Derivations on CR manifolds

Abstract

We studied the relation between the tangentialCauchy-Riemann operator overline partial _b on CR-manifoldsand the derivation d^{pi^{0,1}} associated to the naturalprojection map pi^{0,1}:TM otimes {mathbb C}=T^{1,0} oplusT^{0,1} to T^{0,1}. We found that these two differentialoperators agree only on the space of functions Omega^0(M),unless T^{1,0} is involutive as well. We showed that thedifference is a derivation, which vanishes on Omega^0(M), andit is induced by the Nijenhuis tensor associated to pi^{0,1}.