Non-compact Einstein manifolds with unimodular isometry group

TL;DR

This paper proves that certain negatively curved Einstein manifolds with a unimodular isometry group decompose into a product of symmetric and compact Einstein manifolds, using advanced geometric and Lie group techniques.

Contribution

It establishes a splitting theorem for negative Einstein manifolds with unimodular isometry groups, extending understanding of their geometric structure.

Findings

Manifolds split into symmetric and compact Einstein factors.

Proper isometric actions lead to isometric product decompositions.

The proof combines polar actions, Lie theory, and maximum principles.

Abstract

We show that a negative Einstein manifold admitting a proper isometric action of a connected unimodular Lie group with compact, possibly singular, orbit space splits isometrically as a product of a symmetric space and a compact negative Einstein manifold. The proof involves the theory of polar actions, Lie-theoretic arguments and maximum principles.