# An Inverse Problem for Fractional Connection Laplacians

Chun-Kai Kevin Chien

###### Abstract.

Consider a fractional operator $P^{s}$, $0<s<1$, for connection Laplacian
$P:=\nabla^{*}\nabla+A$ on a smooth Hermitian vector bundle over a closed,
connected Riemannian manifold of dimension $n\geq 2$. We show that local
knowledge of the metric, Hermitian bundle, connection, potential, and source-
to-solution map associated with $P^{s}$ determines these structures globally.
This extends a result known for the fractional Laplace-Beltrami operator.

##  0\. Introduction

###  0.1. Background

The classical Calderón problem seeks to recover the coefficient $\gamma$ in
the elliptic equation

$\div(\gamma\grad u)=0$ (0.1)

on an open set $U\subset\RR^{n}$ from its Dirichlet-to-Neumann map $\Lambda$,
which takes prescribed boundary values $u|_{\partial U}=f$ to
$\gamma\partial_{\nu}|_{\partial U}u$ . As outlined in [Uhl09], this problem
is motivated by real-world applications such as electrical impedance
tomography, but the resolution of this problem and its analogues have seen
numerous geometric applications. For example, recovering the metric $g$ up to
a conformal factor from $\Lambda$ associated with

$\Delta_{g}u=0$ (0.2)

instead of (0.1) in two dimensions is a critical part of the proof of boundary
rigidity in [PU05].

One can generalize Calderón’s problem by considering bundle valued operators.
A version of this is found in [Alb+13], where the authors replaced (0.2) with

$\nabla^{*}\nabla+A=0,$ (0.3)

with connection $\nabla$ on a Hermitian bundle over a Riemann surface with
boundary $E\to M$ and potential $A\in L^{\infty}(M;\operatorname{End}(E))$. It
was shown that appropriate boundary Cauchy data fixes $\nabla$ and $A$ up to
unitary isomorphism. This recently was applied by [Bao+19] to the AdS/CFT
correspondence in physics, where this result was used to argue that the bulk
metric is fixed by entanglement entropies of regions on the boundary CFT.

Instead of local operators, one can consider nonlocal operators such as the
fractional Laplacian, whose formulation as a suitably extended Dirichlet-to-
Neumann map was given in the seminal paper [CS07]. The geometric version of
this extension involving the fractional Laplace-Beltrami operator was provided
in [CM11] in the context of asymptotically hyperbolic manifolds, and its
relation to the fractional Yamabe problem was explored in [GQ13]. There has
also been recent interest in fractional Laplacians and related fractional
operators from an inverse problems perspective. In the Euclidean setting,
[GSU20] showed how one recovers the $L^{\infty}$ potential $q$ from a nonlocal
exterior Dirichlet-to-Neumann map associated with

$\displaystyle\begin{cases}(\Delta^{s}+q)u=0\,\text{ on }U,\\\ u=f\,\text{ on
}\RR^{n}\setminus\overline{U}.\end{cases}$ (0.4)

This result was extended for lower regularity potentials in [RS20]. Work of
[CLR20] and [Cov+22] showed similar results for more general local
perturbations of the fractional Laplacian, and [BGU21] considered the related
inverse problem for a “regional”-type nonlocal perturbation. Work of [GLX17]
dealt with the inverse problem associated with (0.4) with $\Delta$ replaced by
elliptic $-\nabla\cdot(A(x)\nabla)$.

In a geometric setting, [Fei+21] recovered the Riemannian metric $g$ from a
fractional source-to-solution map $\mathcal{L}^{\operatorname{frac}}$
associated with

$\Delta_{g}^{s}u=f$ (0.5)

on a closed and connected manifold $M$. Their strategy, which will also be
used in this paper, was to relate source-to-solution data to a local equality
of heat kernels; this then transformed the problem into an inverse problem for
the wave equation, for which there are well-established techniques involving
finite propogation speed and controllability to recover the metric structure.
Work by [QU22] used similar techniques to recover $g$ and additional Clifford
bundle structures from $\mathcal{L}^{\operatorname{frac}}$ associated with a
fractional Dirac operator and also discussed how such fractional operators
arise in a physical setting.

Our work is a generalization of [Fei+21], where we replace (0.5) with a
fractional power of a symmetric Laplace-type operator, $P^{s}$, to be defined
below. The operator $P$ has the form of a lower order perturbation of a bundle
Laplacian, and we show that a suitably-defined source-to-solution map
$\mathcal{L}^{\operatorname{frac}}$ determines the geometric structures on a
Hermitian bundle $E\to M$, or equivalently, the bundle-valued coefficients of
$P$ up to gauge.

###  0.2. Fractional connection Laplacians and source-to-solution maps

Let $(M,g)$ be a smooth, closed Riemannian manifold equipped with a smooth
Hermitian vector bundle $(E,\g{\cdot}{\cdot}_{E})$. Recall that if $\tilde{E}$
is an arbitrary tensor product $E^{\otimes m}\otimes T^{k,l}M$, then the
appropriate tensor product of $\g{\cdot}{\cdot}_{E}$ and $g$ together
determine a bundle metric $\g{\cdot}{\cdot}_{\tilde{E}}$ on $\tilde{E}$. This
induces an $L^{2}$-product on smooth sections $C^{\infty}(\tilde{E})$ given by

$(u,v)_{L^{2}(\tilde{E})}:=\int_{M}\langle u,v\rangle_{\tilde{E}}\,dV_{g},$

and $L^{2}(M;\tilde{E})$ is the completion of $C^{\infty}(M;\tilde{E})$ under
the induced norm. If $U\subset M$ is an open set, we denote the compactly
supported smooth sections of $\tilde{E}$ inside $U$ by
$C_{0}^{\infty}(U;\tilde{E})$. We will drop the reference to the bundle
$\tilde{E}$ whenever it is contextually clear.

Consider any smooth connection $\nabla:C^{\infty}(M;E)\to
C^{\infty}(M;E\otimes T^{*}M)$ compatible with the Hermitian structure on $E$,
that is $d\g{u}{v}=\g{\nabla u}{v}+\g{u}{\nabla v}$. The Sobolev spaces
$H^{k}(M;E)$ of nonnegative integer order $k$ are given by the completion of
$C^{\infty}(M;E)$ under the norm induced by the inner product

$(u,v)_{H^{k}}:=(u,v)_{L^{2}}+(\nabla u,\nabla
v)_{L^{2}}+\cdots+(\nabla^{k}u,\nabla^{k}v)_{L^{2}}.$

Since $M$ is compact, the spaces $L^{2}$ and $H^{k}$ as defined do not depend
on $g$, $\g{\cdot}{\cdot}_{E}$, or $\nabla$, though the inner products do.

With respect to $({\cdot},\,{\cdot})_{L^{2}}$, the connection $\nabla$ has
formal adjoint $\nabla^{*}$ in the sense that $(\nabla
u,v)_{L^{2}}=(u,\nabla^{*}v)_{L^{2}}$ for $u\in C^{\infty}(M;E)$, $v\in
C^{\infty}(E\otimes T^{*}M)$. Their composition is the bundle Laplacian
$\nabla^{*}\nabla=-\tr_{g}\nabla^{2}=:\Delta^{E}$. By elliptic regularity,
there is a unique self-adjoint extension
$\Delta^{E}:\mathcal{D}(\Delta^{E})\subset L^{2}(M;E)\to L^{2}(M;E)$ with
domain $\mathcal{D}(\Delta^{E})=H^{2}(E)$. Taking a symmetric potential $A\in
L^{\infty}(M;\operatorname{End}(E))$ in the sense that
$({u},\,{Av})_{L^{2}}=({Au},\,{v})_{L^{2}}$, we now consider the (generalized)
connection Laplacian

$P:=\Delta^{E}+A$ (0.6)

which is also self-adjoint with domain
$\mathcal{D}(P)=\mathcal{D}(\Delta^{E})=H^{2}(M;E)$. The spectral theorem for
self-adjoint operators yields an orthogonal decomposition
$L^{2}=\bigoplus_{k=1}^{\infty}V_{k}$ in terms of the spectrum
$-c=\lambda_{1}<\lambda_{2}<\cdots\to+\infty$, $c\geq 0$, of $P$ (which is
discrete), where each $V_{k}:=\operatorname{Ker}(P-\lambda_{k})$ is the finite
dimensional eigenspace for eigenvalue $\lambda_{k}$. In terms of the
$L^{2}$-projectors $\pi_{k}:L^{2}\to V_{k}$, we define:

######  Definition 1.

Let $P$ as in (0.6) be nonnegative (i.e. $\lambda_{1}\geq 0$). The fractional
connection Laplacian $P^{s}:\mathcal{D}(P^{s})\subset L^{2}\to L^{2}$ for
$0<s<1$ is given by

$P^{s}u:=\sum_{k\geq 1}\lambda_{k}^{s}\pi_{k}u$ (0.7)

with domain $\mathcal{D}(P^{s})=H^{2s}:=\\{u\in L^{2}:\sum_{k\geq
1}\lambda_{k}^{2s}\|u\|^{2}_{L^{2}}<\infty\\}$.

Turning our attention to the restriction of $P^{s}$ to
$\mathcal{D}(P^{s})\cap\operatorname{Ker}(P)^{\perp}$ in the case that $P$ has
a zero eigenvalue, we see that $P^{s}$ has inverse

$P^{-s}f:=\sum_{k\geq 1}\lambda_{k}^{-s}\pi_{k}f,\,\hskip 4.0pt0<s<1,$ (0.8)

where the sum is taken over the nonzero eigenvalues. That is, if
$f\in\operatorname{Ker}(P)^{\perp}$, then $u=P^{-s}f$ is the unique solution
to $P^{s}u=f$ on $M$. This motivates the following:

######  Definition 2.

The local fractional source-to-solution map
$\mathcal{L}^{\operatorname{frac}}_{P,U}:C_{0}^{\infty}(U;E)\cap\operatorname{Ker}(P)^{\perp}\to
C^{\infty}(U;E)$ associated with $P^{s}$ over an open set $U\subset M$ is
given by $f\mapsto P^{-s}f|_{U}$.

We now examine how $\mathcal{L}^{\operatorname{frac}}_{P,U}$ behaves under
pullback. Consider smooth, closed, connected Riemannian manifolds equipped
with respective Hermitian vector bundles, compatible connections, and
symmetric potentials:

$\mathcal{M}_{i}:=(M_{i},g_{i},E_{i},\g{\cdot}{\cdot}_{E_{i}},\nabla^{E_{i}},A^{E_{i}}),\,i=1,2.$
(0.9)

######  Definition 3.

Suppose we have a smooth vector bundle isomorphism

${E_{2}}$${E_{1}}$${M_{2}}$${M_{1}}$$\scriptstyle{\Psi}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\psi}$
(0.10)

We say $\Psi:\mathcal{M}_{2}\to\mathcal{M}_{1}$ is a structure-preserving
isomorphism if the following conditions are satisfied:

  1. (1)

The map $\psi$ is an isometry: $g_{2}=\psi^{*}g_{1}$.

  2. (2)

The Hermitian metric
$\g{\cdot}{\cdot}_{E_{2}}=\Psi^{*}\g{\cdot}{\cdot}_{E_{1}}$, where
$\Psi^{*}\g{u_{2}}{\tilde{u}_{2}}_{E_{1}}=\g{u_{1}}{\tilde{u}_{1}}_{E_{1}}$
for $u_{2}=\Psi^{*}u_{1}$, $\tilde{u}_{2}=\Psi^{*}\tilde{u}_{1}$. Note that
the sections $u_{2}\in C^{\infty}(M_{2};E_{2})$ are in bijective
correspondence with $u_{1}\in C^{\infty}(M_{1};E_{1})$ by the pullback
$u_{2}=\Psi^{*}u_{1}:=\inv\Psi\circ u_{1}\circ\psi$.

  3. (3)

The $\g{\cdot}{\cdot}_{E_{2}}$-compatible connection is
$\nabla^{E_{2}}=\Psi^{*}\nabla^{E_{1}}$, with the pullback connection acting
on $u\in C^{\infty}(M_{2};E_{2})$, $\nu\in C^{\infty}(TM_{2})$ by

$(\Psi^{*}\nabla^{E_{2}})_{\nu}u:=\Psi^{*}(\nabla^{E_{1}}_{d\psi(\nu)}({\inv\Psi}^{*}u)).$

  4. (4)

The symmetric potential satisfies
$A^{E_{2}}=\Psi^{*}A^{E_{1}}:=\Psi^{*}A^{E_{1}}{\inv\Psi}^{*}$.

We note that a structure-preserving isomorphism naturally arises from any
diffeomorphism $\psi:M_{2}\to M_{1}$ by defining $g_{2}:=\psi^{*}g_{1}$,
$E_{2}:=\psi^{*}E_{1}$, and pulling back the remaining structures using
$\Psi$; in this case, the bundle isomorphism $\Psi$ restricts to the identity
on each fibre.

Letting $P_{i}:=\Delta^{E_{i}}+A^{E_{i}}$ for $i=1,2$, we find
$P_{2}(\Psi^{*}u)=\Psi^{*}(P_{1}u)$ for $u\in C^{\infty}(M_{1};E_{1})$ and
furthermore
$\operatorname{Ker}(P_{2}-\lambda_{k})=\Psi^{*}\operatorname{Ker}(P_{1}-\lambda_{k})$
by looking at pullbacks of $L^{2}$-eigensections. For sections not in
$\operatorname{Ker}(P_{1})$, we therefore find
$\Psi^{*}(P_{1}^{-s}u)=P_{2}^{-s}(\Psi^{*}u)$. Moreover, restricting to $f\in
C_{0}^{\infty}(U_{1};E_{1})\cap\operatorname{Ker}(P_{1})^{\perp}$, setting
$U_{2}=\inv\psi(U_{1})$ and letting
$\mathcal{L}_{i}^{\operatorname{frac}}=\mathcal{L}^{\operatorname{frac}}_{P_{i},U_{i}}$
for $i=1,2$, we have

$\Psi^{*}\mathcal{L}_{1}^{\operatorname{frac}}f=\mathcal{L}_{2}^{\operatorname{frac}}\Psi^{*}f.$
(0.11)

###  0.3. Main result and outline

Our discussion above implies that (0.11) holds whenever
$\Psi:\mathcal{M}_{2}\to\mathcal{M}_{1}$ is a structure-preserving
isomorphism. This motivates our main theorem, which is the following rigidity
result:

######  Theorem 0.1.

Let $\mathcal{M}_{1},\mathcal{M}_{2}$ be smooth, closed, connected Riemannian
manifolds of dimension $n\geq 2$ equipped with respective Hermitian vector
bundles, compatible connections, and symmetric potentials as in (0.9) such
that $P_{1},P_{2}\geq 0$. For $i=1,2$, consider open sets $U_{i}\subset M_{i}$
such that there is a structure-preserving isomorphism
$\mathcal{M}_{1}|_{U_{1}}\cong\mathcal{M}_{2}|_{U_{2}}$:

${E_{2}}$${E_{1}}$${U_{2}}$${U_{1}}$$\scriptstyle{\tilde{\Psi}}$$\scriptstyle{\pi_{2}}$$\scriptstyle{\pi_{1}}$$\scriptstyle{\tilde{\psi}}$
(0.12)

If
$\tilde{\Psi}^{*}\mathcal{L}_{1}^{\operatorname{frac}}=\mathcal{L}_{2}^{\operatorname{frac}}\tilde{\Psi}^{*}$
on $C_{0}^{\infty}(U_{1};E_{1})\cap\operatorname{Ker}(P_{1})^{\perp}$, then
for any open set $\mathcal{O}_{2}\Subset U_{2}$ there exists a global
structure-preserving isomorphism $\Psi:\mathcal{M}_{2}\to\mathcal{M}_{1}$ with
$\Psi|_{\mathcal{O}_{2}}=\tilde{\Psi}|_{\mathcal{O}_{2}}$ and
$\psi|_{\mathcal{O}_{2}}=\tilde{\psi}|_{\mathcal{O}_{2}}$.

Our result generalizes that of [Fei+21], which concerned the trivial line
bundle $\CC\to M$ with Euclidean connection and $A=0$ so that $P$ is the
Laplace-Beltrami operator. As any Laplace-type operator on $E\to M$ is locally
of form

$P=-g^{ij}\operatorname{Id}_{E}\partial_{i}\partial_{j}+b^{j}\partial_{j}+c,\,\text{
for }b^{j},c\in C^{\infty}(U;\operatorname{End}(E)),$ (0.13)

our result can also be interpreted as recovering the coefficients of (0.13) up
to gauge for symmetric Laplace-type operators from local fractional source-to-
solution data.

The outline of our proof—and this paper—is as follows. In Section One, we
adapt the argument used in [Fei+21] to show that
$\mathcal{L}_{P,U}^{\operatorname{frac}}$ determines the heat kernel of $P$ on
$U$. By a variant of the Kannai transmutation formula, such a determination
then yields the local source-to-solution map for a linear wave equation for
$P$. We review propagation and unique continuation results for the wave
equation in Section Two as well as recall some useful properties of cut times
on Riemannian manifolds. Section Three proves our main theorem in two parts.
The first uses the argument in [Hel+18] to recover the Riemannian structure on
$M$, while the second adapts the argument in [KOP18] to recover the remainder
of the structure-preserving isomorphism.

### Acknowledgements

The author would like to thank Gunther Uhlmann for suggesting this problem and
his support throughout; and Hadrian Quan for helpful references.

##  1\. Transformation into a linear wave inverse problem

###  1.1. Local identification of the heat kernel

On $\mathcal{M}$ and associated $P$, we start with an alternative expression
for (0.8) in terms of the Gamma function

$\Gamma(s)=\int_{0}^{\infty}t^{s-1}e^{-t}\,dt$

via spectral mapping. A substitution $t\mapsto t/\lambda$ yields

$\lambda^{-s}=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}e^{-t\lambda}\,dt,$

and therefore

$P^{-s}u=\frac{1}{\Gamma(s)}\int_{0}^{\infty}t^{s-1}e^{-tP}u\,dt,\,\hskip
4.0pt0<s<1,$ (1.1)

where $e^{-tP}$ is the heat semigroup generated by $P$. Recall that for $f\in
L^{2}(M;E)$, the heat semigroup gives the smooth solution to the heat equation

$\displaystyle\begin{cases}(\partial_{t}+P)u(t,x)=0\,\text{ on
}(0,\infty)\times M,\\\ u|_{t=0}=f.\end{cases}$ (1.2)

by setting $u(t,x):=e^{-tP}f$. Furthermore, its kernel satisfies the
following:

######  Lemma 1.1.

On $\mathcal{M}$, the heat kernel $e^{-tP}(x,y)$ satisfies the pointwise bound

$|e^{-tP}(x,y)|<Ct^{-(n+1)}e^{-d_{g}(x,y)^{2}/t}\,\text{ for }t<1$ (1.3)

and the operator bound

$\|e^{-tP}\|_{L^{1}\to L^{\infty}}\leq Ct^{-n/2}\,\text{ for }t>0.$ (1.4)

###### Proof.

The first inequality can be found in [Lud19], while the second follows from
Sobolev embedding and [Var85]. ∎

These bounds are crucial in showing the following:

######  Proposition 1.2.

Let $E\to M$ be a smooth vector bundle over a closed manifold, and consider
geometric structures $(g_{i},\g{\cdot}{\cdot}_{i},\nabla_{i},A_{i})$, $i=1,2$,
on this bundle and manifold that coincide over an open set $U\subset M$.
Consider the local source-to-solutions maps
$\mathcal{L}^{\operatorname{frac}}_{i}:=\mathcal{L}^{\operatorname{frac}}_{P_{i},U}$
associated with these structures. If
$\mathcal{L}^{\operatorname{frac}}_{1}=\mathcal{L}^{\operatorname{frac}}_{2}$,
then we have the local equality of heat kernels

$e^{-tP_{1}}(x,y)=e^{-tP_{2}}(x,y)\,\text{ for }(t,x,y)\in(0,\infty)\times
U\times U.$ (1.5)

###### Proof.

For integer $k>0$, we have $P^{k}f:=P_{1}^{k}f=P_{2}^{k}f\in
C_{0}^{\infty}(U;E)$ for $f\in C_{0}^{\infty}(U;E)$, hence
$\mathcal{L}^{\operatorname{frac}}_{1}P^{k}f(x)=\mathcal{L}^{\operatorname{frac}}_{2}P^{k}f(x)$
for $x\in\Omega_{1}$. Consider open sets $\Omega_{1},\Omega_{2}\Subset U$
whose closures are disjoint, and choose $f\in C_{0}^{\infty}(\Omega_{1};E)$.
Expressing the fractional power by (1.1), we find for $x\in U$

$\int_{0}^{\infty}t^{s-1}(e^{-tP_{1}}-e^{-tP_{2}})P^{k}f(x)\,dt=0.$ (1.6)

Since $e^{-tP_{i}}$ is generated by $P_{i}$ and $e^{-tP_{i}}f$ is smooth, we
have

$e^{-tP_{i}}P^{k}f=P^{k}e^{-tP_{i}}f=(-\partial_{t})^{k}e^{-tP_{i}}f,$ (1.7)

on $U$, and (1.6) implies

$\int_{0}^{\infty}t^{s-1}\partial_{t}^{k}(e^{-tP_{1}}-e^{-tP_{2}})f(x)\,dt=0.$
(1.8)

Consider any point $x\in\Omega_{2}$. We proceed to integrate (1.8) by parts
$k$ times to find

$\int_{0}^{\infty}t^{-(k+1)}\frac{(e^{-tP_{1}}-e^{-tP_{2}})f(x)}{t^{-s}}\,dt=0.$
(1.9)

This follows because the boundary terms in the integration by parts disappear,
as seen from using the heat kernel to write

$(-\partial_{t})^{k}(e^{-tP_{1}}-e^{-tP_{2}})f(x)=\int_{\Omega_{1}}(e^{-tP_{1}}(x,y)-e^{-tP_{2}}(x,y))P^{k}f(y)\,dV(y).$
(1.10)

and observing the decay provided by Lemma 1.1: decay at zero is given by (1.3)
and $d_{g}(\Omega_{1},\Omega_{2})>0$, and decay at infinity is given by (1.4).

Substituting $t\mapsto 1/t$ in the above and re-indexing $k$ shows that

$\int_{0}^{\infty}t^{k}\frac{(e^{-\frac{1}{t}P_{1}}-e^{-\frac{1}{t}P_{2}})f(x)}{t^{s}}\,dt=0$
(1.11)

on $\Omega_{2}$ for all $k\geq 0$. Let

$\varphi(t):=\chi_{(0,\infty)}\frac{(e^{-\frac{1}{t}P_{1}}-e^{-\frac{1}{t}P_{2}})f(x)}{t^{s}}.$

The exponential bound in (1.3) implies that the Fourier transform
$\widehat{\varphi}(\xi)$ is holomorphic for $\operatorname{Im}(\xi)<c$, $c>0$,
and (1.11) implies that $\widehat{\varphi}(\xi)$ vanishes to infinite order at
$\xi=0$. We conclude that $(e^{-tP_{1}}-e^{-tP_{2}})f(x)=0$ for $t>0$ and
$x\in\Omega_{2}$. In fact, by unique continuation of the heat equation (1.2),
we have $(e^{-tP_{1}}-e^{-tP_{2}})f(x)\equiv 0$ on all of $(0,\infty)\times U$
(see [Lin90]). Since this holds for any $\Omega_{1}$ and $f\in
C_{0}^{\infty}(\Omega_{1};E)$, we see that $e^{-tP_{1}}(x,y)=e^{-tP_{2}}(x,y)$
on $(0,\infty)\times U$. ∎

We adapt this proposition to the setting of Theorem 0.1 by pulling back
$P_{1}$ and the kernel $e^{-tP_{1}}(x,y)$ locally on $U_{2}$ to find:

######  Proposition 1.3.

Let $\mathcal{M}_{1},\mathcal{M}_{2}$ satisfy the assumptions of Theorem 0.1
with local structure-preserving isomorphism $\tilde{\Psi}$ as in (0.12). If
$\tilde{\Psi}^{*}\mathcal{L}_{1}^{\operatorname{frac}}=\mathcal{L}_{2}^{\operatorname{frac}}\tilde{\Psi}^{*}$
on $C_{0}^{\infty}(U_{1};E_{1})\cap\operatorname{Ker}(P_{1})^{\perp}$, then we
have the local equality of heat kernels

$\tilde{\Psi}^{*}e^{-tP_{1}}(x,y)=e^{-tP_{2}}(x,y)\,\text{ for
}(t,x,y)\in(0,\infty)\times U_{2}\times U_{2}.$ (1.12)

###  1.2. The wave source-to-solution map

Having shown that $\mathcal{L}^{\operatorname{frac}}_{P,U}$ determines the
heat kernel of $P$ locally, we now show that knowledge of the heat kernel
gives the source-to-solution map for a wave equation. Namely, consider

$\displaystyle\begin{cases}(\partial_{t}^{2}+P)w^{f}=f\,\text{ on
}(0,\infty)\times M,\\\
w^{f}|_{\\{t=0\\}}=\partial_{t}w^{f}|_{\\{t=0\\}}=0.\\\ \end{cases}$ (1.13)

For smooth sections $f\in C^{\infty}((0,\infty)\times M;E)$, the unique
solution to (1.13) is given by

$w^{f}(t,x)=\int_{0}^{t}G(t-s,P)f(s,x)\,ds,$ (1.14)

where $G(s,P)$ is the wave kernel defined via the functional calculus with

$G(s,\lambda):=\sum_{k=0}^{\infty}\frac{s^{2k+1}\lambda^{k}}{(2k+1)!}=\begin{cases}\sin{(s\sqrt{\lambda})}/\sqrt{\lambda}\text{
for }\lambda>0,\\\ \sinh{(s\sqrt{-\lambda})}/\sqrt{-\lambda}\text{ for
}\lambda<0.\end{cases}$ (1.15)

######  Definition 4.

The local wave source-to-solution map
$\mathcal{L}_{P,U}^{wave}:C_{0}^{\infty}((0,\infty)\times U;E)\to
C^{\infty}((0,\infty)\times U;E)$ is given by
$\mathcal{L}_{P,U}^{wave}f:=w^{f}|_{(0,\infty)\times U}$, with $w^{f}$ as in
(1.14).

The link between $\mathcal{L}^{\operatorname{frac}}_{P,U}$ and
$\mathcal{L}_{P,U}^{wave}$ is given by Proposition 1.2 and the following:

######  Proposition 1.4.

Let $E\to M$, $(g_{i},\g{\cdot}{\cdot}_{i},\nabla_{i},A_{i})$ be as in
Proposition 1.2, and let
$\mathcal{L}^{wave}_{i}:=\mathcal{L}_{P_{i},U}^{wave}$ for $U\subset M$.
Equality of the heat kernels $e^{-tP_{1}}(x,y)=e^{-tP_{2}}(x,y)$ on
$(0,\infty)\times U\times U$ implies that
$\mathcal{L}^{wave}_{1}=\mathcal{L}^{wave}_{2}$.

###### Proof.

By (1.14), it suffices to show that $G(s,P_{1})u(x)=G(s,P_{2})u(x)$ for all
$u\in C_{0}^{\infty}(U;E)$, $x\in U$. To do so, we exploit a version of the
Kannai transmutation formula as used in [Fei+21]. By Theorem 2.1 in [Lud19],
we have

$e^{-tP}u=\frac{1}{(4\pi
t)^{1/2}}\int_{-\infty}^{\infty}e^{-s^{2}/4t}G^{\prime}(s,P)\,ds$

Because $e^{-s^{2}}G^{\prime}(s,\lambda)\to 0$ as $s\to\pm\infty$, we
integrate by parts to find

$e^{-tP}u=\frac{1}{4\pi^{1/2}t^{3/2}}\int_{0}^{\infty}e^{-s/4t}G(s,P)u\,ds.$

For smooth sections $u\in C_{0}^{\infty}(U;E)$, we have
$e^{-tP_{1}}u=e^{-tP_{2}}u$ on $U$ by assumption. Therefore, using the Laplace
transform $\mathcal{L}$, we find on $U$:

$\mathcal{L}(G(\cdot,P_{1})u)(1/4t)=\int_{0}^{\infty}e^{-s/4t}G(s,P_{1})u\,ds=\int_{0}^{\infty}e^{-s/4t}G(s,P_{2})u\,ds=\mathcal{L}(G(\cdot,P_{2})u)(1/4t).$

Taking the inverse Laplace transform gives $G(s,P_{1})u(x)=G(s,P_{2})u(x)$ for
$s>0$, $x\in U$. ∎

Once again, computing with the local pullback $\tilde{\Psi}^{*}$ yields the
following:

######  Proposition 1.5.

Let $\mathcal{M}_{1}$, $\mathcal{M}_{2}$ satisfy the assumptions of Theorem
0.1 with local structure-preserving isomorphism $\tilde{\Psi}$ as in (0.12).
Consider $\mathcal{L}^{wave}_{i}:=\mathcal{L}_{P_{i},U}^{wave}$ for $i=1,2$.
If the local equality of heat kernels (1.12) holds, then
$\tilde{\Psi}^{*}\mathcal{L}^{wave}_{1}=\mathcal{L}^{wave}_{2}\tilde{\Psi}^{*}$.

##  2\. Wave propagation and cut times

We have now reduced our fractional inverse problem to one involving a linear
wave equation and its associated $\mathcal{L}^{wave}_{P,U}$ for which we have
techniques used in [Hel+18],[KOP18]. We proceed to recount these and other
necessary tools in this section.

###  2.1. Unique continuation and approximate controllability

The results in this subsection hold for all complete manifolds $M$ and are the
main tools we use to recover geometric structure from the Dirichlet-to-Neumann
data.

The first fact we will use carries over from the scalar setting:

######  Proposition 2.1.

(Finite speed of propagation) Let $f\in L^{2}(\RR\times M;E)$, and suppose $u$
is a solution to

$\displaystyle\begin{cases}(\partial_{t}^{2}+P)u=f\,\text{ on
}(0,\infty)\times M,\\\ f|_{C(T,p)}=0,\\\
u|_{B(p,T)\times\\{t=0\\}}=\partial_{t}u|_{B(p,T)\times\\{t=0\\}}=0.\end{cases}$
(2.1)

Then $u\equiv 0$ on $C(T,p)$.

Consider the “diamond set” $C(T,U):=\\{(t,p)\in(0,2T)\times
M:d_{g}(p,U)<\min(t,2T-t)\\}$ containing $U\subset M$. We also denote
$M(T,U):=\\{p\in M:d_{g}(p,U)\leq T\\}$. Because $P$ is principally scalar,
the following is due to the Carleman estimates of [Ell+02] and the method of
proof outlined in [KKL01], Section 2.5:

######  Proposition 2.2.

(Unique continuation) Suppose $U\subset M$ is a bounded open set. If $u\in
C_{0}^{\infty}(\RR\times M;E)$ solves $(\partial_{t}^{2}+P)u=0$ in
$(0,2T)\times M(T,U)$ and $u|_{(0,2T)\times U}\equiv 0$, then $u\equiv 0$ on
$C(T,U)$.

These two propositions above combine to yield two important lemmas.

######  Lemma 2.3.

(Approximate controllability) Let $U\subset M$ be a bounded open set. For any
$T>\eps>0$, the set of solutions $\mathcal{W}_{T}:=\\{w^{f}(T,\cdot):f\in
C_{0}^{\infty}((T-\eps,T)\times U;E),\,w^{f}\text{ solves \eqref{eq_wave}}\\}$
is dense in $L^{2}(M(\eps,U);E)$.

###### Proof.

We show that $\mathcal{W}_{T}^{\perp}=\\{0\\}$ in $L^{2}(M(\eps,U);E)$, from
which the claim follows. Let $\phi\in\mathcal{W}_{T}^{\perp}\subset
L^{2}(M(\eps,U);E)$ and $u\in C^{\infty}((0,\infty)\times M)$ be the solution
in $(0,T)\times M$ to

$\displaystyle\begin{cases}(\partial_{t}^{2}+P)u=0\\\
u|_{t=T}=0,\partial_{t}u|_{t=T}=\phi.\end{cases}$ (2.2)

For any $\psi\in\mathcal{C}_{0}^{\infty}((T-\eps,T)\times U;E)$, we have

$(u,\psi)_{L^{2}((0,T)\times
M;E)}=(u,(\partial_{t}^{2}+P)w^{\psi})_{L^{2}((0,T)\times
M;E)}=((\partial_{t}^{2}+P)u,w^{\psi})_{L^{2}((0,T)\times M;E)}=0,$

where the boundary terms at $t=0,T$ vanish in part due to
$(\phi,w^{\psi}(T,\cdot))_{L^{2}}=0$ by assumption. (In the case that $M$ is
not closed but merely complete, we further use Proposition 2.1 in the above
calculation.) By density, we conclude $u=0$ on $(T-\eps,T]\times U$.

Using the odd reflection $-u(2T-t,x)$, we also construct another solution to
(2.2) on $(T,2T)\times M$ with the same initial conditions at $t=T$. These two
solutions combine to give a solution $\tilde{u}$ to (2.2) on $(0,2T)\times M$
satisfying $\tilde{u}=0$ on $(T-\eps,T+\eps)\times U$. After translating in
time, Proposition 2.2 implies $\tilde{u}=0$ on

$\\{(t,p)\in(T-\eps,T+\eps)\times M:d_{g}(p,U)<\min(t-(T-\eps),(T+\eps)-t)\\}$

and in particular $\partial_{t}\tilde{u}|_{t=T}=\phi|_{M(\eps,U)}=0$ as
needed. ∎

We now define $\mathcal{F}(T,U):=\\{f\in C_{0}^{\infty}(\RR\times
M;E):supp(f)\subset(0,T)\times U\\}$ and consider the time averaging operator
$J\phi(t):=\frac{1}{2}\int_{t}^{2T-t}\phi(s)\,ds$.

######  Lemma 2.4.

(Blagovestchenskii identity) Let $U\subset M$ be a bounded open set and $T>0$.
For any $f,h\in\mathcal{F}(2T,U)$, we have:

$({w^{f}(T,\cdot)},\,{w^{h}(T,\cdot)})_{L^{2}}=({f},\,{(J\mathcal{L}_{P,U}^{wave}-{\mathcal{L}_{P,U}^{wave}}^{*}_{U}J)h})_{L^{2}}$

###### Proof.

Observe that
$({\partial_{t}^{2}w^{f}(t,\cdot)},\,{w^{h}(s,\cdot)})_{L^{2}}=({f(t,\cdot)},\,{\mathcal{L}_{P,U}^{wave}h(s,\cdot)})_{L^{2}}-({Pw^{f}(t,\cdot)},\,{w^{h}(s,\cdot)})_{L^{2}}$
because

$supp(f(t,\cdot))\subset U.$

Consider $W\in C^{\infty}((0,\infty)^{2};\RR)$ given by
$W(t,s):=({w^{f}(t,\cdot)},\,{w^{h}(s,\cdot)})_{L^{2}}$ and
$F(t,s):=(\partial_{t}^{2}-\partial_{s}^{2})W(t,s)$. By our above observation,
we find that $W$ solves the wave equation

$\displaystyle\begin{cases}(\partial_{t}^{2}-\partial_{s}^{2})W=F\,\text{ on
}(0,2T)\times(0,\infty),\\\ W|_{\\{t=0\\}}=\partial_{t}W|_{\\{t=0\\}}=0,\\\
\end{cases}$

hence is given by

$W(t,s)=\frac{1}{2}\int_{0}^{t}\int_{\tau}^{2T-\tau}F(\tau,y)\,dyd\tau.$

Changing variables yields
$W(T,T)=({f},\,{J\mathcal{L}_{P,U}^{wave}h})_{L^{2}}-({\mathcal{L}_{P,U}^{wave}f},\,{Jh})_{L^{2}}$
as needed. ∎

###  2.2. Cut times and geodesically transported neighborhoods

We review some basic properties of cut times which will be of use in metric
reconstruction, and we further construct special neighborhoods that are
adapted for our proof of bundle reconstruction. In this subsection, $(M,g)$ is
a complete, connected, smooth Riemannian manifold.

Recall that the cut time $t^{*}:SM\to(0,\infty]$ is a continuous function on
the unit tangent bundle given by
$t^{*}(x,v):=\inf\\{t>0:\gamma_{x,v}([0,t])\text{ is length minimizing}\\}$,
where $\gamma_{x,v}$ is the unit speed geodesic with initial point and
direction $(x,v)$. The following lemma finds $t^{*}(x,v)$ in terms of
containment of balls:

######  Lemma 2.5.

Let $x,y\in M$ and $s:=d_{g}(x,y)>0$. Suppose $\gamma_{x,v}$ is a geodesic
starting at $x$ whose restriction to the interval $[0,s]$ is the length
minimizer joining $x$ to $y$. Let $\mathcal{I}(x,y)$ be the collection of
times where the following containment is possible:

$\displaystyle\mathcal{I}(x,y):=\\{r>s:\text{ there exists }\veps(r)>0\text{
such that }B(y,r-s+\veps(r))\subset B(x,r)\\}.$

Then the cut time $t^{*}(x,v)=\inf\mathcal{I}(x,y)$

###### Proof.

Let $r\in\mathcal{I}(x,y)$ and $z:=\gamma_{x,v}(r+\delta)$ for any
$\delta<\veps(r)$. Since $d_{g}(y,z)\leq\ell(\gamma_{x,v}([s,r+\delta])$, we
find $z\in B(y,r-s+\veps(r))\subset B(x,r)$, hence $\gamma_{x,v}$ is not
length minimizing from $x$ to $z$. Therefore, $t^{*}(x,v)<r+\delta$, and we
conclude $t^{*}(x,v)\leq\inf\mathcal{I}(x,y)$. Conversely, let
$r^{\prime}=t^{*}(x,v)+\delta$. In order to show
$r^{\prime}\in\mathcal{I}(x,y)$, it suffices to show $\partial
B(y,r^{\prime}-s)\subset B(x,r^{\prime})$. Let $z^{\prime}\in\partial
B(y,r^{\prime}-s)$, so $d_{g}(x,z^{\prime})\leq r^{\prime}$. Observe that if
equality is achieved, then $z^{\prime}=\gamma_{x,v}(r^{\prime})$ because any
other broken geodesic passing through $y$ can be shortened by smoothing.
However, this contradicts $r^{\prime}>t^{*}(x,v)$, so we must have
$d_{g}(x,z^{\prime})<r^{\prime}$, thereby concluding the proof. ∎

The cut locus of $x$ is given by
$\operatorname{Cut}(x):=\\{q=\gamma_{x,v}(t^{*}(x,v))$, and the injectivity
domain of the exponential map at $x$ is given by
$\mathcal{D}^{\mathrm{inj}}(x):=\\{sv\in T_{x}M:s<t^{*}(x,v)\\}\subset
T_{x}M$. A proof of the following can be found in [Lee18]:

######  Lemma 2.6.

For any $x\in M$, we have the following:

  1. (a)

$\operatorname{Cut}(x)\subset M$ has measure zero.

  2. (b)

The restricted exponential map $\exp_{x}:\mathcal{D}^{\mathrm{inj}}(x)\to
M\setminus\operatorname{Cut}(x)$ is a diffeomorphism.

For points within $\operatorname{Cut}(x)$, we construct the following pairs of
open sets that shrink to points as $k\to\infty$ by transporting polar
neighborhoods of $x$ along distance minimizing geodesic segments.

######  Lemma 2.7.

Suppose $y=\gamma_{x,v}(s)$ for some $0<s<t^{*}(x,v)$, and let $\pi:E\to M$ be
a smooth vector bundle. For sufficiently large $k>k^{*}>0$, there are
neighborhoods $(X_{k})_{k\geq k^{*}},(Y_{k})_{k\geq k^{*}}$ of $x,y$
respectively such that

  1. (1)

$\lim_{k\to\infty}\operatorname{Diam}(X_{k})=0=\lim_{k\to\infty}\operatorname{Diam}(Y_{k})$;

  2. (2)

$d_{g}(X_{k},Y_{k})<s+1/k$;

  3. (3)

Both $E|_{X_{k}}$ and $E|_{Y_{k}}$ are trivial.

###### Proof.

By extending the geodesic segment $\gamma_{x,v}([0,s])$ past $x$, we can find
$(x^{\prime},v^{\prime})\in SM$ such that
$x=\gamma_{x^{\prime},v^{\prime}}(r^{\prime})$ and
$y=\gamma_{x^{\prime},v^{\prime}}(s+r^{\prime})$ with
$s+r^{\prime}<t^{*}(x^{\prime},v^{\prime})=t^{*}(x,v)$. Taking $r^{\prime}>0$
small enough so that $x$ is contained within a polar normal chart
$(\mathcal{O}^{\prime},(r,\theta^{j}))$ centered at $x^{\prime}$, we see
$x=(r^{\prime},\theta^{j})$ in these polar normal coordinates. By continuity
of $t^{*}$, we see that for small enough $\eps,\delta>0$,

$\mathcal{U}(x,\eps,\delta):=(r^{\prime}-\eps,r^{\prime}+\eps)\times(\theta^{j}-\delta,\theta^{j}+\delta)\subset\mathcal{O}^{\prime}$
(2.3)

is a neighborhood of $x$ such that

$\mathcal{V}(y,\eps,\delta):=\exp_{x^{\prime}}(s\mathcal{U}(x,\eps,\delta))\subset
M\setminus\operatorname{Cut}(x^{\prime}),$ (2.4)

where we regarded $\mathcal{U}(x,\eps,\delta)$ as a subset of
$T_{x^{\prime}}M$ in (2.4). Observe that with this construction, we can also
turn $\mathcal{V}(y,\eps,\delta)$ into a coordinate chart by using the same
polar coordinates associated with $\mathcal{O}^{\prime}$ on
$\mathcal{U}(x,\eps,\delta)$ and translating the radial coordinate by $s$.
This corresponds to how every point in $\mathcal{V}(y,\eps,\delta)$ is joined
to a unique point in $\mathcal{U}(x,\eps,\delta)$ via a length $s$ segment of
a radial geodesic emanating from $x^{\prime}$. Moreover, both (2.3) and (2.4)
are contractible, hence any vector bundle over those respective sets are
trivial.

Choosing $k>0$ large enough such that $1/k=\eps=\delta$ satisfies the above
allows us to construct $X_{k}$ using (2.3) and $Y_{k}$ using
$\eqref{eq_shrinky}$. ∎

##  3\. Proof of Theorem

###  3.1. Recovery of the metric

We adapt the method of [Hel+18] to our bundle-valued setting to determine the
metric $g$. Let $S(x,\veps,\tau):=(T-\tau,T)\times B(x,\veps)$. Due to finite
speed of propagation and approximate controllability of the wave equation, we
have the following characterization of containment of balls using solutions to
(1.13):

######  Lemma 3.1.

For $x,y,z\in M$ and $\tau_{x},\tau_{y},\tau_{z}>\delta>0$, the following
conditions are equivalent:

  1. (1)

$B(x,\tau_{x})\subsetneq{B(y,\tau_{y})\cup B(z,\tau_{z})}$

  2. (2)

For all $f\in C_{0}^{\infty}(S(x,\delta,\tau_{x}-\delta);E)$, there is a
sequence $(f_{j})_{j=1}^{\infty}\subset
C_{0}^{\infty}(S(y,\delta,\tau_{y}-\delta)\cup S(z,\delta,\tau_{z}-\delta);E)$
whose respective solutions to (1.13) approximate the solution for $f$, that
is:

$\lim_{j\to\infty}\|w^{f}(T,\cdot)-w^{f_{j}}(T,\cdot)\|_{L^{2}(M;E)}=0$

###### Proof.

For the forward implication, choose $f\in
C_{0}^{\infty}(S(x,\delta,\tau_{x}-\delta);E)$, so in particular the
restriction to backward cones $f|_{C(T,p)}=0$ for all points satisfying
$d_{g}(p,B(x,\delta))\geq\tau_{x}-\delta$. Lemma 2.1 then yields
$supp(w^{f}(T,\cdot))\subset B(x,\tau_{x})\subset{B(y,\tau_{y})\cup
B(z,\tau_{z})}$, the latter containment following by assumption. Let
$\chi_{y}$ be the characteristic function for $B(y,\tau_{y})$, and rewrite
$w^{f}(T,\cdot)=\chi_{y}w^{f}(T,\cdot)+(1-\chi_{y})w^{f}(T,\cdot)$. We see
$\chi_{y}w^{f}(T,\cdot)\in
L^{2}(B(y,\tau_{y});E)=L^{2}(M(\tau_{y}-\delta,B(y,\delta));E)$, hence Lemma
2.3 implies $\chi_{y}w^{f}(T,\cdot)$ can be approximated in
$L^{2}(B(y,\tau_{y});E)$ by $w^{\phi^{(y)}_{j}}(T,\cdot)$ for
$\phi^{(y)}_{j}\in S(y,\delta,\tau_{y}-\delta)$. Similarly, one can find a
suitable sequence approximating $(1-\chi_{y})w^{f}(T,\cdot)$, and the sequence
constructed from their sum gives the result.

Conversely, suppose (1) does not hold and take a nonempty open set $U\subset
B(x,\tau_{x})\setminus{B(y,\tau_{y})\cup B(z,\tau_{z})}$. By Lemma 2.1,
$w^{\tilde{f}}(T,\cdot)=0$ on $U$ for all $\tilde{f}\in
C_{0}^{\infty}(S(y,\delta,\tau_{y}-\delta)\cup
S(z,\delta,\tau_{z}-\delta);E)$, whereas we can find $f\in
C_{0}^{\infty}(S(x,\delta,\tau_{x}-\delta);E)$ such that
$\|w^{f}(T,\cdot)\|_{L^{2}}>0$ by Lemma 2.3. Therefore, (2) also does not
hold.

∎

Our approach requires the restricted distance function $d_{g}|_{U\times U}$.
Observe that while we have knowledge of $(U,g|_{U})$, we do not assume
$U\subset M$ is geodesically convex, nor do we initially have the information
to shrink to such an appropriate subset of $U$. Instead, we use properties of
the wave equation to prove the following:

######  Proposition 3.2.

The topological structure of $U$ and the wave source-to-solution data
$\mathcal{L}_{P,U}^{wave}$ determine the local distance map
$d_{g}(\cdot,\cdot)|_{U\times U}$.

###### Proof.

The topology of $U\subset(M,g)$ is metrizable, so we may fix a compatible
metric on $U$ and consider the metric space $(U,d_{0})$. Let $B_{0}(p,r)$
denote the open metric balls with respect to $d_{0}$. For $x,y\in U$ and
$\eps>0$, consider the sets

$D(x,y,k):=\\{t>0:\text{ there exists }f\in C_{0}^{\infty}((0,\infty)\times
B_{0}(x,1/k))\text{ such that }supp(w^{f}(t,\cdot))\cap
B_{0}(y,1/k)\neq\varnothing\\}$

that can be determined by $\mathcal{L}_{P,U}^{wave}$. Using Lemma 2.3 and
similar reasoning as in the proof of Lemma 3.1, we find
$d_{g}(x,y)=\liminf_{k\to\infty}D(x,y,k)$. ∎

######  Proposition 3.3.

Let $U\subset M$ be a nonempty open set. The restricted Riemannian structure
$(U,g|_{U},d_{g}|_{U\times U})$ and the wave source-to-solution data
$\mathcal{L}_{P,U}^{wave}$ determines on $U$ the family of distance functions

$\mathcal{R}_{U}(M):=\\{d_{g}(p,\cdot)|_{U}:p\in M\\}.$

###### Proof.

We first note that because $U$ is open and nonempty, any point in $M$ can be
reached by a unit speed geodesic emanating from $U$ before its cut time.
Indeed, if this were not true, $U$ would be contained in the cut locus of a
point, which has measure zero, leading to a contradiction. Therefore, it
suffices to show that we can determine $t^{*}$ on $SM|_{U}$ and, further, find
$d(p,\cdot)|_{U}$ for any $p=\gamma_{x,v}(r^{\prime})$ with $(x,v)\in
SM|_{U}$, $r^{\prime}<t^{*}(x,v)$.

To this end, for any $(x,v)\in SM|_{U}$ we choose $s>0$ small enough such that
$\gamma_{x,v}([0,s])\subset U$. Letting $y:=\gamma_{x,v}(s)$, we see
$\mathcal{L}_{P,U}^{wave}$ determines $\mathcal{I}(x,y)$ via Lemma 2.4,
polarization, and Condition 2 of Lemma 3.1. Therefore, we know $t^{*}(x,v)$ by
the characterization of Lemma 2.5.

For $z\in U$, we consider the set

$\displaystyle\mathcal{I}(x,y,z):=\\{r>0:\text{ there exists }\veps(r)>0\text{
such that }B(y,r^{\prime}-s+\veps(r))\subset B(x,r^{\prime})\cup B(z,r)\\}.$

We claim that $d(p,z)=\inf\mathcal{I}(x,y,z)$. Indeed, $d(p,x)=r^{\prime}$, so
for $r\in\mathcal{I}(x,y,z)$, we have $p\in B(z,r)$ since $p\in
B(y,r^{\prime}-s+\veps(r))$; thus, $d(p,z)\leq\inf\mathcal{I}(x,y,z)$. Now,
let $\tilde{r}:=d(p,z)+\delta$ for some $\delta>0$. If
$\tilde{r}\notin\mathcal{I}(x,y,z)$, then by passing to a subsequence we may
find points

$p_{k}\in B(y,r^{\prime}-s+1/k)\setminus(B(x,r^{\prime})\cup
B(z,\tilde{r}))\subset\overline{B(y,r^{\prime}-s+1)}$

such that $p_{k}\to p^{\prime}\in\partial B(y,r^{\prime}-s)$. Observe that
$d(x,p^{\prime})\leq r^{\prime}$, with equality achieved if and only if
$p=p^{\prime}$ by a smoothing argument as in the proof of Lemma 2.5. The
equality $p=p^{\prime}$ and the fact $\tilde{r}>d(p,z)$ would contradict the
fact that our limiting sequence stays outside $B(z,\tilde{r})$; however,
$d(x,p^{\prime})<r^{\prime}$ would imply our limiting sequence does not stay
outside $B(x,r^{\prime})$. Therefore, we find $\inf\mathcal{I}(x,y,z)\leq
d(p,z)$ and our claim follows.

Because $\mathcal{I}(x,y,z)$ can be determined from our initial data by
similar methods as those used for $\mathcal{I}(x,y)$ earlier, we are able to
deduce $d(p,z)$ and ultimately determine $\mathcal{R}_{U}(M)$. ∎

######  Proposition 3.4.

Let $\mathcal{M}_{1}$, $\mathcal{M}_{2}$ satisfy the assumptions of Theorem
0.1 with local structure-preserving isomorphism $\tilde{\Psi}$ as in (0.12).
Choose any $\mathcal{O}_{2}\Subset U_{2}$. If
$\tilde{\Psi}^{*}\mathcal{L}^{wave}_{1}=\mathcal{L}^{wave}_{2}\tilde{\Psi}^{*}$,
then there is an isometry $\psi:M_{2}\to M_{1}$ with
$\psi|_{\mathcal{O}_{2}}=\tilde{\psi}|_{\mathcal{O}_{2}}$.

###### Proof.

Choosing $\mathcal{O}_{2}=\tilde{\psi}(\mathcal{O}_{1})$ as necessary, we are
able to determine

$\mathcal{R}_{i}(M_{i}):=\\{d_{g_{i}}(p,\cdot)|_{\overline{\mathcal{O}_{i}}}:p\in
M_{i}\\}\,\text{ for }i=1,2$ (3.1)

by our previous proposition. As shown in [Hel+18], the map

$R_{i}:M_{i}\ni p\mapsto
d_{g_{i}}(p,\cdot)|_{\overline{U}_{i}}\in\mathcal{R}_{i}(M)$

is a diffeomorphism when $\mathcal{R}_{i}(M)$ is equipped with a smooth
structure constructed explicitly using the injectivity domain of points in
$\overline{U}_{i}$, which is known from $\mathcal{L}^{wave}_{i}$ using
Proposition 3.3. Equipping $\mathcal{R}_{i}(M)$ with the pullback of $g_{i}$
using $\inv R_{i}$ makes $R_{i}$ an isometry, and by Proposition 5 in [Hel+18]
the coordinate representation of this metric on $\mathcal{R}_{i}(M)$ is
determined by Proposition 3.3. Because
$\tilde{\Psi}^{*}\mathcal{L}^{wave}_{1}=\mathcal{L}^{wave}_{2}\tilde{\Psi}^{*}$,
the map $\psi_{R}:\mathcal{R}_{1}(M_{1})\to\mathcal{R}_{2}(M_{2})$ defined
using the pullback of distance functions by $\tilde{\psi}$ is an isometry.
Setting $\psi:=\inv R_{1}\inv\psi_{R}R_{2}$ yields a global isometry with the
required properties. ∎

By Propositions 1.3, 1.5, and 3.4, the existence of $\psi$ in Theorem 0.1
follows.

###  3.2. Recovery of the bundle and operator

Recovery of our remaining geometric quantities follows the procedure of
[KOP18] in the setting of closed manifolds.

######  Lemma 3.5.

Let $U_{1},U_{2}\subset M$ be open sets and $\eps_{1},\eps_{2}>0$. Suppose
there exists a sequence $(f_{j})_{k=1}^{\infty}\subset
C_{0}^{\infty}((T-\eps_{1},T)\times U_{1})$ such that:

  1. (1)

The sequence of solutions $(w^{f_{j}}(T,\cdot))_{j=1}^{\infty}$ converges
weakly to $\phi\in L^{2}(M;E)$;

  2. (2)

For all $h\in C_{0}^{\infty}((T-\eps_{2},T)\times U_{2})$, we have
$\lim_{j\to\infty}({w^{f_{j}}(T,\cdot)},\,{w^{h}(T,\cdot)})_{L^{2}}=0$.

Then $supp(\phi)\subset M(\eps_{1},U_{1})\setminus M(\eps_{2},U_{2})^{int}$.

###### Proof.

This follows from the same argument used in the proof of Lemma 3.1. ∎

######  Lemma 3.6.

Consider an open set $U\subset M$. For any $T>0$ and point $p\in M$ such that
$d_{g}(p,U)<T$, there exists a tuple
$(h_{\ell})_{\ell=1}^{rk(E)}\subset\mathcal{F}(2T,U)$ such that
$(w^{h_{\ell}}(T,\cdot))_{\ell=1}^{rk(E)}$ is a smooth local frame for $E$ in
a neighborhood $x\in V_{x}\subset M$ and is orthonormal at $x$.

###### Proof.

It suffices to show $w^{h}(T,x)$, $h\in\mathcal{F}(T,U)$, spans $E_{x}$; our
claims then follow by Gram-Schmidt and smoothness of $w^{h}(T,\cdot)$. To this
end, we show that if $e\in E_{x}$ is such that $\g{e}{w^{h}(T,x)}_{E}=0$, then
$e=0$. If we let $\psi:=e\delta_{x}$ in (2.2), our assertion follows from the
same reasoning as the proof of Lemma 2.3. ∎

The following two lemmas identify certain double sequences of sections with
vectors in the fibre $E_{y}$, where $y$ may not be contained in the support of
the sequence. For convenience, we define $s_{k}:=s+1/k$ where
$\gamma_{x,v}(s)=y$, $s<t^{*}(x,v)$. In the case $s>0$, we construct $X_{k}$,
$Y_{k}$ as in (2.3) and (2.4); otherwise, set $Y_{k}=X_{k}$ as defined in
$\eqref{eq_shrinkx}$. Observe that in either case, $Y_{k}\subset
M(s_{k},X_{k})$.

######  Lemma 3.7.

Let $U\subset M$ be open, and consider $y=\gamma_{x,v}(s)\in M$ for $(x,v)\in
SM|_{U}$, $s<t^{*}(x,v)$. Suppose we have a double sequence of sections
$\Phi^{y}:=(f_{jk})_{j,k=1}^{\infty}$ with $f_{jk}\in
C_{0}^{\infty}((T-s_{k},T)\times X_{k};E)$ such that:

  1. (1)

For each $k$, $(w^{f_{jk}}(T,\cdot))_{j=1}^{\infty}$ converges weakly in
$L^{2}(M;E)$ to a section $u_{k}$ supported in $Y_{k}$;

  2. (2)

There is a uniform $C>0$ such that
$\|w^{f_{jk}}(T,\cdot)\|_{L^{2}}<C(Vol(Y_{k}))^{-1/2}$;

  3. (3)

For any $h\in\mathcal{F}(2T,U)$,
$\lim_{k\to\infty}\lim_{j\to\infty}({w^{jk}(T,\cdot)},\,{w^{h}(T,\cdot)})_{L^{2}}$
exists.

Then there is a vector $e(\Phi^{y})\in E_{y}$ depending on $\Phi^{y}$ such
that for all $\phi\in C^{\infty}(M;E)$,

$\lim_{k\to\infty}\lim_{j\to\infty}({w^{f_{jk}}(T,\cdot)},\,{\phi})_{L^{2}}=\g{e(\Phi^{y})}{\phi}_{E}.$
(3.2)

###### Proof.

By taking $k$ large enough, we may assume without loss of generality that
$X_{k}\subset U$ and $Y_{k}$ is contained within a fixed coordinate chart
$(W,\tilde{y})$ with $\tilde{y}=0$ at $y$. Let
$b_{\ell}(\cdot):=w^{h_{\ell}}(T,\cdot)$ denote the sections of the local
frame for $E$ constructed using Lemma 3.6 which are orthonormal at $E_{y}$.
Any $\phi\in C^{\infty}(M;E)$ can be written locally using a linearization as

$\phi(\tilde{y})=\sum_{\ell=1}^{rk(E)}\g{\phi}{b_{\ell}}_{E}b_{\ell}=c^{\ell}b_{\ell}(\tilde{y})+\tilde{y}^{i}\psi_{i}(\tilde{y})$

where $\tilde{\psi}_{k}\in C^{\infty}(W;E)$, $1\leq i\leq n$, and
$c^{\ell}\in\CC$. Taking the inner product with the weak limit in (1) yields

$({u_{k}},\,{\phi})_{L^{2}}=\overline{c^{\ell}}({u_{k}},\,{b_{\ell}})_{L^{2}}+r_{k}$

where

$\displaystyle|r_{k}|$
$\displaystyle<n\max_{i}\|\psi_{i}\|_{C(W)}\operatorname{Diam}(Y_{k})\int_{Y_{k}}|u_{k}|\,dV$
$\displaystyle<n\max_{i}\|\psi_{i}\|_{C(W)}\operatorname{Diam}(Y_{k})|(Vol(Y_{k}))^{1/2}\|u_{k}\|_{L^{2}}.$

Since $Y_{k}\to y$ by construction, (2) implies $r_{k}\to 0$, and (3) implies
we can set $a^{\ell}:=\lim_{k\to\infty}({u_{k}},\,{b_{\ell}})_{L^{2}}$.
Letting $e_{y}^{S}:=a^{\ell}b_{\ell}(y)$, we find

$\lim_{k\to\infty}\lim_{j\to\infty}({w^{f_{jk}}(T,\cdot)},\,{\phi})_{L^{2}}=\lim_{k\to\infty}({u_{k}},\,{\phi})_{L^{2}}=\sum_{\ell=1}^{rk(E)}\overline{c^{\ell}}a^{\ell}=\g{e(\Phi^{y})}{\phi}_{E}.$

∎

Conversely, we also have:

######  Lemma 3.8.

Let $U\subset M$ be open, and consider $y=\gamma_{x,v}(s)\in M$ for $(x,v)\in
SM|_{U}$, $s<t^{*}(x,v)$. For any $e_{y}\in E_{y}$, there is a double sequence
$\Phi^{y}=(f_{jk})_{j,k=1}^{\infty}$ with $f_{jk}\in
C_{0}^{\infty}((T-s_{k},T)\times X_{k};E)$ satisfying conditions (1)–(3) of
Lemma 3.7, with $e(\Phi^{y})=e_{y}$.

###### Proof.

For any smooth section $\tilde{e}\in C^{\infty}(M;E)$ with
$\tilde{e}(y)=e_{y}$, we construct

$u_{k}\in Vol(Y_{k})^{-1}\chi_{k}\tilde{e}\in L^{2}(M(s_{k},X_{k});E)$

with $\chi_{k}$ the indicator function for $Y_{k}$. Using Lemma 2.3 for each
$k$, we find our double sequence $\Phi^{y}$ to approximate each $u_{k}$
strongly, and Condition (1) follows. Condition (2) is also immediate with
$C=\|\tilde{e}\|_{L^{\infty}}$, and Condition (3) and (3.2) follow by direct
calculation. ∎

######  Proposition 3.9.

Let $\mathcal{M}$ be as in (0.9). If we know the Riemannian structure $(M,g)$,
the local Hermitian bundle $(E,\g{\cdot}{\cdot}_{E})|_{U}$, and
$\mathcal{L}_{P,U}^{wave}$, then we can determine the structures
$(E,\g{\cdot}{\cdot}_{E},\nabla,A)$ globally on $M$.

###### Proof.

Shrinking $U$ as necessary, we assume $U$ is a coordinate ball that
trivializes $E$. Observe that due to compactness of $M$ and Lemma 2.6, we can
cover $E$ with finitely many charts of the form

$\mathcal{A}=({}^{(0)}\mathcal{A},{}^{(1)}\mathcal{A},\ldots,{}^{(N)}\mathcal{A}):=(U,{}^{(1)}\mathcal{V},\ldots,{}^{(N)}\mathcal{V}),$

with ${}^{(j)}\mathcal{V}$ constructed using (2.4). Therefore, $E$ is trivial
over each ${}^{(j)}\mathcal{A}$. For each $y\in^{(j)}\mathcal{A}$, we can use
$\mathcal{L}_{P,U}^{wave}$ to detect sequences $(\Phi^{y}_{\alpha})_{\alpha}$
satisfying the conditions of Lemma 3.7 by Lemmas 2.3, 2.4, and 3.5. We are
able to test for smoothness of $y\mapsto e(\Phi^{y})$ by checking the
smoothness of $y\mapsto\g{e(\Phi^{y})}{w^{h}(T,\cdot)}_{E}$ for
$h\in\mathcal{F}(2T,U)$, per Condition 3 of Lemma 3.7 and Lemma 3.6. Further,
due to the existence of an orthonormal frame $(e_{\ell})_{\ell=1}^{rk(E)}$
over ${}^{(j)}\mathcal{A}$ and Lemma 3.8, we can select for sequences
$(\Phi^{y}_{\ell})_{\ell=1}^{rk(E)}$ such that
$e_{\ell}(y)=e(\Phi^{y}_{\ell})$ by verifying

$\lim_{k\to\infty}\lim_{j\to\infty}({\chi_{{}^{(j)}\mathcal{A}}e_{\kappa}(\Phi^{y})},\,{w^{f^{y}_{jk,\ell}}(T,\cdot)})_{L^{2}}=\g{e_{\kappa}(y)}{e_{\ell}(y)}_{E}=\delta_{\kappa\ell}.$
(3.3)

The Hermitian metric $\g{\cdot}{\cdot}_{E}$ on this trivialization over
${}^{(j)}\mathcal{A}$ is then
$\g{v}{w}_{E}=\delta_{\kappa\ell}a^{\kappa}\overline{c^{\ell}}$ for
$v=a^{\kappa}e_{\kappa}$, $w=c^{\ell}e_{\ell}$. Given a similar orthonormal
frame $(\tilde{e}_{\ell})_{\ell=1}^{rk(E)}$ and associated sequence over
${}^{(k)}\mathcal{A}$, we apply the method of (3.3) to calculate
$\g{e_{\kappa}}{\tilde{e}_{\ell}}_{E}$ to find the transition functions
$\tau_{jk}:{}^{(j)}\mathcal{A}\cap{}^{(k)}\mathcal{A}\to GL(rk(E),\CC)$.

Having determined the Hermitian bundle structure $(E,\g{\cdot}{\cdot}_{E})\to
M$, we now show how we can recover the connection $\nabla$ and potential $A$.
From our reasoning above, we know the components of $w^{h}(T,\cdot)$ for
$h\in\mathcal{F}(2T,U)$ on ${}^{(j)}\mathcal{A}$, and by a time translation we
know the components of $w^{h}(t,\cdot)$ in the same chart, for $t\in(0,T)$ and
$h\in\mathcal{F}(T,U)$. We therefore know the left hand side of

$-({\partial_{t}^{2}w^{h}(T,\cdot)},\,{\phi})_{L^{2}}=({Pw^{h}(T,\cdot)},\,{\phi})_{L^{2}}=({w^{h}(T,\cdot)},\,{P\phi})_{L^{2}}.$
(3.4)

for any $\phi\in C_{0}^{\infty}({}^{(j)}\mathcal{A};E)$, $h\in
C_{0}^{\infty}((0,T)\times U;E)$. By density of $w^{h}(T,\cdot)$ in
$L^{2}({}^{(j)}\mathcal{A};E)$, we deduce $P\phi$ from (3.4).

Since $\nabla=d+B$ for skew-Hermitian $B\in C^{\infty}(M;E\otimes E^{*}\otimes
T^{*}M)$, we have:

$(P-A)\phi=\nabla^{*}\nabla\phi=d^{*}d\phi-2\tr_{g}(B\otimes
d\phi)+(d^{*}B)\phi-\tr_{g}(B\otimes B\phi)$ (3.5)

where the contractions are between the $E^{*}\otimes T^{*}M$ and $E\otimes
T^{*}M$ entries. Locally on ${}^{(j)}\mathcal{A}$, if we choose
$\phi=\phi^{\ell}e_{\ell}$ such that $\phi(0)=0$,
$\partial_{k}\phi^{\ell}(0)=1$, then combined with our knowledge of $g$ we may
calculate $d^{*}d\phi$ and find

$2\tr(B\otimes d\phi)=2g^{ik}e_{\ell}(B_{i})$

from (3.5). This yields $B$ and therefore the connection $\nabla$. Subtracting
$P\phi$ from (3.5) then gives $A$, as needed. ∎

By Propositions 1.3, 1.5, 3.4, and 3.9, we have recovered the full structure-
preserving isomorphism $\Psi$.

## References

  * [Alb+13] Pierre Albin, Colin Guillarmou, Leo Tzou and Gunther Uhlmann  “Inverse boundary problems for systems in two dimensions”  In _Annales Henri Poincaré_ 14.6, 2013, pp. 1551–1571  Springer
  * [Bao+19] Ning Bao, ChunJun Cao, Sebastian Fischetti and Cynthia Keeler  “Towards bulk metric reconstruction from extremal area variations”  In _Classical and Quantum Gravity_ 36.18 IOP Publishing, 2019, pp. 185002
  * [BGU21] Sombuddha Bhattacharyya, Tuhin Ghosh and Gunther Uhlmann  “Inverse problems for the fractional-Laplacian with lower order non-local perturbations”  In _Transactions of the American Mathematical Society_ 374.5, 2021, pp. 3053–3075
  * [CLR20] Mihajlo Cekić, Yi-Hsuan Lin and Angkana Rüland  “The Calderón problem for the fractional Schrödinger equation with drift”  In _Calculus of Variations and Partial Differential Equations_ 59.3 Springer, 2020, pp. 1–46
  * [CM11] Sun-Yung Alice Chang and Maria Mar Gonzalez  “Fractional Laplacian in conformal geometry”  In _Advances in Mathematics_ 226.2 Elsevier, 2011, pp. 1410–1432
  * [Cov+22] Giovanni Covi, Keijo Mönkkönen, Jesse Railo and Gunther Uhlmann  “The higher order fractional Calderón problem for linear local operators: Uniqueness”  In _Advances in Mathematics_ 399 Elsevier, 2022, pp. 108246
  * [CS07] Luis Caffarelli and Luis Silvestre  “An extension problem related to the fractional Laplacian”  In _Communications in partial differential equations_ 32.8 Taylor & Francis, 2007, pp. 1245–1260
  * [Ell+02] Matthias Eller, Victor Isakov, Gen Nakamura and Daniel Tataru  “Uniqueness and stability in the Cauchy problem for Maxwell and elasticity systems”  In _Nonlinear partial differential equations and their applications: Collège de France seminar. Volume XIV_ , 2002, pp. 329–349  Elsevier
  * [Fei+21] Ali Feizmohammadi, Tuhin Ghosh, Katya Krupchyk and Gunther Uhlmann  “Fractional anisotropic Calderón problem on closed Riemannian manifolds”  arXiv, 2021  DOI: 10.48550/ARXIV.2112.03480
  * [GLX17] Tuhin Ghosh, Yi-Hsuan Lin and Jingni Xiao  “The Calderón problem for variable coefficients nonlocal elliptic operators”  In _Communications in Partial Differential Equations_ 42.12 Taylor & Francis, 2017, pp. 1923–1961
  * [GQ13] María del Mar González Nogueras and Jie Qing  “Fractional conformal Laplacians and fractional Yamabe problems”  In _Analysis & PDE_ 6.7 Mathematical Sciences Publishers, 2013, pp. 1535–1576
  * [GSU20] Tuhin Ghosh, Mikko Salo and Gunther Uhlmann  “The Calderón problem for the fractional Schrödinger equation”  In _Analysis & PDE_ 13.2 Mathematical Sciences Publishers, 2020, pp. 455–475
  * [Hel+18] Tapio Helin, Matti Lassas, Lauri Oksanen and Teemu Saksala  “Correlation based passive imaging with a white noise source”  In _Journal de Mathématiques Pures et Appliquées_ 116 Elsevier, 2018, pp. 132–160
  * [KKL01] Alexander Kachalov, Yaroslav Kurylev and Matti Lassas  “Inverse boundary spectral problems”  ChapmanHall/CRC, 2001
  * [KOP18] Yaroslav Kurylev, Lauri Oksanen and Gabriel P Paternain  “Inverse problems for the connection Laplacian”  In _Journal of Differential Geometry_ 110.3 Lehigh University, 2018, pp. 457–494
  * [Lee18] John M Lee  “Introduction to Riemannian manifolds”  Springer, 2018
  * [Lin90] Fang Hua Lin  “A uniqueness theorem for parabolic equations”  In _Communications on Pure and Applied Mathematics_ 43.1 Wiley-Liss Inc., 1990, pp. 127–136
  * [Lud19] Matthias Ludewig  “Strong short-time asymptotics and convolution approximation of the heat kernel”  In _Annals of Global Analysis and Geometry_ 55.2 Springer, 2019, pp. 371–394
  * [PU05] Leonid Pestov and Gunther Uhlmann  “Two dimensional compact simple Riemannian manifolds are boundary distance rigid”  In _Annals of mathematics_ JSTOR, 2005, pp. 1093–1110
  * [QU22] Hadrian Quan and Gunther Uhlmann  “The Calderón problem for the fractional Dirac operator”  arXiv, 2022  DOI: 10.48550/ARXIV.2204.00965
  * [RS20] Angkana Rüland and Mikko Salo  “The fractional Calderón problem: low regularity and stability”  In _Nonlinear Analysis_ 193 Elsevier, 2020, pp. 111529
  * [Uhl09] Gunther Uhlmann  “Electrical impedance tomography and Calderón’s problem”  In _Inverse problems_ 25.12 IOP Publishing, 2009, pp. 123011
  * [Var85] N Th Varopoulos  “Hardy-Littlewood theory for semigroups”  In _Journal of functional analysis_ 63.2 Elsevier, 1985, pp. 240–260