Curvature and Lagrangian submanifolds of the homogeneous nearly K\"ahler $\mathbb{C}P^3$

TL;DR

This paper provides explicit descriptions of the nearly K"ahler structure on , analyzes its curvature, isometry groups, and classifies Lagrangian submanifolds, including their rigidity and curvature properties.

Contribution

It introduces a tractable definition of the nearly K"ahler structure on , computes curvature tensors, and classifies Lagrangian submanifolds with rigidity results.

Findings

Explicit formulas for Riemann curvature tensors of 

Classification of extrinsically homogeneous Lagrangian submanifolds

Nonexistence of Lagrangians with constant sectional curvature

Abstract

A tractable definition of the homogeneous nearly K\"ahler structure on $\mathbb{C}P^3$ is given via the Hopf fibration, facilitating explicit computations and analysis. The description extends to all homogeneous metrics on $\mathbb{C}P^3$, providing expressions for their Riemann curvature tensors and full isometry groups. Rigid immersions are presented for all extrinsically homogeneous Lagrangian submanifolds in the nearly K\"ahler $\mathbb{C}P^3$, and the nonexistence of Lagrangians with constant sectional curvature is established.