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Flat Hermitian Lie algebras are Kähler

Dongmei Zhang, Fangyang Zheng

TL;DR

The paper addresses whether a flat Hermitian Lie algebra must be Kähler. Building on Milnor’s flatness criterion and the BDF refinement for flat Lie algebras with Hermitian structures, it analyzes 2-step solvable Lie algebras using admissible frames and the Chern connection. The main result proves that any flat Hermitian Lie algebra has vanishing Chern torsion, hence is Kähler, aligning with the known Kähler-flat classification. This establishes a strong rigidity: flatness in the Riemannian sense enforces Kähler geometry for left-invariant Hermitian structures on Lie groups. The work leverages Hermitian geometry of 2-step solvable groups (FSFS2, CZ) and contributes to understanding Chern/Bismut-flat questions in unimodular settings.

Abstract

In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author.

Flat Hermitian Lie algebras are Kähler

TL;DR

The paper addresses whether a flat Hermitian Lie algebra must be Kähler. Building on Milnor’s flatness criterion and the BDF refinement for flat Lie algebras with Hermitian structures, it analyzes 2-step solvable Lie algebras using admissible frames and the Chern connection. The main result proves that any flat Hermitian Lie algebra has vanishing Chern torsion, hence is Kähler, aligning with the known Kähler-flat classification. This establishes a strong rigidity: flatness in the Riemannian sense enforces Kähler geometry for left-invariant Hermitian structures on Lie groups. The work leverages Hermitian geometry of 2-step solvable groups (FSFS2, CZ) and contributes to understanding Chern/Bismut-flat questions in unimodular settings.

Abstract

In 1976, Milnor classified all Lie groups admitting a flat left-invariant metric. They form a special type of unimodular 2-step solvable groups. Considering Lie groups with Hermitian structure, namely, a left-invariant complex structure and a compatible left-invariant metric, in 2006, Barberis-Dotti-Fino obtained among other things full classification of all Lie groups with Hermitian structure that are Kähler and flat. In this note, we examine Lie groups with a Hermitian structure that are flat, and show that they actually must be Kähler, or equivalently speaking, a flat Hermitian Lie algebra is always Kähler. In the proofs we utilized analysis on the Hermitian geometry of 2-step solvable Lie groups developed by Freibert-Swann and by Chen and the second named author.

Paper Structure

This paper contains 3 sections, 5 theorems, 32 equations.

Table of Contents

  1. Introduction and statement of result
  2. Preliminaries
  3. Proof of Theorem \ref{['thm3']}

Key Result

Theorem 1

Let $({\mathfrak g}, g)$ be a Lie algebra with a metric. Then $g$ is flat if and only if ${\mathfrak g}$ is the orthogonal direct sum ${\mathfrak h} \oplus {\mathfrak z} \oplus {\mathfrak g}'$, where ${\mathfrak z}$ is the center of ${\mathfrak g}$, ${\mathfrak g}'=[{\mathfrak g}, {\mathfrak g}]$ is

Theorems & Definitions (9)

  • Theorem 1: BDF
  • Theorem 2: BDF
  • Theorem 3
  • Lemma 1: CZ
  • Definition 1: admissible frames CZ
  • Lemma 2: CZ
  • proof : Proof of Theorem \ref{['thm3']}:
  • proof : Proof of Claim 1:
  • proof : Proof of Claim 2: