Explicit expression of Cartan's connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere (Q1935625)

Let \(M^3 \subset {\mathbb{C}}^2\) be a \(C^6\)-smooth Levi-nondegenerate hypersurface defined as the graph of a function \(v = \varphi(x,y,u)\). The authors study the Cartan geometry of such manifolds, and present a so-called curvature function of a related Tanaka-type normal connection. All elements of the connection are completely effective in terms of the graphing function \(\varphi\). The authors prove that the vanishing of this curvature function explicitly characterizes the local biholomorphic equivalence of such \(M\) and the Heisenberg sphere. These \(M\) are then necessarily real-analytic.