X-RAY TRANSFORM ON DAMEK-RICCI SPACES

Damek-Ricci spaces, also called harmonic NA groups, make up a large class of harmonic Riemannian manifolds including all hyperbolic spaces. We prove here an inversion formula and a support theorem for the X-ray transform, i.e. integration along geodesics, on those spaces. Using suitably chosen totally geodesic submanifolds we reduce the problems to similar questions on low-dimensional hyperbolic spaces.


Introduction
Trying to reconstruct a function on a manifold knowing its integrals over a certain family of submanifolds is one of the main problems of integral geometry.In the framework of Riemannian manifolds a natural choice is the family of all geodesics: the simple example of lines in Euclidean space has suggested naming X-ray transform the corresponding integral operator, associating to a function f its integrals Rf (ξ) along all geodesics ξ of the manifold.
But few explicit formulas are known to recover f from Rf , only valid (to the best of my knowledge) for Riemannian symmetric spaces: see Helgason [6] for Euclidean spaces, hyperbolic spaces or spheres, Helgason [7] or the author's paper [10] for general Riemannian symmetric spaces of the noncompact type.
We prove here an inversion formula and a support theorem for the X-ray transform on Damek-Ricci spaces (also called harmonic N A groups).Introduced in 1992 [3] as a negative answer to a 1944 question by Lichnérowicz: "is a harmonic Riemannian manifold necessarily a symmetric space?", those spaces were actively studied in the 90's.They make up a large class of harmonic Riemannian manifolds including all hyperbolic spaces.
Though the main results in analysis on Damek-Ricci spaces (see [4], [8]) look quite similar to their hyperbolic space analogs, their proofs are more delicate however for lack of the familiar compact isotropy group of the origin in the hyperbolic case.This difficulty arises here too; we circumvent it by reducing our problem to some totally geodesic submanifold, isometric to the complex hyperbolic space H 2 (C).
In Sections 2 to 4 we develop the necessary tools.In Section 2 we recall, with slight change, the inversion formula obtained in [10] for Riemannian symmetric spaces (Theorem 2.1).Section 3 contains a brief summary about Damek-Ricci spaces and a close look at the special case of hyperbolic spaces.Section 4 is devoted to the construction of totally geodesic subgroups of a Damek-Ricci space.Our main results (Theorems 5.1 and 6.1) are proved in Sections 5 and 6.

X-ray transform on symmetric spaces
The X-ray transform f → Rf on a complete Riemannian manifold M is defined by where f is a function on M , ξ is an arbitrary geodesic and dm ξ is the arc length measure on ξ.The integral converges if f is, for instance, continuous and rapidly decreasing with respect to the Riemannian distance on M .
An inversion formula, reconstructing f from Rf , was proved for M = G/K, a Riemannian symmetric space of the noncompact type, by Helgason [7] and independently in [10].Theorem 2.1 below is a restatement of the main result of [10], in a slightly modified form better suited to our needs.
Let us recall first a few classical semisimple notations; see [5] for more details.In this section G denotes a connected noncompact real semisimple Lie group with finite center, K a maximal compact subgroup of G, g = k ⊕ p the corresponding Cartan decomposition of the Lie algebra (eigenspaces of the Cartan involution θ), a a maximal abelian subspace of p, and α a root of the pair (g, a) i.e. a nonzero linear form on a such that the joint eigenspace The Killing form B(X, Y ) = tr (ad X ad Y ) of g gives the invariant scalar product < X, Y >= −B(X, θY ) of X, Y ∈ g and the norm |X| = −B(X, θX).Let H α ∈ a be the dual vector to α defined by B(H, H α ) = α(H) for all H ∈ a; a different normalization of H α was chosen in [10] but this is insignificant.We set |α| = |H α |.
The manifold G/K is equipped with the G-invariant Riemannian metric defined by this scalar product on p (identified to the tangent space to G/K at the origin o).As usual exp : g → G will denote the exponential mapping of the group G and Exp the exponential mapping of G/K at the origin, a global diffeomorphism of p onto G/K.
Let α be a fixed root.Taking ξ 0 = Exp RH α as the origin in the space of geodesics the X-ray transform of a function f on G/K satisfies, for g ∈ G, where dots denote the natural action of G on G/K.We also need the shifted dual transform of a function ϕ on the space of geodesics, defined by where the shift γ is a given element of G and dk is the normalized Haar measure on K. Roughly speaking R * γ integrates ϕ over a set of geodesics at a given distance from the point x ∈ G/K.When γ is the identity element R * γ is the classical dual transform.
Note that the group G does not act transitively on the space of all geodesics unless G/K has rank one.In the next theorem, having chosen ξ 0 = Exp RH α as the origin, we use a shift arising from the Lie subalgebra g α .
Theorem 2.1.Let G/K be a Riemannian symmetric space of the noncompact type and let α be any root of the pair (g, a).Taking ξ 0 = Exp RH α as the origin in the space of geodesics, let R denote the X-ray transform obtained by integrating over geodesics in a family containing all g • ξ 0 , g ∈ G.
For any nonzero X ∈ g α this transform is inverted by The formula now extends to an arbitrary X ∈ g α , since changing t into λt with λ > 0 in the integral amounts to replacing X with λX.

Damek-Ricci spaces
A Damek-Ricci space, or harmonic N A group, is a simply connected Lie group S with Lie algebra s satisfying the following assumptions 1 to 4: 1. Direct sum decomposition.s is a direct sum of vector subspaces s = v ⊕ z ⊕ a, with a one-dimensional.Let H be some fixed nonzero vector in a (see 2 below); elements of s will be written as 3. Scalar product.v and z are equipped with scalar products < , >, extended to a scalar product on s as The group S is equipped with the left-invariant Riemannian metric defined by this scalar product on s.Let .denote the corresponding norm on s; thus H = 1. 4. The maps J Z .For Z ∈ z let J Z : v → v denote the linear map defined by From (3.3) and (3.4) it is easily shown that From assumptions 1 to 3 it follows that n = v⊕z is a nilpotent Lie algebra, [s, s] ⊂ n, and s is solvable with orthogonal decomposition s = v ⊕ z ⊕ a into eigenspaces of ad H with respective eigenvalues 1/2, 1, 0. At the group level we have S = N A, a semi-direct product, where N and A are the Lie subgroups of S with Lie algebras n and a.
Damek-Ricci spaces generalize rank one symmetric spaces of the noncompact type (hyperbolic spaces).The latter are, in the notation of Section 2, the spaces G/K with G of rank one i.e. dim a = 1.The classical Iwasawa decomposition G = N AK, where N , resp.A, is a nilpotent, resp.one-dimensional abelian, Lie subgroup of G, gives a diffeomorphism ϕ : na → naK of the solvable group N A onto G/K, intertwining the left translation by x ∈ N A with the natural action on G/K of the same element x ∈ G.A G-invariant Riemannian metric on G/K therefore corresponds via ϕ with a left invariant metric on N A. Forgetting G and K the Riemannian manifold G/K can thus be studied as the solvable group N A with this left invariant metric.
The next proposition links both points of view precisely.Let n = g α ⊕ g 2α where α and 2α denote the positive roots of the pair (g, a) with respective eigenspaces g α , g 2α ; let p = dim g α , q = dim g 2α (with q = 0 if 2α is not a root).The abovementioned Iwasawa decomposition of G is given by its Lie subgroups N , A with Lie algebras n, a.On g we consider the scalar product the Killing form B is here normalized so that H ∈ a defined by α(H) = 1/2 is a unit vector.In this notation we have the following (with Y , Y ∈ n, H , H ∈ a), the Lie algebra s = n ⊕ a satisfies assumptions 1 to 4 with v = g α , z = g 2α and The group S = N A is a Damek-Ricci space, isometric to the hyperbolic space G/K equipped with the G-invariant metric given by the scalar product induced on p by < , > g .
Proof.Easy by means of the linear isomorphism 1 2 (I − θ) of s onto p (the projection parallel to k in g).See [9] §6.1 for details.General Damek-Ricci spaces are also solvable groups N A with one-dimensional A, but not necessarily arising from Iwasawa decompositions of semisimple Lie groups; excepting the hyperbolic spaces they are not symmetric spaces.For more details we refer to Damek and Ricci [3][4], Cowling et al. [1][2] or to our expository notes [8][9].

Totally geodesic subgroups
A (connected) submanifold M of a (connected) Riemannian manifold M is said to be totally geodesic in M if each geodesic of M (with respect to the metric induced by M ) is a geodesic of M .Let us recall the following general Proposition 4.1.(Cowling et al. [1]) Let S be a Lie group with the left invariant Riemannian metric defined by a scalar product < , > on its Lie algebra s.Let S 0 be a Lie subgroup of S with Lie algebra s 0 .Then S 0 is totally geodesic in S if and only if < X, [X, Y ] >= 0 for all X ∈ s 0 and all Y ∈ s ⊥ 0 (the orthogonal complement of s 0 in s).
Inverse Problems and Imaging Volume X, No. X (200X), X-XX notations related to G * will bear the subscript * .Let g * = n * ⊕ a * ⊕ k * be the Iwasawa decomposition arising from a * = RH * with The bracket relations (4.1) mean that the mapping (4.2) with v, w, z, t ∈ R, is a Lie algebra isomorphism of s 0 onto the solvable subalgebra This gives a Lie group isomorphism of S 0 onto the subgroup S * = N * A * of G * hence, composing with na → naK * , a diffeomorphism of S 0 onto the homogeneous space G * /K * = H 2 (C).In order to apply Proposition 3.1 (with p = 2, q = 1) we endow g * with the metric (3.6) where t means transpose.Indeed B * (X , X ) = 6 tr (X X ) and θ * X = − t X for the Lie algebra g * = su(1, 2).The corresponding scalar product on s * is, by Proposition Applying this to matrices given by (4.2) (with t = t = 0 for Y , Y , and H = t H * , H = t H * ) yields v v + w w + z z + t t , in agreement with the scalar product in s 0 with respect to its orthonormal basis (V, W, Z, H).The diffeomorphism of S 0 onto H 2 (C) is therefore an isometry, as claimed.

Inversion formula for Damek-Ricci spaces
For lack of a compact group K inversion formulas for the X-ray transform on a Damek-Ricci space cannot be directly obtained by the method of Section 2. The difficulty can be circumvented however by means of totally geodesic submanifolds, drawing inspiration from [7] and [10].
On a Damek-Ricci space S = N A let exp denote the exponential mapping of the group S and Exp the exponential mapping of the Riemannian manifold S at the origin o (the identity element).They are (distinct) global diffeomorphisms of s onto S ([9] p. 18 and 24).Let ξ 0 = Exp (RH) = A be the geodesic tangent to H at o. Theorem 5.1.Let S be a Damek-Ricci space and V be any nonzero vector in v.The X-ray transform R on S is inverted by where dots mean here the natural action of G * on G * /K * .Going back to S 0 we obtain Indeed the mapping exp for V and the constant factor in (5.1) is −1/π √ 3 V .We finally introduce the following analog of the shifted dual transform of a function ϕ on the set of all geodesics of S: Let us explain the notation: ξ 0 is a geodesic of S and S 0 , γ • ξ 0 is the geodesic of S 0 (and S) obtained by left action of γ in S 0 .With k• as above, k • γ • ξ 0 is a geodesic of S 0 , therefore of S. Finally x • k • γ • ξ 0 is the geodesic of S obtained by left action of x ∈ S. The integral, taken with respect to the normalized Haar measure dk on the compact group K Thus the integral over K * in (5.1) is R * exp tV Rf (o), and the theorem follows by left action of x.Proof.(i) implies (ii): obvious.(ii) implies (i).Let x ∈ S with d(o, x) ≥ R. Since Exp is a global diffeomorphism there exists X ∈ s = v ⊕ z ⊕ a such that x = Exp X (and X = d(o, x)).This vector decomposes as X = vV + zZ + tH where V ∈ v, Z ∈ z are unit vectors and v, z, t ∈ R .Let s 0 = RV ⊕ RJ Z V ⊕ RZ ⊕ RH.

Remark. When written down explicitly our inversion formula only involves the geodesics
By Proposition 4.3 the corresponding S 0 is totally geodesic in S, contains x and is isometric to the hyperbolic space H 2 (C).Considering the restriction of f to S 0 our assumption implies Rf (ξ) = 0 for any geodesic ξ of S 0 such that d(o, ξ) ≥ R. Helgason's Corollary 4.1 of [6] p. 120 then applies to S 0 , hence f (x) = 0.
Remark.Helgason's result is also proved by restriction to a totally geodesic submanifold, this time isometric to the two-dimensional hyperbolic space H 2 (R).

Proposition 3 . 1 .
(Hyperbolic spaces as Damek-Ricci spaces [9]) Equipped with the Lie bracket induced by g and the scalar product where R * is defined by (5.2) below.Proof.Let S 0 be the totally geodesic subgroup of S given by Proposition 4.3 from the given V and any nonzero Z ∈ z.We work at the origin first, restricting f to S 0 .By Proposition 4.3 and its proof we have a Lie group isomorphism S 0 → S * = N * A * onto a solvable sugbroup of G * = SU (1, 2) and an isometry of S * onto Inverse Problems and Imaging Volume X, No. X (200X), X-XX G * /K * = H 2 (C).Using for objects tranferred from S 0 to G * /K * by the composed diffeomorphism, or from s 0 to s * ⊂ g * , we have by Theorem 2.1 at the origin of S 0 (restriction to s 0 of exp for S) corresponds with exp for S * (restriction to S * of exp for G * ) in view of the Lie isomorphism (4.2), and the left translation by exp tV in S 0 with the action of exp t V on G * /K * .Abusing notations we have still denoted by k • (• • • ) the isometry of S 0 corresponding with the action of k ∈ K * on the hyperbolic space.The norm used in (5.1) is given by | V | 2 = −B * ( V , θ * V ) therefore, by (3.6) with p = 2, q = 1 and Proposition 3.1 for g

√ 2 V 1 √ 2 |√ 6 V . 6 .
arg sh t from x. Indeed the distance used here on S 0 is 1/ √ 6 times the distance in[10], which was |Y | arg sh t = V | arg sh t (see[10], final Remark), and | V | = Support theorem on Damek-Ricci spaces On a Riemannian manifold M with origin o and distance d we shall say (following [6] p. 120) that a function f is exponentially decreasing if e kd(o,x) f (x) is bounded on M for any k ≥ 0. Because of the triangle inequality this definition does not depend on the choice of o.Theorem 6.1.Let S be a Damek-Ricci space and let R > 0. For an exponentially decreasing continuous function f on S the following are equivalent: (i) f (x) = 0 for any point x ∈ S such that d(o, x) ≥ R (ii) Rf (ξ) = 0 for any geodesic ξ of S such that d(o, ξ) ≥ R.