Almost Isotropic Kaehler Manifolds

TL;DR

This paper classifies almost isotropic simply connected Kähler manifolds, showing they are either complex projective space, complex hyperbolic space, or foliated by complex Euclidean leaves.

Contribution

It provides a classification of almost isotropic Kähler manifolds, extending classical results on isotropic manifolds to a broader almost isotropic setting.

Findings

Manifolds are either complex projective or hyperbolic spaces.

Existence of foliations by complex Euclidean spaces.

Extension of Schur's classical result to almost isotropic manifolds.

Abstract

Let $M$ be a complete Riemannian manifold and suppose $p\in M$. For each unit vector $v \in T_p M$, the $\textit{Jacobi operator}$, $\mathcal{J}_v: v^\perp \rightarrow v^\perp$ is the symmetric endomorphism, $\mathcal{J}_v(w) = R(w,v)v$. Then $p$ is an $\textit{isotropic point}$ if there exists a constant $\kappa_p \in \mathbf{R}$ such that $\mathcal{J}_v = \kappa_p \textit{ Id}_{v^\perp}$ for each unit vector $v \in T_pM$. 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 $\textit{almost isotropic manifolds}$, 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 $\text{rank}(\mathcal{J}_v - \kappa_p \textit{Id}_{v^\perp}) \leq 1$ for each unit vector $v \in T_pM$. Our main theorem classifies the almost isotropic simply connected K\"ahler manifolds, proving that those of dimension $d=2n \geq 4$ are either isometric to complex projective space or complex hyperbolic space or are totally geodesically foliated by leaves isometric to $\mathbf{C}^{n-1}.$