[Paper Review] Metrics of positive Ricci curvature on quotient spaces

Lorenz Schwachhoefer, Wilderich Tuschmann|ArXiv.org|Mar 7, 2003

Geometric Analysis and Curvature Flows27 references4 citations

TL;DR

This paper establishes new classes of closed manifolds admitting invariant Riemannian metrics with positive Ricci curvature and almost nonnegative sectional curvature by taking quotients of compact homogeneous spaces and cohomogeneity one manifolds under free isometric group actions. The key result is that such quotients admit these curvature properties if and only if their fundamental group is finite, yielding new examples of manifolds with strong curvature and topological constraints.

ABSTRACT

We show that any closed biquotient with finite fundamental group admits metrics of positive Ricci curvature. Also, let M be a closed manifold on which a compact Lie group G acts with cohomogeneity one, and let L be a closed subgroup of G which acts freely on M. We show that the quotient N := M/L carries metrics of nonnegative Ricci and almost nonnegative sectional curvature. Moreover, if N has finite fundamental group, then N admits also metrics of positive Ricci curvature. Particular examples include infinite families of simply connected manifolds with the rational cohomology rings and integral homology of complex and quaternionic projective spaces.

Motivation & Objective

• To identify new classes of closed manifolds that admit Riemannian metrics with positive Ricci curvature.
• To investigate the curvature behavior of quotient spaces obtained from free isometric group actions on manifolds with positive or nonnegative Ricci curvature.
• To determine topological and geometric conditions—particularly finiteness of the fundamental group—under which such quotients inherit positive Ricci curvature.
• To extend known results on metrics of positive Ricci and almost nonnegative sectional curvature to new families of manifolds, including biquotients and quotients of Brieskorn manifolds.
• To construct explicit examples of simply connected manifolds with rational cohomology rings of complex and quaternionic projective spaces, admitting such curvature metrics.

Proposed method

• Analyzing the Ricci curvature of quotient manifolds M/L where M is a compact homogeneous space or cohomogeneity one manifold and L acts freely by isometries.
• Applying the Gysin sequence to compute the integral cohomology rings of quotient spaces, particularly for S1 and Sp(1) quotients of Brieskorn manifolds.
• Using invariant metric constructions via the normalizer of the group action to ensure invariance and control curvature properties.
• Leveraging known results on cohomogeneity one manifolds and biquotients to establish existence of metrics with positive Ricci and almost nonnegative sectional curvature.
• Constructing explicit families of manifolds via quotients of Brieskorn manifolds by free circle and 3-sphere actions, with cohomology rings isomorphic to those of CP^{2m-1} and HP^{2m-1}.
• Verifying cohomogeneity two actions of S1×SU(m) and S1×Sp(m) on the quotient manifolds and confirming their isometric invariance under these actions.

Experimental results

Research questions

• RQ1 Under what conditions does the quotient of a compact homogeneous space or cohomogeneity one manifold under a free isometric action admit a metric of positive Ricci curvature?
• RQ2 Can metrics of positive Ricci curvature and almost nonnegative sectional curvature coexist on quotient manifolds, and what topological obstructions exist?
• RQ3 Do biquotients of compact Lie groups with finite fundamental group admit metrics of positive Ricci and almost nonnegative sectional curvature?
• RQ4 What is the cohomology structure of quotients of Brieskorn manifolds by free S1 and Sp(1) actions, and how does it relate to known spaces like CP^{2m-1} and HP^{2m-1}?
• RQ5 Are there new examples of simply connected manifolds with rational cohomology rings of CP^n and HP^n that admit both positive Ricci and almost nonnegative sectional curvature?

Key findings

• The quotient of a compact homogeneous space or cohomogeneity one manifold under a free isometric action admits a metric of positive Ricci curvature if and only if its fundamental group is finite.
• All biquotients G//H of compact Lie groups G with finite fundamental group admit invariant metrics of positive Ricci and almost nonnegative sectional curvature.
• Quotients of Brieskorn manifolds by free S1 and Sp(1) actions yield simply connected manifolds with rational cohomology rings isomorphic to those of CP^{2m-1} and HP^{2m-1}, respectively, and admit metrics of positive Ricci and almost nonnegative sectional curvature.
• For d ≥3, the manifolds N^{4m-2}_d and ˜N^{8m-4}_d are not diffeomorphic to any biquotient and do not admit Lie group actions of cohomogeneity less than two.
• The quotients N^{4m-2}_d and ˜N^{8m-4}_d admit fiber bundles S^2 → N^{8m-2}_d → ˜N^{8m-4}_d that are Riemannian fibrations under the constructed metrics.
• For m=1, the quotient ˜N^4_d is diffeomorphic to S^4, and the corresponding S^3-bundles over S^4 are classified by their Euler class d, with all nontrivial ones realized as quotients of Brieskorn manifolds.