Abstract: There are several fundamental notions of "flatness" and "curvature" in CR-geometry (that is, the biholomorphically invariant geometry of real submanifolds of complex space). Roughly speaking, these correspond to type conditions and nondegeneracy conditions. Type conditions play an important role in partial differential equations. The type controls the boundary behaviour of solutions to the (inhomogeneous) Cauchy-Riemann equations with (Neumann) boundary conditions. They also control extension properties of CR functions. Nondegeneracy conditions, on the other hand, control the geometric behaviour of a real object. They can be phrased in terms of properties of reflections, and are thus related to extension properties of CR mappings. We will discuss how these conditions are characterized in a geometric way, and will show how non-flatness corresponds to rigidity, and how non-rigidity gives rise to flatness.

 

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