Almost isotropic Kahler manifolds

Let M be a complete Riemannian manifold and suppose p is an element of M. For each unit vector v is an element of TpM, the Jacobi operator, J v : v(perpendicular to) -> v(perpendicular to) is the symmetric endomorphism, Jv.w/ D R.w; v/v. Then p is an isotropic point if there exists a constant k(p) is an element of R such that J(v) = kappa(p) Idv(perpendicular to) for each unit vector v is an element of TpM. If all points are isotropic, then M is said to be isotropic; it is a classical result of Schur that isotropic manifolds of dimension at least 3 have constant sectional curvatures. In this paper we consider almost isotropic manifolds, i.e. manifolds having the property that for each p is an element of M, there exists a constant kappa(p) is an element of R such that the Jacobi operators J v satisfy rank. (J(v) = kappa(p) Idv(perpendicular to)) <= 1 for each unit vector v is an element of TpM. Our main theorem classifies the almost isotropic simply connected Kahler manifolds, proving that those of dimension d = 2n >= 4 are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to Cn-1.