A horosphere in hyperbolic space is a hypersurface orthogonal to all geodesics approaching a fixed ideal point. Equivalently, it is a limit of spheres whose centers approach that point on the ideal boundary.
In the upper half-space model, horospheres centered at the point at infinity are horizontal Euclidean hyperplanes;
the others are Euclidean spheres tangent to the ideal
boundary. When the ambient sectional curvature
is 
, a horosphere has all principal
curvatures equal to 1 for one choice of unit
normal vector and 
for the other, so it is a constant mean
curvature surface.
