Almost isotropic Kähler manifolds

Abstract

Abstract Let M be a complete Riemannian manifold and suppose p ∈ M {p\in M} . For each unit vector v ∈ T p ⁢ M {v\in T_{p}M} , the Jacobi operator, 𝒥 v : v ⊥ → v ⊥ {\mathcal{J}_{v}:v^{\perp}\rightarrow v^{\perp}} is the symmetric endomorphism, 𝒥 v ⁢ ( w ) = R ⁢ ( w , v ) ⁢ v {\mathcal{J}_{v}(w)=R(w,v)v} . Then p is an isotropic point if there exists a constant κ p ∈ ℝ {\kappa_{p}\in{\mathbb{R}}} such that 𝒥 v = κ p ⁢ Id v ⊥ {\mathcal{J}_{v}=\kappa_{p}\operatorname{Id}_{v^{\perp}}} for each unit vector v ∈ T p ⁢ M {v\in T_{p}M} . 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 ∈ M {p\in M} , there exists a constant κ p ∈ ℝ {\kappa_{p}\in\mathbb{R}} such that the Jacobi operators 𝒥 v {\mathcal{J}_{v}} satisfy rank ⁡ ( 𝒥 v - κ p ⁢ Id v ⊥ ) ≤ 1 {\operatorname{rank}({\mathcal{J}_{v}-\kappa_{p}\operatorname{Id}_{v^{\perp}}}% )\leq 1} for each unit vector v ∈ T p ⁢ M {v\in T_{p}M} . Our main theorem classifies the almost isotropic simply connected Kähler manifolds, proving that those of dimension d = 2 ⁢ n ⩾ 4 {d=2n\geqslant 4} are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to ℂ n - 1 {{\mathbb{C}}^{n-1}} .