DERIVATIONS ON CR MANIFOLDS

Abstract

We studied the relation between the tangential Cauchy-Riemann operator <TEX>${\={\partial}}_b$</TEX> CR-manifolds and the derivation <TEX>$d^{{\pi}^{0,\;1}}$</TEX> associated to the natural projection map <TEX>${\pi}^{0.1}\;:\;TM\;{\bigotimes}\;{\mathbb{C}}\;=\;T^{1,0}\;{\bigoplus}\;T^{0,\;1}\;{\rightarrow}\;T^{0,\;1}$</TEX>. We found that these two differential operators agree only on the space of functions <TEX>${\Omega}^0(M),\;unless\;T^{1,\;0}$</TEX> is involutive as well. We showed that the difference is a derivation, which vanishes on <TEX>${\Omega}^0(M)$</TEX>, and it is induced by the Nijenhuis tensor associated to <TEX>${\pi}^{0.1}$</TEX>.