Finite field theory on noncommutative geometries

The propagator is calculated on a noncommutative version of the flat plane and the Lobachevsky plane with and without an extra (Euclidean) time parameter. In agreement with the general idea of noncommutative geometry it is found that the limit; when the two "points" coincide is finite and diverges only when the geometry becomes commutative. The flat four-dimensional case is also considered. This is at the moment less interesting since there has been no curved case developed with which it can be compared.