[Paper Review] Negatively curved left-invariant metrics on Lie groups

Sigbjørn Hervik|arXiv (Cornell University)|Feb 10, 2010

Advanced Differential Geometry Research7 references3 citations

TL;DR

This paper investigates negatively curved left-invariant Riemannian metrics on Lie groups, showing that Ricci nilsolitons arise as solutions to higher-curvature gravity theories. It establishes general criteria for negative Ricci curvature and demonstrates that these metrics satisfy zero-action solutions in certain gravity models, linking geometric structures to theoretical physics via higher-curvature actions.

ABSTRACT

We discuss negatively curved homogeneous spaces admitting a simply transitive group of isometries, or equivalently, negatively curved left-invariant metrics on Lie groups. Negatively curved spaces have a remarkably rich and diverse structure and are interesting from both a mathematical and a physical perspective. As well as giving general criteria for having left-invariant metrics with negative Ricci curvature scalar, we also consider special cases, like Einstein spaces and Ricci nilsolitons. We point out the relevance these spaces play in some higher-dimensional theories of gravity. In particular, we show that the Ricci nilsolitons are Riemannian solutions to certain higher-curvature gravity theories.

Motivation & Objective

• To identify general conditions under which Lie groups admit left-invariant metrics with negative Ricci curvature scalar.
• To analyze special cases such as Einstein spaces and Ricci nilsolitons within the context of negatively curved homogeneous spaces.
• To establish the relevance of these geometric structures in higher-dimensional gravity theories.
• To demonstrate that Ricci nilsolitons are solutions to specific higher-curvature gravity actions with vanishing action.

Proposed method

• Utilizes left-invariant frames and orthonormal Cartan co-fibrations to describe Riemannian metrics on Lie groups.
• Applies structure constants of Lie algebras and their GL(n) and O(n) orbits to classify isometric and isomorphic metric structures.
• Employs the Ricci curvature tensor derived from the Levi-Civita connection and Killing vector fields to analyze curvature properties.
• Derives the condition for negative Ricci curvature using the Ricci tensor expression in terms of structure constants.
• Constructs a higher-curvature gravity action S(α,β) and shows that Ricci nilsolitons satisfy the field equations with S(α,β)[Hm,n] = 0.
• Analyzes the relationship between isometries, conformal boundaries, and the AdS/CFT correspondence in negatively curved spaces.

Experimental results

Research questions

• RQ1 Under what conditions does a Lie group admit a left-invariant metric with negative Ricci curvature scalar?
• RQ2 How do Ricci nilsolitons arise as solutions in higher-curvature gravity theories?
• RQ3 What is the role of the O(n) orbit of structure constants in classifying isometric left-invariant metrics on a Lie group?
• RQ4 Can Einstein metrics on solvable Lie groups be characterized within the framework of higher-curvature gravity actions?
• RQ5 To what extent do the symmetries of negatively curved homogeneous spaces relate to their conformal boundaries, particularly in the context of AdS/CFT?

Key findings

• Ricci nilsolitons on Lie groups are shown to be solutions to a class of higher-curvature gravity theories with vanishing action, S(α,β)[Hm,n] = 0.
• The paper derives a condition for negative Ricci curvature in terms of the structure constants of the Lie algebra and the metric components.
• For solvable Lie groups, the field equations of the higher-curvature action (28) select a particularly symmetric metric, indicating a distinguished geometric structure.
• The action S(α,β) vanishes on Ricci nilsolitons, suggesting they are exact solutions in these gravity models.
• The paper establishes that Einstein metrics on semisimple Lie groups arise as solutions when β > 0 in the higher-curvature action.
• The study reveals that the isometry group of real hyperbolic space coincides with the conformal group of its boundary, suggesting potential generalizations to other negatively curved spaces.