Almost isotropic Kähler manifolds

Abstract

AbstractLetMbe a complete Riemannian manifold and suppose{p\in M}. For each unit vector{v\in T_{p}M}, theJacobi operator,{\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}}is the symmetric endomorphism,{\mathcal{J}_{v}(w)=R(w,v)v}. Thenpis anisotropic pointif there exists a constant{\kappa_{p}\in{\mathbb{R}}}such that{\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}}for each unit vector{v\in T_{p}M}. If all points are isotropic, thenMis 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 consideralmost isotropic manifolds, i.e. manifolds having the property that for each{p\in M}, there exists a constant{\kappa_{p}\in\mathbb{R}}such that the Jacobi operators{\mathcal{J}_{v}}satisfy{\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1}for each unit vector{v\in T_{p}M}. Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension{d=2n\geqslant 4}are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to{{\mathbb{C}}^{n-1}}.