This paper provides upper and lower bounds on the kissing number of congruent radius \(r > 0\) spheres in hyperbolic \(\mathbb {H}^n\) and spherical \(\mathbb {S}^n\) spaces, for \(n\ge 2\). For that purpose, the kissing number is replaced by the kissing function \(\kappa _H(n, r)\), resp. \(\kappa _S(n, r)\), which depends on the dimension n and the radius r.
After we obtain some theoretical upper and lower bounds for \(\kappa _H(n, r)\), we study their asymptotic behaviour and show, in particular, that \(\kappa _H(n,r) \sim (n-1) \cdot d_{n-1} \cdot B(\frac{n-1}{2}, \frac{1}{2}) \cdot e^{(n-1) r}\), where \(d_n\) is the sphere packing density in \(\mathbb {R}^n\), and B is the beta-function. Then we produce numeric upper bounds by solving a suitable semidefinite program, as well as lower bounds coming from concrete spherical codes. A similar approach allows us to locate the values of \(\kappa _S(n, r)\), for \(n= 3,\, 4\), over subintervals in \([0, \pi ]\) with relatively high accuracy.
