Given a strictly pseudoconvex hypersurface M ⊂ Cn+1, we discuss the problem of classifying all local CR diffeomorphisms between open subsets N, N′ ⊂ M. Our method exploits the Tanaka-Webster pseudohermitian invariants of a contact form ϑ on M, their transformation formulae, and the Chern-Moser invariants. Our main application concerns a class of generalized ellipsoids where we classify all local CR mappings.
"Pseudohermitian invariants and classification of CR mappings in generalized ellipsoids." J. Math. Soc. Japan 64 (1) 153 - 179, January, 2012. https://doi.org/10.2969/jmsj/06410153