The fundamental theorems for curves and surfaces in 3d Heisenberg group

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.