# DERIVATIONS ON CR MANIFOLDS

Ryu, Jeong-Seog;Yi, Seung-Hun

• Published : 2004.01.01
• 51 8

#### Abstract

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

#### Keywords

derivation;tangential Cauchy-Riemann operator;CR-manifold

#### References

1. Natural operations in differential geometry I.Kolar;P.W.Michor;J.Slovak
2. J. Korean Math. Soc. v.24 Curvatures and complex structures H.J.Kim
3. Trans. Amer. Math. Soc. v.92 Natural operators on differential forms R.Palais https://doi.org/10.1090/S0002-9947-1959-0116352-7