Real hypersurfaces of non-flat complex space forms in terms of the Jacobi structure operator (Q2834179)

Let \(M_n(c)\) be an \(n\)-dimensional complex space form of constant holomorphic sectional curvature \(c\). The induced almost contact metric structure of a real hypersurface \(M\) in \(M_n(c)\) will be denoted by \((\phi,\xi,\eta,g)\). Let \(A\) and \(l\) be the shape operator of \(M\) and the Jacobi structure operator \(lX=R(X,\xi)\xi\), respectively, \(R\) being the curvature tensor. The main result can be stated as follows.NEWLINENEWLINE Theorem. Let \(M\) be a real hypersurface of \(M_n(c)\), with \(n>2\) and \(c\neq0\), satisfying \(\phi l=l\phi\) on \({\text ker}(\eta)\). If one of conditions (i) \(\nabla_\xi l=0\) or (ii) \(lA=Al\) holds on \(\text{ker}(\eta)\) or on \(\text{span}\{\xi\}\), then \(M\) is a Hopf hypersurface. Furthermore, if \(\eta(A\xi)\neq0\), then \(M\) is locally congruent to a model space of type \(A\).NEWLINENEWLINE Model spaces of type \(A\) are given in papers by \textit{S. Montiel} and \textit{A. Romero} [Geom. Dedicata 20, 245--261 (1986; Zbl 0587.53052)], and \textit{M. Okumura} [Trans. Am. Math. Soc. 212, 355--364 (1975; Zbl 0288.53043)]. In \(\mathbb{C}P^n\), these spaces are tubes of radius \(r\), \(0<r<\pi/2\), over (\(A_1\)) a hyperplane \(\mathbb{C}P^{n-1}\) or (\(A_2\)) a totally geodesic \(\mathbb{C}P^k\), \(1\leq k\leq n-2\). In \(\mathbb{C}H^n\), they are (\(A_0\)) horospheres, (\(A_1\)) geodesic hyperspheres or tubes over a hyperplane \(\mathbb{C}H^{n-1}\), or (\(A_2\)) tubes over a totally geodesic \(\mathbb{C}H^k\), \(1\leq k\leq n-2\).