Abstract. We prove a relation between the fc-dimensional totally geodesic Radon transforms on the various constant curvature spaces using the geodesic correspondence between the spaces. Then we use this relation to obtain improved support theorems for these transforms.
IntroductionLet J7n be an «-dimensional simply connected Riemannian manifold of constant curvature k . Normalizing the metric so that k = -1, 0, or +1 we must deal only with the hyperbolic space H" (k = -1), the Euclidean space Rn (k = 0), and the sphere Sn (k = +1 ).For a fixed k (1 < k < n -1) let ¿; be an arbitrary totally geodesic submanifold of J7n of dimension k . The fc-dimensional totally geodesic Radon transform Rf of f e L2(J7n) is defined bywhere dx is the surface element on £ induced by the metric of J7n [6]. Our first goal in this paper is to prove a link between these Radon transforms taken on different constant curvature spaces. The proof is based on the geodesic correspondence between the spaces [10]. The idea to use projection from constant curvature spaces to Euclidean spaces also appeared in [3,11].The connection established in Theorem 2.1 allows one to transpose results from one space to the other; hence, it can be used to get inversion formulas, range characterizations [1], and so on. We shall use it to obtain support theorems.Roughly speaking a support theorem states that if the function / is in a suitable function space of J7n and the support of Rfi, supp Rf, is bounded, then supp f ç P supp Rf, where P maps the set of total geodesies into Jt"1 so that the total geodesies correspond to their point closest to the origin. For more information about the applications of support theorems we refer to [4, 5, ]]■_
Let C1 and C2 be convex closed domains in the plane with C 2 boundaries 0C1 and cqd2 intersecting each other in nonzero angles. Assume the two strictly convex bodies .T1 and ~'2 with C 2 boundaries in the interior of C1 f3 C~ subtend equal visual angles at each point of 0C1 and 0C2. Then fi'l and 3r2 coincide. Generalizations are also discussed. (1991): 52A10.
Mathematics Subject Classification
Abstract. A correspondence among the totally geodesic Radon transformsas well as among their duals-on the constant curvature spaces is established, and is used here to obtain various range characterizations.
The Radon transform that integrates a function in H", the n-dimensional hyperbolic space, over totally geodesic submanifolds with codimension 1 and the dual Radon transform are investigated in this paper. We prove inversion formulas and an inclusion theorem for the range.
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