It is well known that the sphericity of a strictly pseudoconvex real hypersurface in a complex manifold amounts to the vanishing of its Chern-Moser tensor. The latter is built from the principal curvature components of the respective Cartan connection. The curvature construction here appeals to the 6-jet of the hypersurface at a point, and thus requires regularity of the hypersurface of class at least C^6. The problem of checking the sphericity in lower regularity than C^6 is open.
In our joint work with Zaitsev, we apply our recent theorem on the analytic regularizability of a strictly pseudoconvex hypersurface to find a necessary and sufficient condition for the sphericity of a strictly pseudoconvex hypersurfaces of arbitrary regularity starting with C^3.
Further, we obtain a simple condition for the analytic regularizability of hypersurfaces of the respective classes.
Finally, in a recent work with Vinicius Da Silva we show applications to general involutive structures.
Surprisingly, despite of the seemingly analytic nature of the problem, our technique is geometric and is based on the Reflection Principle in SCV.