Biholomorphic equivalence to totally nondegenerate model CR manifolds

Abstract

Applying Elie Cartan’s classical method, we show that the biholomorphic equivalence problem to a totally nondegenerate Beloshapka’s model of CR dimension one and codimension $$k> 1$$ , whence of real dimension $$2+k$$ , is reducible to some absolute parallelism, namely to an $$\{e\}$$ -structure on a certain prolonged manifold of real dimension either $$3+k$$ or $$4+k$$ . The proof relies upon a careful weight analysis on the structure equations associated with the mentioned problem of equivalence. As one of the applications of the achieved results, we also reconfirm in CR dimension one Beloshapka’s maximum conjecture on the holomorphic rigidity of his models in certain lengths equal or greater than three.