It is known that the \(n\)-center problem in a convected 3-dimensional smooth manifold \(Q\) is the Lagrangian system \[ {d\over dt} \dot q=-\text{grad} W(q), \tag{1} \] on \(T(Q\setminus P)\) with the Lagrangian \[ L(q, \dot q)=\frac 12\|\dot q\|^2_g -V(q), \tag{2} \] where \(P=\{p_1, \dots, p_n\}\) is a finite set in \(Q\), \(g\) is a fix \(C^4\) Riemannian metric on \(Q\), and \(W\) is a \(C^3\) function in a neighborhood \(U_i\) of \(p_i\), \[ W(q)=-{b_i(q)\over \text{dist} (q,p_i)}+ c_i(q), \tag{3} \] with \(b_i\), \(c_i\in C^3(U_i)\) and \(b_i (p_i) >0\). The distance is measured in terms of the Riemannian metric. Here the authors describe a global version of the KS regularization of the \(n\)-center problem on \(Q\). As an application, they show that the \(n\)-center problem in \(S^3\) has positive topological entropy for \(n\geq 5\) and energy greater than the maximum of the potential energy. To this end the authors use the results of Gromov and Paternian on the topological entropy of geodesic flows.