Integral geometry of equidistants in hyperbolic space

We generalize the classical formulas of integral geometry, by getting integral geometric formulas for the intersection of a fixed compact hypersurface of hyperbolic space and a moving totally umbilical hypersurface. In particular we compute the mean value of the volume, the total mean curvatures and the Euler characteristic of these intersections when the totally umbilical hypersurface moves over all the intersecting positions. Analogous formulas are given for totally umbilical hypersurfaces contained in totally geodesic planes of ℍⁿ. Work partially supported by MECD grant number EX2003-0987, and MCYT grant number BFM2003-03458.