[Paper Review] Homogeneous Metrics with nonnegative curvature

Lorenz Schwachhöfer, Kristopher Tapp|ArXiv.org|Apr 23, 2008

Geometric Analysis and Curvature Flows8 references4 citations

TL;DR

This paper investigates G-invariant Riemannian metrics on homogeneous spaces G/H with nonnegative sectional curvature, focusing on curvature-preserving deformations via inverse-linear paths that scale the metric along intermediate subalgebras. The key contribution is a sufficient condition—based on Lie bracket control—under which such deformations preserve nonnegative curvature, with symmetric pairs (K,H) guaranteeing curvature nonnegativity for t ∈ (−∞, 1/4], yielding new examples of nonnegatively curved homogeneous metrics.

ABSTRACT

Given compact Lie groups H\subset G, we study the space of G-invariant metrics on G/H with nonnegative sectional curvature. For an intermediate subgroup K between H and G, we derive conditions under which enlarging the Lie algebra of K maintains nonnegative curvature on G/H. Such an enlarging is possible if (K,H) is a symmetric pair, which yields many new examples of nonnegatively curved homogeneous metrics. We provide other examples of spaces G/H with unexpectedly large families of nonnegatively curved homogeneous metrics.

Motivation & Objective

• To classify G-invariant metrics on G/H with nonnegative sectional curvature beyond the normal homogeneous metric.
• To understand how enlarging the metric along an intermediate subalgebra K (H ⊂ K ⊂ G) affects curvature.
• To derive sufficient conditions under which such metric deformations preserve nonnegative curvature.
• To extend known results on curvature preservation in Riemannian submersions and Cheeger deformations.
• To provide new examples of nonnegatively curved homogeneous spaces using symmetric pair structures.

Proposed method

• Uses inverse-linear paths of inner products on the orthogonal complement 𝔪 ⊕ 𝔰 of 𝔥 in 𝔤, parameterized by t, to deform the metric.
• Applies Cheeger’s method to show the solution space of nonnegatively curved metrics is star-shaped around the normal metric.
• Derives curvature variation formulas inspired by Müter’s power series for inverse-linear paths.
• Analyzes the metric deformation g_t that scales the 𝔪-component by (1−t)⁻¹ while keeping the 𝔰-component fixed.
• Establishes a necessary and sufficient condition for nonnegative curvature: ||[Xᵐ,Yᵐ]ᵐ|| ≤ C·||[X,Y]|| for all X,Y ∈ 𝔭.
• Proves that if (K,H) is a symmetric pair, then g_t has nonnegative curvature for all t ∈ (−∞, 1/4], including t=1/4.

Experimental results

Research questions

• RQ1 Under what conditions does an inverse-linear deformation of the normal metric on G/H preserve nonnegative sectional curvature?
• RQ2 Can the curvature remain nonnegative when the metric is scaled along an intermediate subalgebra 𝔨?
• RQ3 What structural properties of the pair (K,H) ensure that the deformed metric g_t remains nonnegatively curved?
• RQ4 Is there a universal upper bound on the scaling parameter t beyond which curvature becomes negative?
• RQ5 How do symmetric pairs (K,H) contribute to constructing new families of nonnegatively curved homogeneous metrics?

Key findings

• The inverse-linear path g_t preserves nonnegative curvature for small t > 0 if and only if ||[Xᵐ,Yᵐ]ᵐ|| ≤ C·||[X,Y]|| for all X,Y ∈ 𝔭.
• When (K,H) is a symmetric pair, the metric g_t has nonnegative curvature for all t ∈ (−∞, 1/4], with t=1/4 being sharp.
• The scaling factor 4/3 at t=1/4 matches known upper bounds in Hopf fibrations and coset fibrations, suggesting universality.
• For H trivial, nonnegative curvature is preserved under metric deformation only if the semi-simple part of 𝔨 is an ideal in 𝔤.
• The hypothesis ||Xᵐ ∧ Yᵐ|| ≤ C·||[X,Y]|| ensures that arbitrary Ad_H-invariant metrics close to g₀ with adjusted 𝔪-metric remain nonnegatively curved.
• A contradiction argument proves that no sequence of unit vectors can make [X_n,Y_n] → 0 under the given constraints, confirming curvature preservation in the G₂/SU(3) case.