CR-hypersurfaces of a conformal Kenmotsu manifold satisfying certain shape operator conditions
In the reviewed paper the authors introduce conformal Kenmotsu manifolds. A smooth manifold \(M^{2n+1}\) with an almost contact metric structure \((\phi,\eta,\xi,g)\) is called a conformal Kenmotsu manifold if there is a positive smooth function \(f\colon M\to\mathbb R\) such that \(\tilde{g}=\mathrm{exp}(f)g\), \(\tilde{\xi}=(\mathrm{exp}(-f))^{\frac 12}\xi\), \(\tilde{\eta}= (\mathrm{exp}(f))^{\frac 12}\eta\), \(\tilde{\phi}=\phi\), is a Kenmotsu structure on \(M\). Next the authors investigate CR-hypersurfaces of conformal Kenmotsu manifolds. First they consider CR-hypersurfaces with a parallel shape operator. The authors prove that if such hypersurface is tangent to the Lie vector field \(\omega^\sharp=\mathrm{grad}\,f\), then it is totally geodesic, and if it is normal to \(\omega^\sharp\), then it is totally umbilic. Finally the authors construct a 3-dimensional example of a conformal Kenmotsu manifold which is not Kenmotsu.
scientific article; zbMATH DE number 6806100
scientific article