  We study the local equivalence problems of curves and surfaces in three
dimensional Heisenberg group via Cartans method of moving frames and Lie
groups, and find a complete set of invariants for curves and surfaces. For
surfaces, in terms of these invariants and their suitable derivatives, we also
give a Gaussian curvature fromula of the metric induced from the adapted metric
on Heisenberg group, and hence form a new formula for the Euler number of a
closed surface.
