Hypersurfaces with Codazzi-type shape operator for a Tanaka-Webster connection (Q2834188)

In [\textit{J. D. D. Pérez} and \textit{Y. J. Suh}, Publ. Math. 83, No. 4, 707--714 (2013; Zbl 1299.53094)] the authors obtained a classification of real hypersurfaces in the complex projective space whose shape operator is of Codazzi type with respect to a generalized Tanaka-Webster connection with a condition on the principal curvature of the structure vector field. Next they obtained a classification of real hypersurfaces in the complex projective space whose shape operator is generalized Tanaka-Webster parallel with the same condition. In the present paper, the authors complete the above classification obtaining similar results for the complex hyperbolic space \(\mathbb{C}H^n\) as well as for \(\mathbb{C}P^n\). The proof holds for all \(n\geq 2\). After recalling the definition of the g-Tanaka Webster connections, they study some properties of Codazzi tensors (the tensors of Codazzi type) and obtain the following result. Theorem 5. Let \(M^{2n-1}\), \(n\geq 3\), be a real hypersurface in \(\mathbb{C}P^n\) or \(\mathbb{C}H^n\) whose shape operator is of Codazzi type with respect to a g-Tanaka-Webster connection. Then \(M\) must be a Hopf hypersurface.NEWLINENEWLINE The case \(n=2\) is studied separately.NEWLINENEWLINE Theorem 7. Let \(M^3\) be a real hypersurface of \(\mathbb{C}P^2\) or \(\mathbb{C}H^2\) whose shape operator is of Codazzi type with respect to a g-Tanaka-Webster connection. Then \(M\) must be a Hopf hypersurface.NEWLINENEWLINE Finally, the authors study the Hopf hypersurfaces with \(\widehat{\nabla}^{(k)}\)-Codazzi shape operator.NEWLINENEWLINE Theorem 8. Let \(M^{2n-1}\), \(n\geq 2\), be a Hopf hypersurface in \(\mathbb{C}P^n\) or \(\mathbb{C}H^n\) and let \(\widehat{\nabla}^{(k)}\) be a g-Tanaka-Webster connection for which \(2k\neq \alpha\). Then \(M\) is of Codazzi type with respect to \(\widehat{\nabla}^{(k)}\) if and only if \(M\) is an open subset of a type-A hypersuface.