In [Comment. Math. Helv. 55, 593-621 (1990; Zbl 0452.43012)] L. Zalcman and the reviewer stated a very general type of Pompeiu problem as follows. Let \(G\) be a group of transformations of a locally compact space \(X\), \(d\mu\) an invariant measure on \(X\), then an open relatively compact set \(\Omega\subseteq X\) is said to have the Pompeiu property if the map \(P: C(X)\to C(G)\) given by \(Pf(g):=\int_{g(\Omega)} f(x)d\mu(x)\) is injective. In this generality the problem is very hard, but some progress can be made for \(X=G/K\) an irreducible symmetric space of rank one or \(X=\mathbb{R}^ n\), with \(G\) the Euclidean group \(M(n)\). Some of the work of the reviewer on the local Pompeiu property [e.g., C. A. Berenstein and R. Gay, J. Anal. Math. 52, 133-166 (1989; Zbl 0668.30037)] indicates that the existence of a global transformation group \(G\) might not be necessary, similarly the invariance of the measure might also be disposable [e.g., C. Berenstein and D. Pascuas, J. Math. (to appear)]. In that case one may just want to consider a Riemannian manifold and preplace the translates of \(\Omega\) by the collection of all balls of a single radius (or a few radii) [e.g., T. Sunada, Trans. Am. Math. Soc. 267, 483-501 (1981; Zbl 0514.58037), E. T. Quinto (to appear)].
The author proceeds in this direction considering the following idea: Let \(X\) be a Riemannian manifold, \(T_ x\) its tangent space at \(x\in X\). If \(\mu\) is a radial measure with compact support in \(\mathbb{R}^ n\), \(n=\dim X\), then it induces a measure on \(T_ x\) by choosing an orthogonal basis of \(T_ x\) and using it to identify \(T_ x\) to \(\mathbb{R}^ n\), then the Pompeiu transform is \[ P_ \mu(f)(x)= \int_{\mathbb{R}^ n} f(\exp_ x(t))\upsilon_ x(t)d\mu(t), \] where \(\upsilon_ x\) is the Jacobian of \(\exp_ x\). Since \(\mu\) is radial, the definition is independent of the choice of orthonormal basis in \(T_ x\). By taking \(d\mu(t)=\chi_ B(t)dt\), \(B=B/(0,r)\subseteq\mathbb{R}^ n\), we can define the Pompeiu transform with respect to balls. The author uses these ideas to extend the known results about the Pompeiu transform with respect to balls from irreducible symmetric spaces \(\widetilde{X}=G/K\) of rank 1 to locally symmetric spaces \(\Gamma\setminus \widetilde{X}\), for some discrete subgroup \(\Gamma\) of \(G\). He works out in detail a number of interesting examples.