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In this paper we show that the number of distinct distances determined by aset of $n$ points on a constant-degree two-dimensional algebraic variety $V$(i.e., a surface) in $\mathbb R^3$ is at least $\Omega\left(n^{7/9}/{\rmpolylog} \,n\right)$. This bound is significantly larger than the conjecturedbound $\Omega(n^{2/3})$ for general point sets in $\mathbb R^3$.  We also show that the number of unit distances determined by $n$ points on asurface $V$, as above, is $O(n^{4/3})$, a bound that matches the best knownplanar bound, and is worst-case tight in 3-space. This is in sharp contrastwith the best known general bound $O(n^{3/2})$ for points in three dimensions.  To prove these results, we establish an improved upper bound for the numberof incidences between a set $P$ of $m$ points and a set $S$ of $n$ spheres, ofarbitrary radii, in $\mathbb R^3$, provided that the points lie on an algebraicsurface $V$ of constant degree, which does not have linear or sphericalcomponents. Specifically, the bound is $$ O\left( m^{2/3}n^{2/3} +m^{1/2}n^{7/8}\log^\beta(m^4/n) + m + n + \sum_{c} |P_{c}|\cdot |S_{c}| \right), $$ where the constant of proportionality and the constant exponent $\beta$depend on the degree of $V$, and where the sum ranges over all circles $c$ thatare fully contained in $V$, so that, for each such $c$, $P_c = P\cap c$ and$S_c$ is the set of the spheres of $S$ that contain $c$. In addition, $\sum_{c}|P_{c}| = O(m)$ and $\sum_{c} |S_{c}| = O(n)$. This bound too improves uponearlier known bounds. These have been obtained for arbitrary point sets butonly under severe restrictions about the spheres, which are dropped in ourresult. Another interesting application of our result is an incidence bound forarbitrary points and spheres in 3-space, where we improve and generalize theprevious work of Apfelbaum and Sharir[AS].

Distinct and repeated distances on a surface and incidences between points and spheres