Real hypersurfaces of a complex projective space in terms of the Jacobi operators
Let \(M\) be a connected oriented real hypersurface of the complex projective space \((\mathbb{C} P^m,J,\widetilde g)\) with Fubini-Study metric \(\widetilde g\) of constant holomorphic sectional curvature 4, and let \((\phi,\zeta,\xi,g)\) be an almost contact metric structure on \(M\) induced by \((J,\widetilde g)\). Let \(A\) denote the shape operator of \(M\) in \(\mathbb{C} P^n\) and \(\mathcal R_\xi\) the Jacobi operator with respect to the structure vector field \(\xi\). It is proved that if \(\mathcal R_\xi\phi A=A\phi\mathcal R_\xi\), where \(\phi\) is the structure tensor, then \(M\) is locally congruent to a geodesic hypersurface or a tube of radius \(r\) over a totally geodesic \(\mathbb{C} P^k\), \(1\leq k \leq n-2\), \(0<r<\frac\pi 2\). A similar result is derived from the conditions \(\mathcal R_\xi\phi=\phi\mathcal R_\xi\) and \(\mathcal R_\xi A=A\mathcal R_\xi\).
scientific article; zbMATH DE number 1261212