The structure Jacobi operator and the shape operator of real hypersurfaces in \(\mathbb CP^2\) and \(\mathbb CH^2\)
Let \((M, g)\) be a real hypersurface in \(\mathbb{C}P^2\) or \(\mathbb{C}H^2\) and \(A\) its shape operator. \(M\) carries a natural vector field \(\xi \) induced by a fixed unit normal vector field and the Kählerian structure of the ambient manifold; hence we have on \(M\) the structure Jacobi operator \(l=R(\cdot , \xi )\xi \). The two main results of the paper prove that there does not exist \((M, g)\) satisfying \(\mathcal{L}_Xl=\nabla _Xl\) respectively \(\mathcal{L}_XA=\nabla _XA\) for all \(X\in \mathbb{D}\) where \(\mathbb{D}\) is the distribution \(g\)-orthogonal to \(\xi \).
scientific article; zbMATH DE number 6362514
scientific article