Abstract. We show the fundamental theorems of curves and surfaces in the 3-dimensional Heisenberg group and find a complete set of invariants for curves and surfaces respectively. The proofs are based on Cartan's method of moving frames and Lie group theory. As an application of the main theorems, a Crofton-type formula is proved in terms of p-area which naturally arises from the variation of volume. The application makes a connection between CR geometry and integral geometry.
We study the horizontally regular curves in the Heisenberg groups Hn. We show the fundamental theorem of curves in Hn (n ≥ 2) and define the orders of horizontally regular curves. We also show that the curve γ is of order k if and only if, up to a Heisenberg rigid motion, γ lies in H k but not in H k−1 ; moreover, two curves with the same order differ from a rigid motion if and only if they have the same invariants: p-curvatures and contact normality. Thus, combining with our previous work [1] we have completed the classification of horizontally regular curves in Hn for n ≥ 1.
We study the local equivalence problems of curves and surfaces in 3-dimensional Heisenberg group via Cartan's 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 H 1 , and hence form a new formula for the Euler number of a closed surface.
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