  We define positive and strictly positive definite functions on a domain and
study these functions on a list of regular domains. The list includes the unit
ball, conic surface, hyperbolic surface, solid hyperboloid, and simplex. Each
of these domains is embedded in a quadrant or a union of quadrants of the unit
sphere by a distance preserving map, from which characterizations of positive
definite and strictly positive definite functions are derived for these regular
domains.
