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Tensor tomography on surfaces
Gabriel, Paternain; Salo, Mikko; Gunther, Uhlmann
Gabriel, P., Salo, M., & Gunther, U. (2013). Tensor tomography on surfaces.
Inventiones mathematicae, 193(1), 229-247. https://doi.org/10.1007/s00222-012-
0432-1
2013
TENSOR TOMOGRAPHY ON SURFACES
GABRIEL P. PATERNAIN, MIKKO SALO, AND GUNTHER UHLMANN
Abstract. We show that on simple surfaces the geodesic ray transform acting on
solenoidal symmetric tensor fields of arbitrary order is injective. This solves a long
standing inverse problem in the two-dimensional case.
1. Introduction
Let (M, g) be a compact oriented two-dimensional manifold with smooth boundary.
We consider the geodesic ray transform acting on symmetric m-tensor fields on M.
When the metric is Euclidean and m = 0 this transform reduces to the usual X-ray
transform obtained by integrating functions along straight lines. More generally, given
a symmetric (covariant) m-tensor field f = f
i
1
···i
m
dx
i
1
⊗ · · · ⊗ dx
i
m
on M we define
the corresponding function on SM by
f(x, v) = f
i
1
···i
m
v
i
1
· · · v
i
m
.
Here SM = {(x, v) ∈ T M ; |v| = 1} is the unit circle bundle. Geodesics going from
∂M into M are parametrized by the set ∂
+
(SM) = {(x, v) ∈ SM ; x ∈ ∂M, hv, νi ≤ 0}
where ν is the outer unit normal vector to ∂M. For (x, v) ∈ SM we let t 7→ γ(t, x, v)
be the geodesic starting from x in direction v. We assume that (M, g) is nontrapping,
which means that the time τ(x, v) when the geodesic γ(t, x, v) exits M is finite for each
(x, v) in SM.
The ray transform of f is defined by
If(x, v) =
Z
τ(x,v)
0
f(ϕ
t
(x, v)) dt, (x, v) ∈ ∂
+
(SM),
where ϕ
t
denotes the geodesic flow of the Riemannian metric g acting on SM. If h
is a symmetric (m − 1)-tensor field, its inner derivative d
σ
h is a symmetric m-tensor
field defined by d
σ
h = σ∇h, where σ denotes symmetrization and ∇ is the Levi-Civita
connection. A direct calculation in local coordinates shows that
d
σ
h(x, v) = Xh(x, v),
where X is the geodesic vector field associated with ϕ
t
. If additionally h|
∂M
= 0, then
one clearly has I(d
σ
h) = 0. The ray transform on symmetric m-tensors is said to be
s-injective if these are the only elements in the kernel. The terminology arises from
the fact that any tensor field f may be written uniquely as f = f
s
+ d
σ
h, where f
s
is a
symmetric m-tensor with zero divergence and h is an (m − 1)-tensor with h|
∂M
= 0 (cf.
[17]). The tensor fields f
s
and d
σ
h are called respectively the solenoidal and potential
parts of the tensor f . Saying that I is s-injective is saying precisely that I is injective
on the set of solenoidal tensors.
1
2 G.P. PATERNAIN, M. SALO, AND G. UHLMANN
In this paper we will assume that (M, g) is simple, a notion that naturally arises
in the context of the boundary rigidity problem [8]. We recall that a Riemannian
manifold with boundary is said to be simple if the boundary is strictly convex and
given any point p in M the exponential map exp
p
from exp
−1
p
(M) ⊂ T
p
M onto M is a
diffeomorphism. In particular, a simple manifold is nontrapping.
The next result shows that the ray transform on simple surfaces is s-injective for
tensors of any rank. This settles a long standing question in the two-dimensional case
(cf. [13] and [17, Problem 1.1.2]).
Theorem 1.1. Let (M, g) be a simple 2D manifold and let m ≥ 0. If f is a smooth
symmetric m-tensor field on M which satisfies If = 0, then f = d
σ
h for some smooth
symmetric (m − 1)-tensor field h on M with h|
∂M
= 0. (If m = 0, then f = 0.)
The geodesic ray transform is closely related to the boundary rigidity problem of
determining a metric on a compact Riemannian manifold from its boundary distance
function. See [24, 7] for recent reviews. The case m = 0, that is, the integration of a
function along geodesics, is the linearization of the boundary rigidity problem in a fixed
conformal class. The standard X-ray transform, where one integrates a function along
straight lines, corresponds to the case of the Euclidean metric and is the basis of medical
imaging techniques such as CT and PET. The case of integration along more general
geodesics arises in geophysical imaging in determining the inner structure of the Earth
since the speed of elastic waves generally increases with depth, thus curving the rays
back to the Earth surface. It also arises in ultrasound imaging, where the Riemannian
metric models the anisotropic index of refraction. Uniqueness and stability for the case
m = 0 was shown by Mukhometov [9] on simple surfaces, and Fredholm type inversion
formulas were given in [14].
The case m = 1 corresponds to the geodesic Doppler transform in which one inte-
grates a vector field along geodesics. This transform appears in ultrasound tomography
to detect tumors using blood flow measurements and also in non-invasive industrial
measurements for reconstructing the velocity of a moving fluid. In the case m = 1,
s-injectivity was shown by Anikonov and Romanov [1] and sharp stability estimates
were proven in [23].
The integration of tensors of order two along geodesics arises as the linearization of
the boundary rigidity problem and the linear problem is known as deformation bound-
ary rigidity. Sharafutdinov [20] showed s-injectivity in this case for simple surfaces.
Sharafutdinov’s proof follows the outline of the proof by Pestov and Uhlmann [15] of
the non-linear boundary rigidity problem for simple surfaces and it is certainly more
involved than our proof of Theorem 1.1, which is independent of the solution to the
non-linear problem.
The case of tensor fields of rank four describes the perturbation of travel times of
compressional waves propagating in slightly anisotropic elastic media; see Chapter 7
of [17]. For tensor fields of order higher than two, Sharafutdinov gave explicit recon-
struction formulas for the solenoidal part [17] in the case when the underlying metric
is Euclidean. For results obtained under curvature restrictions see [17, 12] and [19, 2]
where non-convex boundaries are also considered.
TENSOR TOMOGRAPHY ON SURFACES 3
In the absence of curvature assumptions, the problem for higher order tensors has
been considered in [18] on certain surfaces of revolution. For general simple manifolds,
it is shown in [21] that I
∗
I is a pseudodifferential operator of order −1 on a slightly
larger simple manifold, and it is elliptic on solenoidal tensor fields. Here I
∗
denotes
the adjoint of I with respect to natural inner products. Thus one can recover the wave
front set of a distribution solution of If = 0. Using this result, the analysis of [25, 23]
and Theorem 1.1 it is straightforward to derive stability estimates.
We give two proofs of Theorem 1.1. These proofs are partially inspired by the proof
of the Kodaira vanishing theorem in Complex Geometry [4]. The theorem states that if
M is a K¨ahler manifold, K
M
is its canonical line bundle and E is a positive holomorphic
line bundle, then the cohomology groups H
q
(M, K
M
⊗ E) vanish for any q > 0. The
positivity of E means that there exists a Hermitian metric on E such that iF
∇
is a
positive differential form, where F
∇
is the curvature of the canonical connection ∇
induced by the Hermitian metric. Via an L
2
energy identity, this positivity of the
curvature implies the vanishing of the relevant harmonic forms. Tensoring with E
will be translated in our a setting as introducing an attenuation given by a suitable
connection into the relevant transport equation. This attenuation will play the role
of ∇ above and a version of the Pestov identity will play the role of the L
2
energy
identity. Of course, this is just an analogy and the technical details are very different
in the two settings, but the analogy is powerful enough to provide the key idea for
dealing with the transport equation. One actually has a choice of different possible
connections as attenuations, and this is what produces the two different proofs. This
approach was already employed in [11], but it was surprising to us to discover that
it could also be successfully used to solve the tensor tomography problem for simple
surfaces. We remark that it is still an open problem to establish Theorem 1.1 when
dim M ≥ 3 and m ≥ 2.
After some preliminaries we provide in Section 3 a new point of view on the Pestov
identity which makes its derivation quite natural. We also explain here why there
is an essential difference between the cases m = 0, 1 and m ≥ 2. In Section 4 we
prove Theorem 1.1 choosing as attenuation a primitive of the area form. We give an
alternative proof in Section 5 choosing as attenuation the Levi-Civita connection. This
alternative proof will also yield a more general result. In order to state it, let I
m
denote
the ray transform acting on symmetric m-tensors and let C
∞
α
(∂
+
(SM)) denote the set
of functions h ∈ C
∞
(∂
+
(SM)) such that the function h
ψ
(x, v) = h(ϕ
−τ(x,−v)
(x, v)) is
smooth on SM. In natural L
2
inner products, the adjoint of I
0
is the operator
I
∗
0
: C
∞
α
(∂
+
(SM)) → C
∞
(M), I
∗
0
h(x) =
Z
S
x
h
ψ
(x, v) dS
x
(v).
Here S
x
= {(x, v) ∈ T M ; |v| = 1} and dS
x
is the volume form on S
x
. For more details
see [15], where it is also proved that I
∗
0
is surjective on any simple manifold.
Theorem 1.2. Let (M, g) be a compact nontrapping surface with strictly convex smooth
boundary. Suppose in addition that I
0
and I
1
are s-injective and that I
∗
0
is surjective.
If f is a smooth symmetric m-tensor field on M, m ≥ 1, which satisfies I
m
f = 0, then
f = d
σ
h for some smooth symmetric (m − 1)-tensor field h on M with h|
∂M
= 0.
4 G.P. PATERNAIN, M. SALO, AND G. UHLMANN
Given this result, it seems natural to conjecture that s-injectivity on tensors should
hold on nontrapping surfaces.
Conjecture 1.3. Let (M, g) be a compact nontrapping surface with strictly convex
boundary. If f is a smooth symmetric m-tensor field on M which satisfies If = 0,
then f = d
σ
h for some smooth symmetric (m − 1)-tensor field h on M with h|
∂M
= 0.
(If m = 0, then f = 0.)
Acknowledgements. M.S. was supported in part by the Academy of Finland, and
G.U. was partly supported by NSF and a Rothschild Distinguished Visiting Fellowship
at the Isaac Newton Institute. The authors would like to express their gratitude to
the Newton Institute and the organizers of the program on Inverse Problems in 2011
where this work was carried out.
2. Preliminaries
Let (M, g) be a compact oriented two dimensional Riemannian manifold with smooth
boundary ∂M. As usual SM will denote the unit circle bundle which is a compact
3-manifold with boundary given by ∂(SM) = {(x, v) ∈ SM : x ∈ ∂M}.
Let X denote the vector field associated with the geodesic flow ϕ
t
. Since M is
assumed oriented there is a circle action on the fibers of SM with infinitesimal generator
V called the vertical vector field. It is possible to complete the pair X, V to a global
frame of T (SM) by considering the vector field X
⊥
:= [X, V ]. There are two additional
structure equations given by X = [V, X
⊥
] and [X, X
⊥
] = −KV where K is the Gaussian
curvature of the surface. Using this frame we can define a Riemannian metric on SM
by declaring {X, X
⊥
, V } to be an orthonormal basis and the volume form of this metric
will be denoted by dΣ
3
. The fact that {X, X
⊥
, V } are orthonormal together with the
commutator formulas implies that the Lie derivative of dΣ
3
along the three vector
fields vanishes, in other words, the three vector fields preserve the volume form dΣ
3
.
We refer the reader to [22] for more details on the assertions in this paragraph.
It will be convenient for later purposes and for the sake of completeness to explicitly
write down the three vector fields locally. We can always choose isothermal coordinates
(x
1
, x
2
) so that the metric can be written as ds
2
= e
2λ
(dx
2
1
+ dx
2
2
) where λ is a smooth
real-valued function of x = (x
1
, x
2
). This gives coordinates (x
1
, x
2
, θ) on SM where
θ is the angle between a unit vector v and ∂/∂x
1
. In these coordinates we may write
V = ∂/∂θ and
X = e
−λ
cos θ
∂
∂x
1
+ sin θ
∂
∂x
2
+
−
∂λ
∂x
1
sin θ +
∂λ
∂x
2
cos θ
∂
∂θ
,
X
⊥
= −e
−λ
− sin θ
∂
∂x
1
+ cos θ
∂
∂x
2
−
∂λ
∂x
1
cos θ +
∂λ
∂x
2
sin θ
∂
∂θ
.
Given functions u, v : SM → C we consider the inner product
(u, v) =
Z
SM
u¯v dΣ
3
.