(2\pi)^{n-1}} \mathrm{Vol}_{g^\circ}(\partial M)\sigma^{n-1} + \mathcal{O}(\sigma^{n-2}),
$$

where $\mathbb{B}^{n-1}$ denotes the unit ball in $\mathbb{R}^{n-1}$.
Consequently, since two isospectral manifolds have the same spectral counting function :

**Corollary II.5.** Given any smooth compact Riemannian manifold of dimension $n \ge 2$ with boundary, the volume of its boundary is a spectral invariant for the Steklov spectrum.

We now introduce the invariants on which this paper is based, provided by the fundamental wave trace formula of Duistermaat-Guillemin [DG75, Theorem 4.5]. The latter, applied to the DN map, is related to the trace of the wave group

$$U(t) := e^{-it\Lambda_g}.$$

It shows in particular that the singularities of

$$Tr(U(t)) := \sum_{k=0}^{+\infty} e^{-i\sigma_k t}$$

as a distribution in time, lie at the periods of the periodic geodesics of the boundary, in other words :

$$\text{Suppsing Tr}(U(t)) \subseteq \mathcal{T},$$

where *Suppsing* denotes the singular support of a distribution (the complement of the set where it is smooth) and $\mathcal{T}$ is the set of periods. Any element of the set $|\mathcal{T}|$ will, of course, be referred to as the length of a periodic orbit.

In order to state this wave trace formula more precisely, we need to recall a few concepts :

* Given a closed orbit $\gamma$ of period $T_\gamma$, its primitive period $T_\gamma^\#$ is the least non-zero period such that $T_\gamma$ is a multiple of $T_\gamma^\#$.

* The linearized Poincaré return map $\mathcal{P}_\gamma$ of the closed orbit $\gamma$ of period $T_\gamma$ starting from $w = (x, \xi) \in S^*N$, is the linear application induced by $d\phi_{T_\gamma}(w)$ in $T_wS^*N/\mathbb{R}X(w)$ where $X$ denotes, of course, the generator of the geodesic flow of $N$. A closed geodesic $\gamma$ is called *non-degenerate* if

$$\det (Id - \mathcal{P}_\gamma) \neq 0.$$

In particular, it is easy to see that this property holds if the underlying manifold $N$ is Anosov, since in this case $\mathcal{P}_\gamma$ may be identified with $d\phi_{T_\gamma}(w)|_{E_s(w) \oplus E_u(w)}$.

* The Morse index $m_\gamma$ of a closed geodesic $\gamma$ is the maximal dimension of a vector subspace on which the second-order differential of the energy functional at $\gamma$ is negative definite. We refer to [Dui76] for details.

For further information on these concepts, we refer more generally to the online book [GM] currently in progress.

**Theorem II.6** (Duistermaat-Guillemin, [DG75]). Suppose that $(\partial M, g^\circ)$ only admits a finite number of closed geodesics $\gamma_1, \dots, \gamma_r$ of period $T$ and that each of these geodesics is nondegenerate.

Then

$$S(t) := \mathrm{Tr}(U(t))$$
is a tempered distribution on $\mathbb{R}$ admitting an isolated singularity in $t = T$ given by a Lagrangian distribution $e_T$ with

$$ \text{Suppsing}(e_T) = \{T\} $$

and complete asymptotic expansion

$$ e_T(t) \sim_{t \to T} c_{T,-1}(t-T+i0)^{-1} + \sum_{j=0}^{+\infty} c_{T,j}(t-T+i0)^j \log(t-T+i0). $$

The leading coefficient is expressed as

$$ c_{T,-1} = \sum_{j=1}^{r} \frac{|T_j^\sharp| e^{im_j \frac{\pi}{2}} e^{-iT \bar{\gamma}_j}}{|\det (Id - \mathcal{P}_j)|^{1/2}}, $$

where

$$ \bar{\gamma}_j := \frac{1}{T} \int_{\gamma_j} \text{sub}(\Lambda_g), $$

and $\mathcal{P}_j, T_j^\sharp, m_j$ are respectively the linearized Poincaré map, the primitive period and the Morse index of $\gamma_j$.

**Remark II.7.** Note that the Morse index is zero for any non-degenerate closed orbit with no conjugate point, which is the case in particular for all closed geodesics of an Anosov manifold. We refer again to [Dui76] for details and, for a proof of this result, the reader can consult the aforementioned [GM].

To the author's knowledge, the latter is the only reference where a proof of this particular point is explicitly made in the Anosov case.

The leading term $c_{T,-1}$ in the previous theorem is called the principal wave trace invariant at $t = T$, and is the term controlling the singularity. In particular, if this term is canceled, then the corresponding period $T$ is no longer a singularity.

Consequently, to take full advantage of this theorem, we systematically consider the general case where $(\partial M, g^\circ)$ is Anosov with the generic assumption that the length spectrum is simple. (i.e. $r = 1$ in the theorem)

It is then possible to formulate a result analogous to [GK80a, Theorem 3] in our context, highlighting two spectral invariants :

**Corollary II.8.** In the class of Riemannian metrics $g$ such that $g^\circ$ is Anosov with simple length spectrum, the Steklov spectrum determines :

(a) the lengths of the closed geodesics on the boundary.

(b) for any period $T \in \mathcal{T}$, the quantity

$$ c_{T,-1} = \frac{|T^\sharp| \exp \left( -iT \int_{\gamma_{g^\circ}} \text{sub}(\Lambda_g) \right)}{|\det (Id - \mathcal{P}_{\gamma_{g^\circ}})|^{1/2}}. $$
**Remark II.9.** As already mentioned, in the standard case of the Laplacean on a closed manifold, the subprincipal symbol is zero and consequently the Anosov property (or the negative curvature in the case of [GK80a]) is sufficient to ensure, in this context, the point (a) of the previous corollary. Indeed in this case, given the expression of the principal wave trace invariant, no cancellation takes place in the wave trace formula i.e. every period contributes to a singularity.

In our case however, the subprincipal symbol is non-zero, which forces us to ensure that there are no such cancellations by adding the simple length spectrum assumption. On the other hand, it also provides additional information, namely point (b).

To conclude this section, note that it is possible to show in a similar way to the spectral problem on the Laplacean (see [GK80a] or [CS98]), that infinitesimal spectral rigidity is actually related to infinitesimal rigidity of the marked length spectrum in the following sense :

If $(g_s)_{0 \le s \le \epsilon}$ is an isospectral deformation (see **Definition I.1**) s.t $g_s^\circ$ is an Anosov metric for all $s$, any closed geodesic for $g^\circ$ deforms smoothly as a closed geodesic for $g_s^\circ$ and its length is a smooth function in $s$. (cf. [DILMM86])

It is then straightforward to see, using the wave trace formula<fnref n="2" /> and more precisely point (a) of **Corollary II.8** above, that the marked length spectrum (see **Remark II.2**) is preserved :

$$ \forall s \in [0, \epsilon] , \quad \mathcal{L}_{g_s^\circ} = \mathcal{L}_{g^\circ} . $$

Consequently, differentiating this equality with respect to $s$, we obtain that the variation

$$ \dot{g}_s^\circ = \frac{d}{dt} \Big|_{t=s} g_t^\circ $$

has to be in the kernel of the X-ray transform $\mathcal{I}_2^{g_s^\circ}$, in other words :

**Proposition II.10.**

$$ \forall s \in [0, \epsilon] , \quad \mathcal{I}_2^{g_s^\circ} \dot{g}_s^\circ = 0 . $$

This interesting point in itself will be used to establish **Proposition A.1**.

### III. RECOVERY OF THE NORMAL DERIVATIVE AT THE BOUNDARY

In this section we show that the information provided at subprincipal order concerns the normal derivative of the conformal factor at the boundary, and we explain how it is possible to recover it from the Steklov spectrum. More precisely, we show here that if $g$ and $cg$ coincide at the boundary and $g^\circ$ is Anosov with simple length spectrum, then

$$ \partial_\nu c|_{\partial M} = 0 . $$

As mentioned in the Introduction, this inverse spectral problem is a priori non-linear and we are actually unable to linearize it using the recursive procedure presented in Section IV, which allows us to deal with the lower-order levels.

In this section, we first formulate a result in the context of isospectral deformations

<footnote n="2">²Assuming the metrics have simple length spectrum.</footnote>
(linear problem) as introduced in **Definition I.1**, the deformation parameter *s* allowing the problem to be linearized by differentiation. (see **Theorem III.2**)
We then prove the result for the general problem, bypassing the absence of a dif-
ferentiation parameter through the non-Abelian Livšic theory. (we refer the reader
to **Proposition III.3** below)

From now on, we therefore assume that $c|_{\partial M} \equiv 1$ and begin by showing that the exploitable quantity that appears when calculating the subprincipal symbol of the DN map $\Lambda_{cg} \in \Psi^1(\partial M)$ is, as mentioned above, the normal derivative of the conformal factor at the boundary :

**Proposition III.1.** Let $c : M \to \mathbb{R}_*^+$ be a smooth function satisfying

$c|_{\partial M} \equiv 1 .$

Then we have the following equality on $T^*\partial M$ :

$$
\mathrm{sub}(\Lambda_{cg}) - \mathrm{sub}(\Lambda_g) = \alpha_n(\partial_\nu c)|_{\partial M}
$$

*Proof.* If we note in any local coordinates

$$
\sigma_{\Lambda_g}^{\text{full}} \sim \sum_{j=-1}^{+\infty} p_{-j} ,
$$

in the sense of (1), then by the result of **Corollary B.4** and the expression of the
subprincipal symbol in coordinates (see the beginning of Section IV), we obtain the
following identity :

$$
\begin{align*}
\mathrm{sub}(\Lambda_{cg}) &= p_0 + \alpha_n(\partial_\nu c)|_{\partial M} - \frac{1}{2i} \sum_{j=1}^{n} \frac{\partial^2 \sigma_{\Lambda_g}}{\partial x_j \partial \xi_j} \\
&= \mathrm{sub}(\Lambda_g) + \alpha_n(\partial_\nu c)|_{\partial M}
\end{align*}
$$

in any coordinate chart $(x, \xi)$ on $T^*\partial M$. $\square$

Combining this with [DG75, Theorem 4.5] (see **Theorem II.6**), we are able to
prove the second part of the infinitesimal spectral rigidity result **Theorem I.4** :

**Theorem III.2.** Let $(M, g)$ be a smooth compact Riemannian manifold with boundary such that $(\partial M, g^\circ)$ is Anosov with simple length spectrum.

Let $(g_s)_{0 \le s \le \epsilon}$ be a Steklov isospectral deformation of the metric $g$ s.t there exists a family of smooth functions $(c_s : M \mapsto \mathbb{R}_*^+)$$_{0 \le s \le \epsilon}$ satisfying for any $s \in [0, \epsilon]$,

$g_s = c_s g$

and

$c_s|_{\partial M} \equiv 1 .$

Then for all $s \in [0, \epsilon]$,

$\partial_{\nu} c_s |_{\partial M} \equiv 0 .$
**Proof.** For all $s \in [0, \varepsilon]$, $g_s^\circ = g^\circ$, so the closed geodesics for $g_s^\circ$ are exactly the closed geodesics for $g^\circ$ and since $g_s^\circ$ and $g^\circ$ are Anosov with simple length spectrum, then by isospectrality, the principal wave trace invariant given by **Theorem II.6** remains constant :

$$ \frac{|T_{\gamma}^{\sharp}|e^{-i \int_{\gamma} \mathrm{sub}(\Lambda_{g_s})}}{|\det(\mathrm{Id} - \mathcal{P}_{\gamma})|^{1/2}} = \frac{|T_{\gamma}^{\sharp}|e^{-i \int_{\gamma} \mathrm{sub}(\Lambda_g)}}{|\det(\mathrm{Id} - \mathcal{P}_{\gamma})|^{1/2}}, $$

for any $\gamma := \gamma_0$ closed orbit of the (co)geodesic flow for $g^\circ$ with period $T_\gamma$. (recall that $T_\gamma^\circ$ is its primitive period)

Thus for all $s \in [0, \varepsilon]$ and any $\gamma$ closed orbit,

$$ \exp \left( -i \int_{\gamma} [\mathrm{sub}(\Lambda_{g_s}) - \mathrm{sub}(\Lambda_g)] \right) = 1 $$

and then by **Proposition III.1** we obtain

$$ \exp \left( -i \alpha_n \int_{\gamma} (\partial_{\nu} c_s) |_{\partial M} \right) = 1. $$

Differentiating with respect to $s$, we finally get for any $[\gamma] \in \mathcal{C}$,

$$ \mathcal{I}_0^{g^\circ} \partial_\nu \dot{c}_s ([\gamma]) = \int_{\gamma} (\partial_\nu \dot{c}_s) |_{\partial M} = 0. $$

From there we can conclude that for all $s \in [0, \varepsilon]$,

$$ (\partial_\nu \dot{c}_s) |_{\partial M} = 0 $$

in the same way as in **Proposition A.1** by the injectivity of the X-ray transform on functions for Anosov manifolds. (**Theorem II.3**)

Hence for all $s \in [0, \varepsilon]$, $\partial_\nu c_s = \partial_\nu c_0 = 0$ on $\partial M$. $\square$

We now consider the non-linear problem. In this case, the information given by the principal wave invariant on the X-ray transform is more delicate to exploit.

First, we use the Livšic theorem for cocycles (see [Pat09, Theorem 2.2] based on [Liv74] and [NT98]) to make this information explicit, more precisely we recall a result analogous to the standard Livšic theorem **Theorem II.1** :

**Proposition III.3.** Let $\mathcal{M}$ be a closed Riemannian manifold, $f: \mathcal{M} \to \mathbb{R}$ be a smooth function, and $(\phi_s)$ a smooth transitive Anosov flow on $\mathcal{M}$ generated by the vector field $X$.

The following propositions are equivalent :

(1) For every periodic orbit $\gamma$ of the flow,

$$ \int_{\gamma} f \in 2\pi\mathbb{Z}. $$
(2) There is $u \in C^\infty(M, \mathbb{S}^1)$ such that

$$f = -i \frac{Xu}{u}.$$

*Proof.* (2) ⇒ (1) follows from an easy calculation.

(1) ⇒ (2) : Assume that for every periodic orbit $\gamma$ of the flow,

$$\int_{\gamma} f \in 2\pi\mathbb{Z}.$$

Consider the smooth cocycle (see [Pat09, Definition 2.1]) $C : \mathcal{M} \times \mathbb{R} \to \mathbb{S}^1$ over $(\phi_s)$ defined by

$$C(x,t) := \exp \left( i \int_0^t f(\phi_s x) ds \right).$$

The above assumption ensures that $C$ satisfies

$$C(x, T) = 1$$

whenever $\phi_T x = x$.

Consequently, by the smooth Livšic theorem for cocycles [Pat09, Theorem 2.2], there exists $u \in C^\infty(M, \mathbb{S}^1)$ such that

$$C(x, t) = u(\phi_t x)u(x)^{-1},$$

for all $(x, t) \in \mathcal{M} \times \mathbb{R}$. Differentiating with respect to $t$ and evaluating at $t = 0$, we get:

$$f = -i \frac{Xu}{u}.$$

□

Of course, it is also possible to prove this result in a more standard way by following the same initial proof as for the Livšic theorem (Theorem II.1), i.e. by first showing the statement in the Hölder $C^\alpha$ regularity and then extending it to smooth regularity. The latter can also be shown using the more recent microlocal approach of [Gui17].

Finally, note that our problem is similar to what appears when considering two transparent connections (see [Pat09, Section 3]) on a Hermitian line bundle over an Anosov manifold as in the proof of [Pat09, Theorem 3.2]. The latter shows that two transparent connections are gauge-equivalent. It is based on the use of a standard algebraic tool called Gysin sequences (see [BT82, Chapter 14, p.177-179]) and on the solenoidal injectivity of $\mathcal{I}_1$ on Anosov manifolds. (see [CS98, Theorem 1.3]) We reason in a similar way to show the next result below, and will therefore use some basics on the so-called vertical Fourier analysis, which the reader can find, for example, in the excellent [PSU23, Section 6.6] or [PSU15, Section 3].
**Theorem III.4.** Let $(N, h)$ be a closed Anosov manifold, $X$ the generator of its geodesic flow and $f$ a smooth real valued function on $N$.

If $u \in C^\infty(SN)$ is a non trivial solution to the following transport equation with potential $f$

$$Xu + ifu = 0,$$

then

$$f \equiv 0.$$

*Proof.* We introduce the closed 1-form

$$\varphi := -\frac{du}{iu} \in C^\infty(SN, T^*SN).$$

By **Lemma C.1**, $\pi^*: H_{\mathrm{DR}}^1(N) \to H_{\mathrm{DR}}^1(SN)$ is an isomorphism since $N$ cannot be the 2-torus. (see **Remark C.2**)

As a consequence, there exists a closed 1-form $\omega$ in $N$ and a smooth function $\psi: SN \to \mathbb{R}$ such that

$$\varphi = \pi^*\omega + d\psi.$$

Applying the vector field $X$ and using that

$$Xu + ifu = 0,$$

we get

$$\pi_0^*f = \pi_1^*\omega + X\psi,$$

then applying Pestov identity on an Anosov manifold (see, for example, [PSU15, Proposition 2.2] and [PSU15, Lemma 4.2]) and using that $\nabla^v \pi_0^*f = 0$, we obtain

$$\|\nabla^v \pi_1^* \omega\|_{L^2(SN)}^2 = \underbrace{\|X\nabla^v \psi\|_{L^2(SN)}^2 - \langle R\nabla^v \psi, \nabla^v \psi \rangle_{L^2(SN)}}_{\ge 0} + (n-1)\|X\psi\|_{L^2(SN)}^2.$$

Moreover, on the one hand,

$$\|\nabla^v \pi_1^* \omega\|_{L^2(SN)}^2 = (n-1)\|\pi_1^* \omega\|_{L^2(SN)}^2$$

and on the other hand by orthogonality of the spherical harmonics expansion,

$$\|X\psi\|_{L^2(SN)}^2 = \|\pi_0^*f\|_{L^2(SN)}^2 + \|\pi_1^*\omega\|_{L^2(SN)}^2.$$

We refer again to [PSU23, Section 6.6] or [PSU15, Section 3] for these specific points.

Then we easily deduce that
$$||\pi_0^* f||_{L^2(SN)} = 0$$

and consequently

$$f \equiv 0.$$

□

We can now show the analog of **Theorem III.2** in the non-linear case :

**Theorem III.5.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$ with smooth boundary. Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum.

Let $c : M \to \mathbb{R}_+^*$ be a smooth function satisfying

$$c|_{\partial M} \equiv 1.$$

If

$$Sp(\Lambda_{cg}) = Sp(\Lambda_g),$$

then

$$\partial_\nu c|_{\partial M} \equiv 0.$$

*Proof.* Since $g$ and $cg$ are two isospectral metrics that coincide at the boundary and $g^\circ$ is Anosov with simple length spectrum, **Theorem II.6** and **Proposition III.1** implies that for any closed geodesic $\gamma$ for $g^\circ$,

$$\exp \left( -i \alpha_n \int_{\gamma} (\partial_{\nu} c) |_{\partial M} \right) = 1$$

in the same way as in the proof of **Theorem III.2**.

Equivalently, if we put $f := \alpha_n(\partial_\nu c)|_{\partial M}$, then for any closed geodesic $\gamma$,

$$\int_{\gamma} f \in 2\pi\mathbb{Z}.$$

Combining **Proposition III.3** and **Theorem III.4**, we obtain

$$f \equiv 0.$$

□
IV. BOUNDARY DETERMINATION OF THE CONFORMAL FACTOR

This section establishes the important results of boundary determination, namely that the spectrum determines the jet of the conformal factor at the boundary. When g and cg are Steklov isospectral, we showed at the end of Section III, under the assumption that Λ$_{cg}$ and Λ$_{g}$ coincide at the principal order, that the subprincipal part of Λ$_{cg}$ − Λ$_{g}$ is zero, which has given the nullity of the normal derivative of c at the boundary.

Here we show more generally that the normal derivative at the boundary at any order $j \ge 2$ is determined by the homogeneous term of order $-j + 1$ in the symbol of $\Lambda_{cg} - \Lambda_g$ and that this term is zero under the assumption of isospectrality of the metrics. This is established recursively on the order, thanks to an adapted version of the Duistermaat-Guillemin trace formula [DG75, Theorem 4.5] generalizing Guillemin's [Gui78, Theorem 4], in combination with the injectivity of the X-ray transform **Theorem II.3**.

In the following, we introduce the necessary formalism and refer the reader to the fundamental works [DH72] and [Hör85b] for details on the theory of Fourier integral operators. (FIO)

First, recall that a pseudodifferential operator $P \in \Psi^m(N)$ acting on half-densities is said to be *classical*, we will note $P \in \Psi_{cl}^m(N)$, if its full symbol $\sigma_P^{full}$ considered in any coordinate chart is *polyhomogeneous* i.e admits an asymptotic expansion

$$ (1) \qquad \sigma_P^{\text{full}}(x, \xi) \sim \sum_{j=0}^{+\infty} p_{m-j}(x, \xi), $$

with $p_{m-j}$ homogeneous of degree $m-j$ in $\xi$. (see for example [Hör85a, Sect.18.1] for these notions)

We will denote $\sigma_P$ the principal symbol of $P$ and $\text{sub}(P)$ the subprincipal symbol respectively given in any local coordinates by $p_m$ and

$$ p_{m-1} - \frac{1}{2i} \sum_{j=1}^{n} \frac{\partial^2 p_m}{\partial x_j \partial \xi_j}. $$

It is well-known that these quantities are invariant to coordinate changes, we refer once again to [Hör85a] and [Hör85b] for details.

In addition, we will note $H_{\sigma_P}$ the symplectic gradient of $\sigma_P$ for the canonical symplectic structure on $T^*N$ and $(\Phi_t^P)$ the Hamiltonian flow generated by $H_{\sigma_P}$.

Integral curves for the vector field $H_{\sigma_P}$ are called *bicharacteristic curves* or *bicharacteristics*. Note that since $\sigma_P$ is constant along the orbits of $(\Phi_t^P)$, the compact submanifold

$$ S^*N := \{(x, \xi) \in T^*N ; \sigma_P(x, \xi) = 1\} \subset T^*N $$

is invariant by this flow. This is obviously a general fact that Hamiltonian flows preserve the energy level hypersurfaces.

Thus, we will always consider $(\Phi_t^P)$ as a flow acting on $S^*N$ and consequently, that the bicharacteristics lie on this hypersurface.
Now as a first step we propose to generalize the approach of Guillemin, namely [Gui78, Theorem 3].

For $k \in \mathbb{N}$, $A \in \Psi_{cl}^1(N)$ and $B \in \Psi_{cl}^{-k}(N)$ self-adjoint operators, we note $U_0(t) := e^{itA}$, $U(t) := e^{it(A+B)}$ the unitary groups generated by $A$ and $A+B$ and the so-called return operator

$$R(t) := U(t)U_0(-t), \quad t \in \mathbb{R}.$$

For each $t \in \mathbb{R}$, [DG75, Theorem 1.1] shows that the Schwartz kernels of $e^{itA}$ and $e^{it(A+B)}$, viewed as distributions on $\mathbb{R} \times N \times N$, are Lagrangian distributions of class $I^{-\frac{1}{4}}(\mathbb{R} \times N \times N, C)$ defined by the same canonical relation

$$C := \{(t, \tau), (x, \xi), (y, \eta) \mid (x, \xi), (y, \eta) \in T^*N \setminus \{0\}, \\ (t, \tau) \in T^*\mathbb{R} \setminus \{0\}, \tau + \sigma_A(x, \xi) = 0, (x, \xi) = \Phi_t^A(y, \eta)\}.$$

This implies in particular that $U(t)$ and $U_0(t)$ are Fourier integral operator of order 0 defined by the canonical transformation $\Phi_t^A$.

Then by composition, $R(t)$ defines a Fourier integral operator with the identity as canonical transformation and with polyhomogeneous symbol of order 0, in other words :

$$\forall t \in \mathbb{R}, \quad R(t) \in \Psi_{cl}^0(N).$$

The following result is a generalization of [Gui78, Theorem 3] :

**Theorem IV.1.** Let $k \in \mathbb{N}^*$, $A \in \Psi_{cl}^1(N)$ and $B \in \Psi_{cl}^{-k}(N)$ self-adjoint operators. Then

$$W(t) := R(t) - \text{Id}$$

is a pseudodifferential operator of order $-k$ on $N$ and its principal symbol is given by :

$$\sigma_{W(t)}(x, \xi) = i \int_{\gamma^t(x, \xi)} \sigma_B ,$$

where $\gamma^t(x, \xi) : [0, t] \ni s \mapsto \Phi_s^A(x, \xi)$ is the bicharacteristic arc of length $t$ passing through $(x, \xi)$.

*Proof.* Differentiating in $t$ the expression of $W(t)$, we get

$$\begin{aligned}
-i\dot{W}(t) &= (A+B)e^{it(A+B)}e^{-itA} - e^{it(A+B)}Ae^{-itA} \\
&= (A+B)R(t) - R(t)A \\
&= (A+B)(W(t)+\text{Id}) - (W(t)+\text{Id})A \\
&= AW(t) - W(t)A + B + BW(t),
\end{aligned}$$

where the second equality comes from the fact that $A$ and $e^{-itA}$ commute. In other words,

$$(2) \qquad \left\{ \begin{array}{l} -i\dot{W}(t) = [A, W(t)] + B + BW(t) \\ W(0) = 0 \end{array} \right.$$
Then let us note $w(t)$ the full symbol of $W(t) \in \Psi_{cl}^0(N)$ in any local coordinates and

$$ (3) \qquad w(t) \sim \sum_{j=0}^{+\infty} w_{-j}(t). $$

If $b$ denotes the full symbol of $B \in \Psi_{cl}^{-k}(N)$ in the same local coordinates, we also note

$$ (4) \qquad b \sim \sum_{j=0}^{+\infty} b_{-k-j}. $$

Inserting (3) and (4) into (2) and collecting homogeneous terms with same order $-j$, one can access the leading term in the symbol of $W(t)$:

* For $j \in \{0, ..., k-1\}$, we get

$$ -i\dot{w}_{-j}(t) = i^{-1}H_{\sigma_A}w_{-j}(t), $$

so that:

$$ \left\{ \begin{array}{l} \dot{w}_{-j}(t) = H_{\sigma_A}w_{-j}(t) \\ w_{-j}(0) = 0 \end{array} \right. $$

Then we can write

$$ w_{-j}(t) = \exp(tH_{\sigma_A}) w_{-j}(0) = 0. $$

* For $j=k$, we obtain:

$$ -i\dot{w}_{-k}(t) = i^{-1}H_{\sigma_A}w_{-k}(t) + b_{-k} + b_{-k}w_0(t), $$

and since $w_0(t) \equiv 0$,

$$ \left\{ \begin{array}{l} \dot{w}_{-k}(t) = H_{\sigma_A}w_{-k}(t) + ib_{-k} \\ w_{-k}(0) = 0 \end{array} \right. $$

Then we can apply Duhamel's principle to get

$$ \begin{align*} w_{-k}(t) &= \exp(tH_{\sigma_A}) w_{-k}(0) + i \int_0^t \exp((t-s)H_{\sigma_A}) b_{-k} ds \\ &= i \int_0^t b_{-k} \circ \Phi_{t-s}^A ds \\ &= i \int_0^t \sigma_B \circ \Phi_s^A ds \end{align*} $$

□

In the same spirit as [Gui78, Theorem 4] and [DG75, Theorem 4.5] (recall that a version of this theorem in the case of the DN map is given in **Theorem II.6**) we can then formulate a trace formula adapted to our problem:
**Theorem IV.2.** Let $k \in \mathbb{N}^*$, $A \in \Psi_{cl}^1(N)$ and $B \in \Psi_{cl}^{-k}(N)$ such that A and $A+B$ are self-adjoint operators. Assume A is elliptic. If A only admits a finite number of closed bicharacteristic curves $\gamma_1, ..., \gamma_r$ of period T and each of these curves are non-degenerate, then

$$S(t) := \mathrm{Tr}(U(t) - U_0(t))$$

is a tempered distribution on $\mathbb{R}$ admitting an isolated singularity in $t = T$ given by a Lagrangian distribution $e_T$ with

$$\mathrm{Suppsing}(e_T) = \{T\}$$

and complete asymptotic expansion

$$e_T(t) \sim_{t \to T} c_{T,-1}(t-T+i0)^{-1} + \sum_{j=0}^{+\infty} c_{T,j}(t-T+i0)^j \log(t-T+i0).$$

The leading coefficient is expressed as

$$c_{T,-1} = \sum_{j=1}^{r} \frac{|T_j^\sharp| e^{im_j \frac{\pi}{2}} e^{-iT \frac{\gamma_j}{\bar{\gamma}_j}}}{|\det(Id - \mathcal{P}_j)|^{1/2}} i \int_{\gamma_j} \sigma_B,$$

where

$$\bar{\gamma}_j := \frac{1}{T} \int_{\gamma_j} \mathrm{sub}(A),$$

and $\mathcal{P}_j$, $T_j^\sharp$, $m_j$ are respectively the linearized Poincaré map, the primitive period and the Maslov index of $\gamma_j$.

*Proof.* The proof is similar to that of [Gui78, Theorem 4], it uses the same arguments as [DG75, Theorem 4.5] on the symbolic calculus for Fourier integral operators.

For the sake of convenience, let us briefly review the main points in our setting :

As we pointed out in the introduction of this section, if we denote $\mu_0(t, x, y)$ the Schwartz kernel of $U_0(t)$,

$$\mu_0 \in I^{-\frac{1}{4}}(\mathbb{R} \times N \times N, C).$$

Moreover, by **Theorem IV.1**, $W(t) \in \Psi^{-k}(N)$ and consequently, the distribution

$$U(t) - U_0(t) = W(t)U_0(t)$$

defines a Fourier integral operator ; more precisely its Schwartz kernel $v$ is a Lagrangian distribution of order $-k - \frac{1}{4}$ on $\mathbb{R} \times N \times N$ :

$$v \in I^{-k-\frac{1}{4}}(\mathbb{R} \times N \times N, C).$$

In addition, it is easy to show that

$$S(t) = \pi_* \Delta^* v,$$
where $\Delta^*$ is the pull-back by the diagonal map $\Delta: \mathbb{R} \times N \to \mathbb{R} \times N \times N$ and $\pi_*$ is the push-forward by the projection map $\pi: \mathbb{R} \times N \to \mathbb{R}$. (see for example [DG75, Lemma 6.4])

Then **Theorem IV.2** follows in the same way as [DG75, Section 6] from *clean intersection theory* developed in [DG75, Section 5], which can be applied here by assumption on the periodic orbits $\gamma_1, \dots, \gamma_r$.

The leading term in the asymptotic expansion is given by the principal symbol of $\pi_* \Delta^* v$ and using notations from [Gui78, Section 4], it can be expressed in terms of the principal symbol $\sigma(v)$ of $v$ as follows:

$$ \sigma_{S(t)} = \pi_* \Delta^* \sigma(v) = \sum_{j=1}^{r} \sigma(v)|_{C_j}, $$

where

$$ C_j := \{(t, \tau), (x, \xi), (y, \eta) \mid (x, \xi) = (y, \eta) \in \gamma_j, t = T\} \subset C. $$

Moreover, we note that the principal symbol of $v$ is given by

$$ \sigma(v)((t, \tau), (x, \xi), (y, \eta)) = \sigma_{W(t)}(x, \xi) \cdot \sigma(\mu_0)((t, \tau), (x, \xi), (y, \eta)), $$

where $\sigma(\mu_0)$ denotes the principal symbol of $\mu_0$.

By [DG75, Theorem 4.5], we know that

$$ \sigma(\mu_0)|_{C_j} = \frac{|T_j^\#| e^{im_j \frac{\pi}{2}} e^{-iT \overline{\gamma_j}}}{|\det(Id - P_j)|^{1/2}}. $$

Finally, by **Theorem IV.1**,

$$ \sigma_{W(t)}|_{C_j} = i \int_{\gamma_j} \sigma_B, $$

which conclude the sketch of proof.

□

**Remark IV.3.** In our case, the Maslov index $m_j$ appearing in this result is the Morse index of the periodic geodesic $\gamma_j$ (or more precisely here the projection of $\gamma_j$) just as in **Theorem II.6**. Recall that these indices are zero in the case where the underlying manifold $N$ is Anosov. (see **Remark II.7**)

In the following, we will use suitable coordinates on the manifold $(M, g)$, called boundary normal coordinates.

If $x' := (x_1, \dots, x_{n-1})$ denotes any local coordinates on the boundary near $p \in \partial M$, we can define the associated boundary normal coordinates $(x', x_n)$ at $p$ by considering $x_n$ to be the distance to the boundary along unit speed geodesics normal to $\partial M$. See for example [LU89, Section 1] for more details.

It follows that in these coordinates, $x_n > 0$ in $M$ and $\partial M$ is locally characterized by $x_n = 0$. The metric in these coordinates has the form
$$g = \sum_{1 \le \alpha, \beta \le n-1} g_{\alpha\beta}(x) dx^\alpha \otimes dx^\beta + dx^n \otimes dx^n.$$

Finally, this local coordinate system naturally defines a diffeomorphism

$$\phi : \mathcal{U}_p \to \mathcal{V}_p,$$

where $\mathcal{U}_p$ is some open set in the half-space

$$\mathbb{R}_+^n := \{(x_1, \dots, x_n) ; x_n \ge 0\}$$

and $\mathcal{V}_p := \phi(\mathcal{U}_p)$ is a neighborhood of $p$ in $M$. For two metrics $g_1$ and $g_2$ on $M$, let $\phi_1$ and $\phi_2$ be such diffeomorphisms, then

$$\phi := \phi_1 \circ \phi_2^{-1}$$

defines a local diffeomorphism sending the unit speed geodesics for $g_1$ normal to $\partial M$ onto the unit speed geodesics for $g_2$ normal to $\partial M$.

Recall that this diffeomorphism can be extended globally into a diffeomorphism that is again denoted $\phi$ and that if $(x', x_n)$ are boundary normal coordinates near a fixed boundary point for the metric $g_1$, they are also boundary normal coordinates for the metric $\phi^*g_2$.

This leads to an important remark that we will use in the following to exploit the formulas in [LU89]:

**Remark IV.4.** Since $\phi$ is a diffeomorphism of $M$ fixing the boundary we have the well-known gauge invariance

$$\Lambda_{\phi^*g_2} = \Lambda_{g_2}$$

and then in particular, assuming that $g_2$-boundary normal coordinates are also $g_1$-boundary normal coordinates does not change the DN map $\Lambda_{g_2}$.

So for our problem, we can always assume that $g$-boundary coordinates are also $cg$-boundary coordinates without loss of generality.

We now have the tools we need to obtain the desired boundary determination results:

First, **Corollary B.4** shows that if the total symbols of $\Lambda_{cg}$ and $\Lambda_g$ coincide at principal and subprincipal order, these two operators differ only by a pseudodifferential operator of order $-1$. So in this case, as already mentioned, we are now able to combine **Theorem IV.2** with the injectivity of the X-ray transform on functions (**Theorem II.3**) to access the homogeneous terms of orders $j \le -1$ contained in the symbol of $\Lambda_{cg} - \Lambda_g \in \Psi^{-1}(\partial M)$ and, more precisely, to show recursively that they are all zero if the metrics $g$ and $cg$ are Steklov isospectral.

This is the subject of the next theorem and, following the work of Lee and Uhlmann [LU89, Proposition 1.3], we begin with a lemma specifying these homogeneous terms of order $j \le -1$:
**Lemma IV.5.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 2$ with smooth boundary and $c: M \to \mathbb{R}_+^*$ be a smooth function satisfying

$$c|_{\partial M} \equiv 1.$$

Let

$$\sigma_{\Lambda_{cg}}^{full} \sim \sum_{j=-1}^{+\infty} \tilde{a}_{-j}$$

and

$$\sigma_{\Lambda_g}^{full} \sim \sum_{j=-1}^{+\infty} a_{-j}$$

be respectively the full symbols of $\Lambda_{cg}$ and $\Lambda_g$ considered in any local coordinate system $x' := (x_1, \dots, x_{n-1})$ on the boundary.

In the $g$-boundary normal coordinates $(x', x_n)$ :

For any $j \ge 1$, if

$$\partial_n^m c|_{x_n=0} = 0, \quad \forall j \ge m \ge 1,$$

then

$$\tilde{a}_{-j} - a_{-j} = \frac{(n-2)}{4} \left(-\frac{1}{2|\xi'|_{g^\circ}}\right)^j \partial_n^{j+1} c|_{x_n=0}.$$

*Proof.* Let $x' := (x_1, \dots, x_{n-1})$ be any local coordinate system on the boundary and $(x', x_n)$ the associated $g$-boundary normal coordinates. By **Remark IV.4**, we can assume that these coordinates are also $cg$-boundary coordinates.

Using [LU89, Proposition 1.3], we denote by $(g^{\alpha\beta})$ the inverse matrix of $(g_{\alpha\beta})$ and introduce for any $1 \le \alpha, \beta \le n-1$,

$$h^{\alpha\beta} := \partial_n g^{\alpha\beta}, \quad h = \sum_{1 \le a, b \le n-1} g_{ab} h^{ab},$$

$$\tilde{h}^{\alpha\beta} := \partial_n (cg)^{\alpha\beta}, \quad \tilde{h} = \sum_{1 \le a, b \le n-1} (cg)_{ab} \tilde{h}^{ab}$$

and the quadratic forms

$$k^{\alpha\beta} := h^{\alpha\beta} - hg^{\alpha\beta}, \quad \tilde{k}^{\alpha\beta} := \tilde{h}^{\alpha\beta} - \tilde{h}(cg)^{\alpha\beta}$$

respectively associated with the metrics $g$ and $cg$.

Noticing that $(cg)^{\alpha\beta} = c^{-1}g^{\alpha\beta}$, a simple calculation gives

$$\tilde{k}^{\alpha\beta} = ((n-1)c^{-1} - 1) \frac{\partial_n c}{c^2} g^{\alpha\beta} + c^{-1} (h^{\alpha\beta} - c^{-1}hg^{\alpha\beta}),$$

then [LU89, Proposition 1.3] gives for $j \ge 1$,

$$\tilde{a}_{-j} = \left(-\frac{1}{2|\xi'|}\right)^{j+2} \sum_{1 \le \alpha, \beta \le n-1} \left(\partial_n^j \tilde{k}^{\alpha\beta}\right) \xi_\alpha \xi_\beta + T_{-j}((cg)_{\alpha\beta}),$$

where $T_{-j}((cg)_{\alpha\beta})$ is an expression involving only the boundary values of $(cg)_{\alpha\beta}$, $(cg)^{\alpha\beta}$ and their normal derivatives of order at most $j$.
Obviously we also have the same formula for $a_{-j}$, just replacing $\tilde{k}^{\alpha\beta}$ by $k^{\alpha\beta}$ and $cg$ by $g$. Now suppose that for any $j \ge 1$,

$$ \partial_n^m c|_{x_n=0} = 0, \quad \forall j \ge m \ge 1. $$

Then we see that on the boundary, for any $j \ge 1$,

$$ \begin{aligned} \partial_n^j \tilde{k}^{\alpha\beta} &= (n-2) (\partial_n^{j+1} c) g^{\alpha\beta} + \partial_n^j h^{\alpha\beta} - \partial_n^j (hg^{\alpha\beta}) \\ &= (n-2) (\partial_n^{j+1} c) g^{\alpha\beta} + \partial_n^j k^{\alpha\beta} \end{aligned} $$

and, of course, the equality of the remainders

$$ T_{-j}((cg)_{\alpha\beta}) = T_{-j}(g_{\alpha\beta}). $$

Thus, noticing that

$$ \sum_{1 \le \alpha, \beta \le n-1} g^{\alpha\beta} \xi_{\alpha} \xi_{\beta} = |\xi'|_{g^{\circ}}^2 $$

and noting

$$ T_{-j} := \sum_{1 \le \alpha, \beta \le n-1} T_{-j}(g_{\alpha\beta}), $$

we get for $j \ge 1$,

$$ \begin{aligned} \tilde{a}_{-j} &= \left(-\frac{1}{2|\xi'|}\right)^{j+2} \sum_{1 \le \alpha, \beta \le n-1} \left[(n-2) (\partial_n^{j+1} c|_{x_n=0}) g^{\alpha\beta} + \partial_n^j k^{\alpha\beta}\right] \xi_{\alpha} \xi_{\beta} + T_{-j} \\ &= \frac{(n-2)}{4|\xi'|_{g^{\circ}}^2} \left(-\frac{1}{2|\xi'|_{g^{\circ}}}\right)^j \partial_n^{j+1} c|_{x_n=0} \sum_{1 \le \alpha, \beta \le n-1} g^{\alpha\beta} \xi_{\alpha} \xi_{\beta} + a_{-j} \\ &= \frac{(n-2)}{4} \left(-\frac{1}{2|\xi'|_{g^{\circ}}}\right)^j \partial_n^{j+1} c|_{x_n=0} + a_{-j} \end{aligned} $$

□

We deduce the boundary determination of the jet of the conformal factor $c$ at any point $p \in \partial M$ in boundary normal coordinates :

**Theorem IV.6.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 3$ with smooth boundary. Assume that $(\partial M, g^{\circ})$ is Anosov with simple length spectrum.

Let $c: M \to \mathbb{R}_*^+$ be a smooth function satisfying

$$ c|_{\partial M} \equiv 1. $$

If

$$ Sp(\Lambda_{cg}) = Sp(\Lambda_g), $$

then for any local coordinate system $x' := (x_1, \dots, x_{n-1})$ on the boundary, the following identity holds in the $g$-boundary normal coordinates $(x', x_n)$:
$$ \partial_n^j c|_{x_n=0} = 0, \forall j \ge 1. $$

Consequently, the Taylor series of $c$ at any point $p \in \partial M$ in $g$-boundary normal coordinates are reduced to the constant function identically equal to 1.

*Proof.* Let us consider $x' := (x_1, \dots, x_{n-1})$ any local coordinate system on the boundary and $(x', x_n)$ the associated $g$-boundary normal coordinates near $\partial M$. We proceed by induction on the order of derivation $j \ge 1$:

* $j = 1$: By **Theorem III.5**,

$$ \partial_n c|_{x_n=0} = 0 $$

since $(\partial M, g^\circ)$ is Anosov with simple length spectrum.

* Let $j \ge 1$ s.t for all $1 \le m \le j$,

$$ \partial_n^m c|_{x_n=0} = 0. $$

Then using **Lemma IV.5**,

$$ \Lambda_{cg} - \Lambda_g \in \Psi^{-j}(\partial M). $$

Now we can apply **Theorem IV.2** to $(N, h) = (\partial M, g^\circ)$,

$$ A = \Lambda_g \in \Psi^1(\partial M) \text{ and } B = \Lambda_{cg} - \Lambda_g \in \Psi^{-j}(\partial M) : $$

Recalling that

- $Sp(\Lambda_{cg}) = Sp(\Lambda_g)$.

- $c|_{\partial M} = 1$.

- $(\partial M, g^\circ)$ is Anosov with simple length spectrum. (which implies, in **Theorem IV.2**, $r = 1$ and the non-degeneracy of closed geodesics as we pointed out in Section II.2)

we obtain that the leading term in the expansion of $S(t)$ is zero, and then using **Theorem IV.2**, we get that for each periodic bicharacteristic $s \mapsto \Phi_s^A(x, \xi)$,

$$ 0 = \int_0^\ell \sigma_B(\Phi_s^A(x, \xi)) ds, $$

where $\ell$ obviously denotes the length of the bicharacteristic. Given the expression for the principal symbol of $B \in \Psi^{-j}(\partial M)$ in **Lemma IV.5**, we conclude as usual by injectivity of the geodesic $X$-ray transform on functions (**Theorem II.3**), that

$$ \partial_n^{j+1} c|_{x_n=0} = 0. $$
It is then easy to see that the Taylor series of the metrics $g$ and $cg$ at any $p \in \partial M$ are identical in $g$-boundary normal coordinates $(x', x_n)$ near this point:

By simply using **Remark IV.4** and [LU89, Proposition 1.3], we know that the knowledge of the full symbols of $\Lambda_{cg}$ and $\Lambda_g$ determine, in $g$-boundary normal coordinates the Taylor series of $cg$ and $g$ respectively.

But with the same notations as in **Lemma IV.5**, we have just shown that in these coordinates,

$$ \tilde{a}_{-j} - a_{-j} = 0, \quad \forall j \ge -1. $$

□

**Remark IV.7.** Under the assumptions of the theorem, we have shown here that the operators $\Lambda_{cg}$ and $\Lambda_g$ coincide modulo a smoothing operator.

By a simple argument of analytic continuation, we then obtain **Corollary I.3**:

*Proof of Corollary I.3.* Let $p \in \partial M$ and consider boundary normal coordinates

$$ \phi : \mathcal{U}_p \to \phi(\mathcal{U}_p) \subset \mathbb{R}_+^n := \{(x_1, \dots, x_n); x_n \ge 0\} $$

defined on an open neighborhood $\mathcal{U}_p$ of $p$ in $M$.

By **Theorem IV.6**, we know that in these coordinates, the Taylor series of $c$ at $p$ is the constant function identically equal to 1.

Now, choose any local real-analytic chart $(\mathcal{V}_p, \tilde{\phi})$ for $M$, and note that the Taylor series of $c$ at $p$ is also reduced to the constant function identically equal to 1 in these coordinates since the nullity of derivatives is obviously a coordinate invariant property:

Derivatives transforms under changes of coordinates by

$$ \frac{\partial}{\partial \tilde{x}_j} = \sum_{i=1}^{n} \left( \frac{\partial x_i}{\partial \tilde{x}_j} \circ \tilde{\phi} \right) \frac{\partial}{\partial x_i}, $$

so that nullity of derivatives in one coordinate system leads to the same in the other coordinate system. As $\tilde{\phi}$ defines an analytic coordinate system and $c$ is analytic, $c$ coincides with the constant function identically equal to 1 in the non-empty open subset $\mathcal{U}_p \cap \mathcal{V}_p \subset \mathcal{V}_p$ of $M$.

Since $M$ is connected, we conclude that $c$ is identically equal to 1 on $M$. □

## V. FURTHER RESULTS AND REMARKS

Here we discuss other similar inverse problems which the author believes could be interesting to study in the line of this paper. In particular, this section establishes **Theorem I.6** using the method developed in Section IV. The result deals with the Steklov spectral inverse problem of recovering a potential $q$ from the Steklov spectrum of a smooth compact Riemannian manifold $M$ of dimension $n \ge 3$ with boundary.

This inverse problem involves the Schrödinger operator $\mathcal{L}_{g,q} := -\Delta_g + q$ where $q$ is a smooth real-valued function on $M$: It is well known that if $0$ is not a Dirichlet eigenvalue of $\mathcal{L}_{g,q}$ in $M$, then the Dirichlet problem
$$ \begin{cases} \mathcal{L}_{g,q} v = 0 \\ v|_{\partial M} = f \end{cases} $$

has a unique solution in $H^1(M)$ for any $f \in H^{1/2}(\partial M)$ and we can define the associated DN map

$$ \Lambda_{g,q} : f \mapsto \partial_{\nu} u|_{\partial M}. $$

We refer, for example, to [Sal13] for details. In the same way as the standard Steklov operator $\Lambda_g$ presented in the Introduction, $\Lambda_{g,q}$ is a positive self-adjoint elliptic and classical (in the sense of (1)) pseudodifferential operator of order 1 on the closed manifold $\partial M$. (see, for example, [DSFKSU09] or [Cek20])

In particular, its spectrum is discrete and is given by a sequence of eigenvalues

$$ 0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \dots \to \infty. $$

Using the expressions for the full symbols of $\Lambda_{g,q}$ in boundary normal coordinates, computed in [DSFKSU09, Lemma 8.6], we can easily show a result analogous to **Lemma IV.5** for this problem:

**Lemma V.1.** Let $q_1$ and $q_2$ two smooth potentials on $M$. Let

$$ \sigma_{\Lambda_{g,q_2}}^{\text{full}} \sim \sum_{j=-1}^{+\infty} \tilde{a}_{-j} $$

and

$$ \sigma_{\Lambda_{g,q_1}}^{\text{full}} \sim \sum_{j=-1}^{+\infty} a_{-j} $$

be respectively the full symbols of $\Lambda_{g,q_2}$ and $\Lambda_{g,q_1}$ considered in any local coordinate system $x' := (x_1, \dots, x_{n-1})$ on the boundary.

In the g-boundary normal coordinates $(x', x_n)$,

$$ \tilde{a}_1 - a_1 = 0 = \tilde{a}_0 - a_0 $$

and

$$ \tilde{a}_{-1} - a_{-1} = (q_2 - q_1)|_{x_n=0}. $$

Furthermore, if

$$ q_1|_{\partial M} = q_2|_{\partial M}, $$

then in the g-boundary normal coordinates $(x', x_n)$:

$$ \tilde{a}_{-j} - a_{-j} = \left(-\frac{1}{2|\xi'|_{g^\circ}}\right)^j \partial_n^{j-1} (q_2 - q_1)|_{x_n=0}, \forall j \ge 2. $$
Note that the proof of this result is actually easier than **Lemma IV.5** since all the terms involving the metric are identical for both operators. Note also that the identity at homogeneity order $-1$ can be seen directly here via the expression of the principal symbol given in **Proposition B.3**.

Now, using the same arguments developed in Section IV, we can show a result similar to **Theorem I.2**, namely that the Steklov spectrum determines the potential in the following sense:

**Theorem V.2.** Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum. Let $q_1$ and $q_2$ two smooth potentials on $M$.

If the Steklov spectrum is preserved

$$Sp(\Lambda_{g,q_1}) = Sp(\Lambda_{g,q_2}),$$

then in $g$-boundary normal coordinates,

$$(q_2 - q_1)|_{x_n=0} = 0$$

and

$$\partial_n^{j-1} (q_2 - q_1)|_{x_n=0} = 0, \forall j \geq 2.$$

Consequently, the Taylor series of $q_1$ and $q_2$ at each point $p \in \partial M$ are equal in boundary normal coordinates and

$$\Lambda_{g,q_1} = \Lambda_{g,q_2}$$

modulo a smoothing operator.

*Proof.* The proof follows exactly the same ideas as **Theorem IV.6**: We proceed by induction on the order of derivation $j \geq 0$ and apply at each step the trace formula **Theorem IV.2** to

$$(N, h) = (\partial M, g^\circ), \quad A = \Lambda_{g,q_1} \in \Psi^1(\partial M), \quad B = \Lambda_{g,q_2} - \Lambda_{g,q_1} \in \Psi^{-j-1}(\partial M)$$

which, combined with the isospectrality assumption, gives us for any periodic bicharacteristic $s \mapsto \Phi_s^A(x, \xi)$,

$$0 = \int_0^\ell \sigma_B (\Phi_s^A (x, \xi)) \, ds,$$

where $\ell$ denotes the length of the bicharacteristic.

Then combining the expression for the principal symbol of $B \in \Psi^{-j-1}(\partial M)$ given by the previous lemma with the injectivity of the X-ray transform (**Theorem II.3**), we obtain

$$\partial_n^j (q_2 - q_1)|_{x_n=0} = 0.$$
Using the notations from the previous lemma, we have just shown that in g-boundary normal coordinates,

$$\tilde{a}_{-j} - a_{-j} = 0, \forall j \ge -1$$

or equivalently that $\Lambda_{g,q_1}$ and $\Lambda_{g,q_2}$ coincide modulo a smoothing operator. To conclude, we can then simply use the calculations of [DSFKSU09, Section 8] or [Cek20, Section 3.2] which show that in g-boundary normal coordinates the Taylor series of $q_1$ and $q_2$ at any point $p \in \partial M$ are determined by the knowledge of the full symbols of $\Lambda_{g,q_1}$ and $\Lambda_{g,q_2}$ respectively. $\square$

**Remark V.3.** Note that initialization here is much simpler, since it is obtained directly by the same argument as at higher orders, unlike the case treated in this paper. In the latter case, initialization consisted in showing that the normal derivative of the conformal factor at the boundary is zero, but this quantity appears at the subprincipal level (via the trace formula Theorem II.6) and then is not directly covered by our method. For this reason, this non-linear case was also treated separately in Section III; the result was stated in Theorem III.5.

To complete the above remark, let us note that this problem is actually made completely linear by the fact that the operators $\Lambda_{g,q_1}$ and $\Lambda_{g,q_2}$ already coincide at principal and subprincipal order and that the information on the potentials is given from order $-1$, which allows us to apply Theorem IV.2 directly. In this sense, the problem is simpler than the case treated in this paper, for which obtaining information in the principal order is a non-linear problem and requires the addition of relatively strong geometric assumptions as discussed in Appendix A.

Finally, let us note that just as Theorem IV.2 is an analogue of [Gui78, Theorem 4], its application Theorem V.2 can be compared with [GK80a, Theorem 2] on the recovery of a potential $q$ from the spectrum of the associated Schrödinger operator. Moreover, it seems worth mentioning here that Theorem IV.2 also gives us in particular, and this is actually equivalent to Theorem V.2, an analogue of [GK80a, Theorem 3(a)], namely:

**Theorem V.4.** Let $(M, g)$ be a compact smooth Riemannian manifold with boundary and $q$ be a smooth potential on $M$. Assume that $(\partial M, g^\circ)$ is Anosov with simple length spectrum. Then the spectrum of $\Lambda_{g,q}$ determines the integrals of $q$ and all its normal derivatives at the boundary over the periodic geodesics of $g^\circ$.

A natural question raised by Theorem V.2 is whether we can explicitly construct an example of Steklov isospectral potentials which are not the same, obviously when the simple length spectrum condition is not verified.

In the case of the Schrödinger operator $\mathcal{L}_{g,q} = -\Delta_g + q$, such examples are shown in [Bro87] via the general method of Sunada [Sun85], which remains probably the most widely used and cited technique to construct isospectral manifolds.

We refer to [Gor09] for details and more specifically to [GHW21] for the Steklov spectrum.

To the author's knowledge, there is no reference for the operator $\Lambda_{g,q}$, but it may be known in folklore that the Sunada construction can be adapted again.
In another direction, recall that in this paper we crucially exploit the fact that the Steklov operator has a non-zero subprincipal symbol which enabled us to use the additional information provided by the principal invariant of the wave trace **Theorem II.6**.

In this way, in **Theorem III.5**, we have recovered the normal derivative of the conformal factor at the boundary which appears as an attenuation factor in a transport-type equation. (see **Theorem III.4**)

To continue in this spirit, we can try to exhibit inverse spectral problems for which the operator has a non-zero subprincipal symbol and which reveal relevant information but more complicated in nature than a simple function, for example a 1-form. One can think of the magnetic Laplacian (which is a first order perturbation of the usual Laplacian) or, in the case of Steklov inverse spectral problem, the magnetic Dirichlet-to-Neumann map $\Lambda_{g,A,q}$, which roughly speaking consists of adding a magnetic potential i.e a smooth 1-form A on M. More precisely, this is the Steklov operator associated with the Dirichlet problem for the Schrödinger operator

$$ \mathcal{L}_{g,A,q} := d_A^* d_A + q, $$

where $d_A := d + iA$ denotes the magnetic differential and $d_A^* := d^* - i\langle A^*, \cdot \rangle$ its formal adjoint. (see, for example, [DSFKSU09] or the recent [CGHP24] for details) We hope to discuss these cases in a forthcoming article, generalizing **Theorem III.4** by the same token. Indeed, Paternain's argument [Pat09] using Gysin sequences (see **Lemma C.1**) still works to get a satisfactory statement but a new formalism in the spirit of Pestov identity can be introduced to handle this equation.

In the problems discussed above in this section, the metric is fixed and the aim is to recover a potential.

But we can also try to generalize the results obtained in this paper in the case where the metrics are no longer in the same conformal class.

This problem obviously poses significant technical issues, since we are no longer reasoning about a function (the conformal factor) but directly about the metric tensor, and it is not clear that we can obtain satisfactory results in the non-linear case. (when isospectral deformations are not considered)

More precisely, consider two metrics $g_1$ and $g_2$ and assume that $(\partial M, g_i^\circ)$, $i \in \{1, 2\}$ is Anosov with simple length spectrum. Then if the Steklov spectrum is preserved

$$ Sp(\Lambda_{g_1}) = Sp(\Lambda_{g_2}), $$

can we show that the Taylor series of $g_1$ and $g_2$ in boundary normal coordinates are equal at each point on the boundary and that $\Lambda_{g_1}$ and $\Lambda_{g_2}$ coincide modulo a smoothing operator ?

Of course, the answer is a priori no, at least without further assumptions.

Indeed, as in the case of conformal metrics, information at principal order poses a significant problem, and therefore it seems necessary here to assume at least that the two metrics are isometric at the boundary, i.e. there is a diffeomorphism $\varphi : \partial M \to \partial M$ such that

$$ g_2^\circ = \varphi^* g_1^\circ. $$
We can also note that, at subprincipal order, the relevant quantity that appears is the second fundamental form of the boundary, since for any metric g,

$$sub(\Lambda_g)(x, \xi) = \frac{1}{2} \left( \frac{\mathcal{L}_g(\xi, \xi)}{|\xi|^2} - H_g \right),$$

where $\mathcal{L}_g(\xi, \xi)$ is the second fundamental form of $\partial M$ in $(M, g)$ and $H_g$ its trace, also called mean curvature of $M$. We refer to [Tay23, Chapter 12, Proposition C.1] for this standard result.

Then we can try to mimick the conformal case and see if we are able to get usable information as well and obtain similar results. In addition, the problem would no longer require the injectivity of the X-ray transform on functions, but rather an injectivity property for the X-ray transform on symmetric tensors of order 2 which, for closed Anosov manifolds, is only known in dimension 2 (see [PSU14, Theorem 1.1] or more generally [Gui17, Theorem 1.4] for tensors of any order), or in any dimension but with non-positive sectional curvature. (see [CS98, Theorem 1.3]) However, we should mention here that the X-ray transform of symmetric tensors of arbitrary degree is generically injective on closed Anosov manifolds of dimension $n \ge 3$, as shown very recently by M. Cekić and T. Lefeuvre in [CL25].

To conclude this paper, note that we may also be interested in establishing stability estimates to complement the results of this paper, even if the techniques used here appear to be non-quantitative. In another direction it would also be interesting to try to generalize some known compactness results for sets of Laplace isospectral metrics, in particular [BPY89, Theorem 2] in the case of conformal metrics, to the Steklov spectrum, this could be done by exploiting heat invariants if possible.

## APPENDIX A. A FEW RESULTS ON THE PRINCIPAL ORDER

Here we establish some simple results on the spectral inverse problem at principal order; more precisely we give examples where the Steklov isospectrality of two conformal metrics $g$ and $cg$ implies their equality at the boundary i.e

$$c|_{\partial M} \equiv 1.$$

As mentioned in this paper, this highly non-linear problem cannot be handled with the techniques used here and it is of course not true in general just as the analogue for the Laplacean is not.

Consequently, we are only able to show this result for the linear problem (i.e considering isospectral deformations) and in the non-linear case under much stronger geometrical assumptions due to the lack of exploitable spectral information. Indeed, for the latter, we only exploit the boundary volume as a spectral invariant, given by the well-known Weyl's law. (see **Proposition II.4**)

To begin with, using the same approach as in [GK80a], it is easy to show that $(\partial M, g^\circ)$ is infinitesimally spectrally rigid in the following sense :

**Proposition A.1.** Let $(M, g)$ be a smooth compact Riemannian manifold with boundary.
Let $(g_s)_{0 \le s \le \varepsilon}$ be a Steklov isospectral deformation of the metric $g$ s.t there exists a family of smooth functions $(c_s : M \mapsto \mathbb{R}_*^+)_{0 \le s \le \varepsilon}$ satisfying for any $s \in [0, \varepsilon]$,

$$g_s = c_s g.$$

Assume that for every $s \in [0, \varepsilon]$, $g_s^\circ$ is Anosov with simple length spectrum. Then for all $s \in [0, \varepsilon]$,

$$g_s^\circ = g^\circ.$$

**Proof.** Using **Proposition II.10** and its notations, we obtain that for all $s \in [0, \varepsilon]$ and $[\gamma] \in \mathcal{C}$,

$$
\begin{align*}
0 = \mathcal{I}_2^{g_s^\circ} \dot{g}_s^\circ ([\gamma]) &= \int_{\gamma_s} \dot{g}_s^\circ = \int_0^\ell \dot{c}_s(\gamma_s(t)) g^\circ(\dot{\gamma}_s(t), \dot{\gamma}_s(t)) dt \\
&= \int_0^\ell \frac{\dot{c}_s}{c_s}(\gamma_s(t)) \underbrace{g_s^\circ(\dot{\gamma}_s(t), \dot{\gamma}_s(t))}_{=1} dt \\
&= \int_{\gamma_s} \frac{\dot{c}_s}{c_s} \\
&= \mathcal{I}_0^{g_s^\circ} \frac{\dot{c}_s}{c_s} ([\gamma]) .
\end{align*}
$$

We conclude the proof with the injectivity of the X-ray transform on functions for Anosov manifolds (**Theorem II.3**) and using the fact that $c_0 = 1$. □

As mentioned above, under certain stronger assumptions, mainly about curvature, we can formulate a first result in the non-linear case by using Weyl's law (see **Proposition II.4**) and adapting arguments<fnref n="3" /> from Besson, Gallot and Courtois initially used to establish a similar spectral rigidity result for the Laplacean [BCG91, Proposition B.2]. However, in our case, the statement is weaker since the total scalar curvature (i.e. the integral of scalar curvature) is not an invariant of the Steklov spectrum :

**Theorem A.2.** Let $(M, g)$ be a smooth compact Riemannian manifold of dimension $n \ge 4$ with boundary such that $(\partial M, g^\circ)$ has a constant negative scalar curvature. Let $c : M \to \mathbb{R}_*^+$ be a smooth function satisfying

$$Sp(\Lambda_{cg}) = Sp(\Lambda_g).$$

If the total scalar curvatures at the boundary coincide

$$S_{(cg)^\circ} = S_{g^\circ},$$

then

$$c|_{\partial M} = 1.$$

<footnote n="3">³Arguments originally put forward by A.El Soufi and S.Ilias in their work on conformal volume.</footnote>
**Proof.** The proof follows exactly that of [BCG91, Proposition B.2].
Let us write $h = cg$ and $c = e^{-2u}$ with $u$ a smooth real-valued function on $M$.
The well-known expression of the scalar curvature after conformal change gives on $\partial M$:

$$ \mathrm{scal}(g^{\circ}) = e^{-2u} (\mathrm{scal}(h^{\circ}) - 2(n-2)\Delta_{h^{\circ}}u - (n-3)(n-2)|du|_{h^{\circ}}^2), $$

then by integrating with respect to the canonical measure of $(\partial M, h^{\circ})$:

$$ \int_{\partial M} e^{2u} (-\mathrm{scal}(g^{\circ})) dv_{h^{\circ}} = \int_{\partial M} (-\mathrm{scal}(h^{\circ})) dv_{h^{\circ}} + (n-3)(n-2) \int_{\partial M} |du|_{h^{\circ}}^2 dv_{h^{\circ}}. $$

Indeed, since the integral of the divergence of a vector field on a closed manifold is zero (by the classical Stokes' theorem), we have:

$$ \int_{\partial M} (\Delta_{h^{\circ}} u) dv_{h^{\circ}} = \int_{\partial M} \mathrm{div}_{h^{\circ}} (\nabla_{h^{\circ}} u) dv_{h^{\circ}} = 0. $$

Thus, as the scalar curvature of $(\partial M, g^{\circ})$ is constant and by the Hölder's inequality applied with exponents $p = \frac{n-1}{2}$ and $q = \frac{n-1}{n-3}$ we get:

$$ \begin{aligned} \int_{\partial M} (-\mathrm{scal}(h^{\circ})) dv_{h^{\circ}} &\le \int_{\partial M} e^{2u} (-\mathrm{scal}(g^{\circ})) dv_{h^{\circ}} \\ &\le -\mathrm{scal}(g^{\circ}) \left( \underbrace{\int_{\partial M} e^{(n-1)u} dv_{h^{\circ}}}_{=\mathrm{Vol}_{g^{\circ}}(\partial M)} \right)^{\frac{1}{p}} \mathrm{Vol}_{h^{\circ}}(\partial M)^{\frac{1}{q}}. \end{aligned} $$

Since $g$ and $h$ are Steklov isospectral and the volume of the boundary $\partial M$ is an invariant for the Steklov spectrum (**Corollary II.5**), we have $\mathrm{Vol}_{g^{\circ}}(\partial M) = \mathrm{Vol}_{h^{\circ}}(\partial M)$ and consequently the previous inequality becomes:

$$ \begin{aligned} \int_{\partial M} (-\mathrm{scal}(h^{\circ})) dv_{h^{\circ}} &\le -\mathrm{scal}(g^{\circ}) (\mathrm{Vol}_{g^{\circ}}(\partial M))^{\frac{1}{p}} \mathrm{Vol}_{h^{\circ}}(\partial M)^{\frac{1}{q}} \\ &\le -\mathrm{scal}(g^{\circ}) \mathrm{Vol}_{g^{\circ}}(\partial M), \end{aligned} $$

obviously the equality holds iff $u|_{\partial M} = 0$.<fnref n="4" />

$$ |g|^q = \lambda^p |f|^p \text{ almost everywhere,} $$

with $\lambda = \frac{\|g\|_q^{q/p}}{\|f\|_p}$. Here we have $f = e^{2u}$ and $g \equiv 1$ on $\partial M$ so we get that the equality holds iff

$$ e^{(n-1)u} \equiv \mathrm{Vol}_{g^{\circ}}(\partial M)/\mathrm{Vol}_{h^{\circ}}(\partial M) = 1 \text{ on } \partial M, $$

iff

$$ u|_{\partial M} = 0. $$

<footnote n="4">⁴Recall that for $f$ and $g$ two integrable functions and $p, q$ conjugate exponents the equality in Hölder's inequality holds iff</footnote>
By assumption, we do have

$$ \int_{\partial M} (-\mathrm{scal}(h^{\circ})) dv_{h^{\circ}} = \int_{\partial M} (-\mathrm{scal}(g^{\circ})) dv_{g^{\circ}} = -\mathrm{scal}(g^{\circ}) \mathrm{Vol}_{g^{\circ}}(\partial M), $$

so the equality in the inequality and therefore $u|_{\partial M} = 0$.

□

In dimension 3, the assumption on total scalar curvatures at the boundary is already verified which gives in particular this elementary statement:

**Proposition A.3.** Let $(M, g)$ be a smooth compact Riemannian 3-manifold with boundary such that $(\partial M, g^{\circ})$ has constant scalar curvature.

Let $h := cg$ be a metric conformal to $g$ and assume that $h^{\circ}$ has constant scalar curvature.<fnref n="5" />

If

$$ \mathrm{Sp}(\Lambda_{cg}) = \mathrm{Sp}(\Lambda_g), $$

then

$$ \mathrm{scal}(h^{\circ}) = \mathrm{scal}(g^{\circ}). $$

In addition, if this scalar curvature is negative,

$$ c|_{\partial M} = 1. $$

*Proof.* As usual let us note $c = e^{2u}$ with $u$ a smooth real-valued function on $M$. The expression of the scalar curvature after conformal change in dimension 2 leads to:

$$ \int_{\partial M} \mathrm{scal}(h^{\circ}) dv_{h^{\circ}} = \int_{\partial M} e^{2u} \mathrm{scal}(g^{\circ}) dv_{g^{\circ}} = \mathrm{scal}(g^{\circ}) \mathrm{Vol}_{g^{\circ}}(\partial M). $$

By isospectrality, $\mathrm{Vol}_{h^{\circ}}(\partial M) = \mathrm{Vol}_{g^{\circ}}(\partial M)$, and since $h^{\circ}$ has constant curvature we obviously deduce that

$$ \mathrm{scal}(h^{\circ}) = \mathrm{scal}(g^{\circ}). $$

Therefore, $\varphi := 2u|_{\partial M}$ satisfies the equation

$$ -\Delta_{h^{\circ}}\varphi + \kappa = \kappa e^{\varphi}, $$

where $\kappa := \mathrm{scal}(g^{\circ})$.

But if $\kappa$ is negative, it is easy to see that $\varphi \equiv 0$ is the unique solution and, equivalently, that there is one and only one metric with scalar curvature $\kappa$ in the conformal class of $g^{\circ}$. We refer, for example, to the proof of [Aub98, Theorem 6.8] and more generally to T.Aubin's work on the prescription of scalar curvature. □

<footnote n="5">⁵By the uniformization theorem it is always possible to find such metrics.</footnote>
With the same idea of using volume as a spectral invariant, we could formulate a second type of result for the non-linear problem, based on the relationship established in [CD04, Theorem 1.2] (via arguments due to Katok [Kat88]) between the (marked) lengths of closed geodesics and volume, but the statement is too unsatisfactory to appear here. In particular, the assumption appearing in [CD04, Theorem 1.2] is a monotonicity condition on the marked length spectrum, and it is not clear how often this condition is true.

## APPENDIX B. A SUITABLE EXPRESSION OF THE DN MAP

Here we provide a useful identity for the calculation of the subprincipal symbol performed in **Proposition III.1**. This easy-to-establish identity is undoubtedly very well known, but the author has found no clear reference for it, so it is reproduced here.

**Lemma B.1.** Let $c: M \to \mathbb{R}_*^+$ be a smooth function satisfying

$$c|_{\partial M} \equiv 1.$$

Then

$$\Lambda_{cg} = \Lambda_{g,-q_c} - \frac{n-2}{4} ((\partial_{\nu}c)|_{\partial M}) Id,$$

where

$$q_c = c^{\frac{n-2}{4}} \Delta_{cg} (c^{-\frac{n-2}{4}}).$$

*Proof.* It is well-known that the Laplace-Beltrami operator transforms under conformal scaling of the metric by

$$\Delta_{cg} = c^{-\frac{n+2}{4}} (\Delta_g + q_c) c^{\frac{n-2}{4}}.$$

Thus, for any $f \in H^{1/2}(\partial M)$, $u$ is solution in $H^1(M)$ of

$$\begin{cases} -\Delta_{cg}u = 0 \\ u|_{\partial M} = f \end{cases}$$

iff

$$v := c^{\frac{n-2}{4}} u$$

is solution in $H^1(M)$ of the following Dirichlet problem

$$\begin{cases} \mathcal{L}_{g,-q_c} v = 0 \\ v|_{\partial M} = f \end{cases},$$

where $\mathcal{L}_{g,-q_c} := -\Delta_g - q_c$.

Therefore,
$$ \begin{aligned} \Lambda_{cg} f &:= \partial_{\nu} u|_{\partial M} = \partial_{\nu} (c^{-\frac{n-2}{4}} v)|_{\partial M} \\ &= -\frac{n-2}{4} \left( (c^{-\frac{n+2}{4}} \partial_{\nu} c) v \right)|_{\partial M} + \left( c^{-\frac{n-2}{4}} \partial_{\nu} v \right)|_{\partial M} \end{aligned} $$

and since $v|_{\partial M} = f$,

$$ \begin{aligned} \Lambda_{cg} f &= -\frac{n-2}{4} ((\partial_{\nu} c)v)|_{\partial M} + \partial_{\nu} v|_{\partial M} \\ &= -\frac{n-2}{4} (\partial_{\nu} c)|_{\partial M} f + \Lambda_{g,-q_c} f . \end{aligned} $$

□

**Remark B.2.** If $M$ is of dimension 2, we easily deduce the well-known conformal invariance of the DN map i.e

$$ \Lambda_{cg} = \Lambda_g, $$

if

$$ c|_{\partial M} = 1. $$

The above formula can be refined with the :

**Proposition B.3.** Let $q \in C^{\infty}(M)$ such that 0 is not a Dirichlet eigenvalue of $-\Delta_g + q$ so that

$$ \Lambda_{g,q} : H^{1/2}(\partial M) \to H^{-1/2}(\partial M) $$

is well defined.

Then

$$ \Lambda_{g,q} = \Lambda_g + R_{-1}, $$

where $R_{-1} \in \Psi^{-1}(\partial M)$ has for principal symbol

$$ \sigma_{R_{-1}}(x, \xi) = \frac{q^{\circ}(x)}{2|\xi|_{g^{\circ}}}. $$

*Proof.* This result is a direct consequence of the expressions in boundary normal coordinates for the full symbols of the pseudodifferential operators $\Lambda_{g,q}$ and $\Lambda_g$ computed, for example, in [DSFKSU09, Lemma 8.6] and [LU89, Section 1] respectively. □

In particular, we deduce the following :

**Corollary B.4.** Let $c : M \to \mathbb{R}_*^+$ be a smooth function satisfying

$$ c|_{\partial M} = 1. $$

Then

$$ \Lambda_{cg} = \Lambda_g + \alpha_n ((\partial_{\nu} c)|_{\partial M}) Id + R_{-1}, $$

where $\alpha_n = -\frac{n-2}{4}$ and $R_{-1} \in \Psi^{-1}(\partial M)$.
APPENDIX C. COHOMOLOGY OF A SPHERE BUNDLE

In this appendix, we show how to use the algebraic tool called Gysin sequences (see [BT82, Chapter 14, p.177-179]), mentioned in Section III and describing the cohomology of a sphere bundle, to prove an important argument used in the proof of **Theorem III.4**. As explained, the argument is actually known and used, for example, by G. Paternain in [Pat09] but, to the author's knowledge, the proof is never really written explicitly in the literature. For convenience, we write this elementary proof here.

In the following, given a smooth manifold $N$, we denote $H_{\mathrm{dR}}^k(N)$ its de Rham cohomology group of degree $k$.

**Lemma C.1.** Assume that the closed Riemannian manifold $(N, h)$ of dimension $n$ satisfies one of the following conditions :

• $n = 2$ and the Euler characteristic satisfies $\chi(N) \neq 0$.

• $n \geq 3$.

If $\pi : SN \to N$ denotes the projection, then

$$ \pi^* : H_{\mathrm{dR}}^1(N) \to H_{\mathrm{dR}}^1(SN) $$

is an isomorphism.

*Proof.* If $n = 2$, the Gysin exact sequence (see [BT82, Proposition 14.33]) for the unit circle bundle $SN \to N$ gives :

$$ 0 \to H_{\mathrm{dR}}^1(N) \xrightarrow{\pi^*} H_{\mathrm{dR}}^1(SN) \xrightarrow{\pi_*} H_{\mathrm{dR}}^0(N) = \mathbb{R} \xrightarrow{\wedge e} H_{\mathrm{dR}}^2(N) = \mathbb{R} \to \dots $$

where $\pi_*$, $\wedge e$, $\pi^*$ are respectively integration along the fiber, multiplication by the Euler class and pullback. Therefore we have

$$ \ker \pi^* = \{0\} $$

as well as

$$ \mathrm{ran} \, \pi^* = \ker \pi_* . $$

Besides we have

$$ \dim H_{\mathrm{dR}}^1(SN) = \dim \ker \pi_* + \dim \mathrm{ran} \, \pi_* = \dim \ker \pi_* + \dim \ker(\wedge e) $$

and since the multiplication by the characteristic class

$$ e = \frac{1}{2\pi} [K_h dv_h] $$

has trivial kernel unless $e$ is exact, i.e. unless the Euler characteristic of the surface satisfies

$$ \chi(N) = \frac{1}{2\pi} \int_N K_h dv_h = 0, $$

this implies
$$ \mathrm{ran} \pi^* = \ker \pi_* = H_{\mathrm{dR}}^1(SN) $$

and proves that $\pi^*$ is an isomorphism.

The case $n \ge 3$ is straightforward, as the Gysin exact sequence takes a simpler form, giving in particular :

$$ 0 \to H_{\mathrm{dR}}^1(N) \xrightarrow{\pi^*} H_{\mathrm{dR}}^1(SN) \to 0. $$

□

**Remark C.2.** The first condition in the previous lemma is equivalent to saying that $N$ is not the 2-torus, this point is verified if $N$ is, for example, an Anosov surface. This follows, for example, from [Ghy84, Théorème A], which characterizes Anosov flows on 3-manifolds that are circle bundles. Applied to the unit fiber bundle of a Riemannian surface, it shows, in particular, that the 2-torus cannot carry any Anosov metric.

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