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Radon Transform on Spaces of Constant Curvature 1 PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 2, February 1997, Pages 455–461 S 0002-9939(97)03570-3 RADON TRANSFORM ON SPACES OF CONSTANT CURVATURE CARLOS A. BERENSTEIN, ENRICO CASADIO TARABUSI, AND ARP´ AD´ KURUSA (Communicated by Peter Li) Abstract. A correspondence among the totally geodesic Radon transforms— as well as among their duals—on the constant curvature spaces is established, and is used here to obtain various range characterizations. 1. Introduction The totally geodesic Radon transform on spaces of constant curvature has been widely studied (see [Hg1], [Hg2], [BC1], [Ku1], to quote only a few). Yet many of the known results, in spite of their similarities, were obtained on each such space independently. The idea of relating these transforms by projecting each space to the Euclidean one appeared independently in [BC2] (for negative curvature) and [Ku4] (for curvature of any sign), and was used to obtain, respectively, range characterizations of the Radon transform on the hyperbolic space and support theorems on all constant curvature spaces. In this paper we extend and exploit further this correspondence, establishing an explicit formula relating the dual Radon transforms on the different constant cur- vature spaces, and proving various range characterizations for the Radon transform and its dual on the Euclidean and elliptic space. The third author wishes to thank the Soros Foundation and MIT for supporting his visit at MIT, during which this paper was conceived. 2. Preliminaries n Let Mκ be an n-dimensional simply connected complete Riemannian manifold of constant curvature κ. Normalizing the metric so that κ equals 1, 0, or +1, we get, respectively, the hyperbolic space Hn, the Euclidean space Rn−, and the sphere Sn—or its two-to-one quotient, the projective space Pn. (The subscripts +1, 1 will henceforth be replaced by +, , respectively.) − − Received by the editors August 8, 1995. 1991 Mathematics Subject Classification. Primary 44A12; Secondary 53C65, 51M10. Key words and phrases. Radon transform, spaces of constant curvature, totally geodesic sub- manifolds, spherical harmonics, moment conditions. The first author was partially supported by NSF grants DMS9225043 and EEC9402384. This research was in part accomplished during the second author’s stay at the University of Maryland, whose hospitality is hereby acknowledged. The third author was partially supported by the Hungarian NSF grants T4427, F016226, W075452, and T020066. c 1997 American Mathematical Society 455 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 456 C. A. BERENSTEIN, E. CASADIO TARABUSI, AND A.´ KURUSA For a fixed k,with1 k<n,letξbe an arbitrary k-dimensional totally geodesic n ≤ submanifold of Mκ.Thek-dimensional Radon transform Rκf (the indication of n n, k will be omitted) of a function f on Mκ whichisintegrableoneachξis given by Rκf (ξ)= f(x)dx, Zξ n n where dx is the volume element on ξ induced by the metric of Mκ.InS, whenever x ξ then also x ξ,sothatR+ cannot separate antipodal points, i.e., it vanishes on∈ odd functions:− ∈ for this reason we will only consider even functions on Sn,or, n n equivalently (except for a factor 2 in the integrals), will take M+ to be P . n,k n Let Γκ be the set of all k-dimensional totally geodesic submanifolds of Mκ, equipped with the natural differentiable structure: if κ = 0 it is the usual affine n,k Grassmannian G of k-planes. The k-dimensional dual Radon transform Rκ∗ φ of, n,k say, a continuous function φ on Γκ is defined as Rκ∗ φ(x)= φ(ξ)dξ, ξ x Z3 n n,k where x Mκ and dξ is the measure on the subset ξ Γκ : ξ x induced ∈ { ∈ n 3 } by the isotropy group of x in the automorphism group of Mκ, normalized as in [Hg1]. n The Riemannian metric of Mκ is determined by the size function νκ, defined on [0,π/2) for κ = +1, and on [0, + )forκ= 1,0, that gives the radius νκ(r)of ∞ − n the Euclidean sphere isometric to the geodesic sphere of radius r in Mκ.Taking n geodesic polar coordinates in Mκ with respect to a fixed reference point o,the function µκ = νκ/(dνκ/dr) n n generates the map pκ : M R given by κ → pκ(Expo rω)=µκ(r)ω, n n where ω is a unit vector in the tangent space ToMκ,identifiedwithR . (Actually n p+ is defined on the complement in P of the projective hyperplane, henceforth n called the distinguished equator eo, orthogonal to o.TheS version of p+ is of course the composition with the canonical quotient map.) The following table gives νκ, µκ explicitly. n Mκ κ νκ µκ Hn 1 sinh r tanh r Rn −0 r r Sn or Pn +1 sin r tan r The map pκ has the property of taking geodesic segments to geodesic segments, although not preserving their arc length: the map Γn,k Gn,k thus induced will κ → also be denoted by pκ.(Ifκ= +1 this is only defined on a Zariski dense open n,k subset of Γ+ .) n Thedistanceofξfrom o in Mκ will be denoted by ξ ;ifk=n 1weshall n | | − parametrize ξ as ξ ω,whereω ToMκ is the unit direction of the closest point of ξ to o. | | ∈ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use RADON TRANSFORM ON SPACES OF CONTANT CURVATURE 457 3. Correspondence among Radon transforms and among their duals The following is proved in [Ku4] and, for the hyperbolic case, in [BC2]. Theorem 3.1. Let f L1(Rn). ∈ n 1 (1) If g is the even function on S (but not defined on eo) given by g(p+− (x)) = (1 + x 2)(k+1)/2 f(x) for all x Rn,then | | ∈ 2 1/2 1 n R0f(ξ)=(1+ ξ )− R+g(p− (ξ)) for every k-plane ξ in R . | | + (2) Assume that f has support in the unit ball Bn of Rn:ifgis the function on n 1 2 (k+1)/2 n H given by g(p− (x)) = (1 x ) f(x) for all x R ,then − −| | ∈ 2 1/2 1 n R0f(ξ)=(1 ξ )− R g(p− (ξ)) for every k-plane ξ in R . −| | − − (These multiplication operators and the Radon transforms are all positive op- erators, and only integrability along k-planes, or totally geodesic submanifolds, is actually required, so f can be taken to belong to a larger function space than L1.) Similar relations hold for the dual Radon transform: it was proved in [BC2] for the hyperbolic case, but the same method can be used in general, as we now show. Theorem 3.2. Let φ (Gn,k). ∈C n,k (1) If ψ is the function on Γ+ (not defined on the totally geodesic submanifolds 1 2 n/2 contained in eo) given by ψ(p+− (ξ)) = (1 + ξ ) φ(ξ) for every k-plane ξ in Rn,then | | 2 (k n)/2 1 n R∗φ(x)=(1+ x ) − R∗ψ(p− (x)) for every x R . 0 | | + + ∈ n,k 1 2 n/2 (2) If ψ is the function on Γ given by ψ(p− (ξ)) = (1 ξ ) φ(ξ) for every k-plane ξ in Rn,then − − −| | 2 (k n)/2 1 n R0∗φ(x)=(1 x ) − R∗ ψ(p− (x)) for every x B . −| | − − ∈ Proof. (1) For κ = +1 consider the functions 1 2 ( k 1)/2 1 2 (k n)/2 ρ(p− (x)) = (1 + κ x ) − − ,ρ0(p−(x)) = (1 + κ x ) − , κ | | κ | | 1 2 1/2 1 2 n/2 σ(p− (ξ)) = (1 + κ ξ )− ,σ0(p−(ξ)) = (1 + κ ξ )− , κ | | κ | | and the maps Φκ,Φκ0,Ψκ,Ψκ0 given by 1 1 Φκ(f pκ)=(ρ p− )f, Φ0 (f pκ)=(ρ0 p− )f, ◦ ◦ κ κ ◦ ◦ κ 1 1 Ψκ(φ pκ)=(σ p− )φ, Ψ0 (φ pκ)=(σ0 p− )φ, ◦ ◦ κ κ ◦ ◦ κ n n,k for f (R )andφ (G ). The Jacobian determinant of pκ is easily seen to ∈D n ∈C1 n be Jpκ =(dνκ/dr)− − (cf. [Ku4, Figure 1]). Therefore, if f 0 is also in (R )and D if g = f pκ and g0 = f 0 pκ,then ◦ ◦ 1 n n Φκg,Φκ0 g0 R = (ρgρ0g0) pκ− = ρρ0gg0 Jpκ = gg0 = g,g0 Mκ , h i Rn ◦ Mn Mn h i Z Z κ Z κ 2 2 (k+1)+(n k)+( n 1) because 1+κ pκ =(dνκ/dr)− , hence ρρ0 Jpκ =(dνκ/dr) − − − =1. | | n Since Φκ is a topological isomorphism of (S ,eo), the space of even functions on n D n S vanishing on a neighborhood of eo,with (R), it follows that the identity D1 1 Φκg,Φ0 g0 Rn = g,g0 Mn implies Φ0 =(Φ∗)− . Similarly we get Ψ0 =(Ψ∗)− . h κ i h i κ κ κ κ κ License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 458 C. A. BERENSTEIN, E. CASADIO TARABUSI, AND A.´ KURUSA 1 With this notation Theorem 3.1 reads R0 =ΨκRκΦκ− ; taking adjoints we obtain 1 1 R0∗ =(Φκ− )∗Rκ∗Ψκ∗ =Φκ0 Rκ(Ψκ0 )− . (2) The argument for κ = 1 is verbatim the same, except that f,f0 must have support in Bn (the expression− of Jp was found in [BC2]), and Φ is a topological isomorphism of (Hn) with its image,− which is strictly contained− in (Rn). D D n Observe that Φ+ is also a topological isomorphism of (S ,eo), the space of even n S n functions on S vanishing on eo to infinite order, with (R ).
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