On the spectrum obtained from packing balls on Riemann manifolds (Q5929664)

Consider packings with non-necessary equal balls in a metric space. Define the \(n\)-packing spectrum \(C_n(M)\) of \(M\) to be the set of \(n\)-tuples \((r_1,\ldots,r_n)\subset {\mathbb R}^n_+\) associated to \(n\) disjoint balls \(B_i\subset M\) of radius \(r(B_i)<r_i\), \(1\leq i\leq n\). Let \(C_n^b\subset C_n(M)\) denote the subset obtained by considering balls \(B(p,r)\) of radius \(r<c(p)\) where \(c(p)\) is the first critical value of the metric \(d(p,*)\). First, the authors compute several examples of \(2\)-spectra for domains in Riemannian manifolds. However, the description of the spectra \(C_n\) for \(n\geq 3\) becomes difficult related to curvature properties of the manifold. Then, the authors study the problem to which extent the knowledge of the packing spectra determines the isometry type of Riemannian manifolds. They show that two domains in a homogeneous manifold with equal packing spectra have identical volume. Another result says that for a given closed \(n\)-manifold \(M\) one finds a metric such that \(C^b_2(M)=C^b_2(S^n)\). Moreover, if \(C^b_2(M)=C^b_2(S^n)\) holds for a complete generic metric on \(M\) and some metric of convex body on \(S^n\), then \(M\) is homeomorphic to \(S^n\).