Let \(X\) be a homogeneous Riemannian manifold, \(G\) the group of isometries of \(X\), \(dx\) the element of Riemannian measure on \(X\). We consider a collection \({\mathcal F}=\{E_i\}_{i=1}^k\) of compact subsets in \(X\) of positive measure. For every open set \({\mathcal U}\subset X\) such that every set \(G_i=\{g\in G: g^{-1}E_i\subset{\mathcal U}\}\), \(i=1,\dots, k\), is nonempty, the Pompeiu transformation \({\mathcal P}_{{\mathcal F};{\mathcal U}}\) maps the set \(C({\mathcal U})\) into the direct product \(C(G_1)\times\dots\times C(G_k)\) as follows: \({\mathcal P}_{{\mathcal F};{\mathcal U}}f=(f_1,\dots,f_k)\), where \(f_i(g)=\int_{g^{-1}E_i} f(x)\, dx\), \(g\in G_i\), \(i=1,\dots, k\). If \({\mathcal U}=X\) then \({\mathcal P}_{{\mathcal F};{\mathcal U}}\) is called the global Pompeiu transformation, and if \({\mathcal U}\neq X\) then \({\mathcal P}_{{\mathcal F};{\mathcal U}}\) is called the local Pompeiu transformation. The following problem is called the Pompeiu problem: Find out whether or not \({\mathcal P}_{{\mathcal F};{\mathcal U}}\) is injective and if not, find its kernel. Let \(B_r\) be the open ball of radius \(r\) and centered at the origin of \(X\), and \(\overline{B}_r\) the closure of \(B_r\). For the case when \({\mathcal U}=B_R\) and \({\mathcal F}=\{\overline{B}_{r_1}, \overline{B}_{r_2}\}\), the necessary and sufficient conditions for the injectivity of \({\mathcal P}_{{\mathcal F};{\mathcal U}}\) are called “local two-radii theorems”. In the paper under review the author obtains a local two-radii theorem in the case when \(X\) is the \(n\)-dimensional sphere \({\mathbb S}^n\).