## Tohoku Mathematical Journal 2014 March SECOND SERIES VOL. 66, NO. 1

 Tohoku Math. J. 66 (2014), 93-105

Title A PARAMETRIZED DE RHAM DECOMPOSITION THEOREM FOR CR MANIFOLDS

Author Atsushi Hayashimoto

(Received June 11, 2012, revised May 9, 2013)
Abstract. The CR equivalence problem between CR manifolds with slice structure is studied. Let $N$ be a connected holomorphically nondegenerate real analytic hypersurface and $M(p)$ a finitely nondegenerate real analytic hypersurface parametrized by $p \in N$. Let $M$ be a totality of $N$ and $M(p)$ with moving $p$ in $N$. Assume that $M$ and $\widetilde{M}$ (with a same structure as $M$) are CR equivalent and that $N$ and $\widetilde{N}$ are also CR equivalent. Then we prove that, for any $p \in N$, there exists $\tilde{p}\in \widetilde{N}$ such that $M(p)$ is CR equivalent to $\widetilde{M}(\tilde{p})$.

Mathematics Subject Classification. Primary 32V20; Secondary 32V35.

Key words and phrases. CR equivalence, Segre mapping, finite nondegeneracy, holomorphic nondegeneracy.