Kissing number in non-Euclidean spaces of constant sectional curvature

Abstract:

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 $n\geq 2$. 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.