TORSION OF GENERALIZED TANAKA-WEBSTER CONNECTIONS ON REAL HYPERSURFACES IN A COMPLEX SPACE FORM

Source abstract

Let $M$ be a real hypersurface in a nonflat complex space form $M_{n}(c)$, $c \neq 0$, endowed with the induced almost contact metric structure $(\phi, \xi, \eta, g)$. We investigate the torsion tensor of the $k$ th generalized Tanaka-Webster connection on $M$. We show that the torsion-free condition forces $M$ to be Hopf and that the shape operator satisfies $A X=k X$ for every vector field $X$ in the holomorphic distribution $D=\operatorname{ker} \eta$. Using the restriction of the torsion tensor to $D$, we obtain $k=0$, contradicting the assumption that $k$ is a nonzero real constant. Consequently, there exists no real hypersurface in a nonflat complex space form whose $k$ th generalized Tanaka-Webster connection is torsion-free.