[Paper Review] Nonnegatively curved 5-manifolds with non-abelian symmetry

Fábio Luiz Borges Simas|arXiv (Cornell University)|Dec 20, 2012

Geometric Analysis and Curvature Flows2 references4 citations

TL;DR

This paper classifies 5-dimensional simply-connected manifolds admitting an effective action by SO(3) or SU(2) that support an invariant metric of nonnegative curvature. Using orbit space analysis, biquotient constructions, and curvature constraints, it shows that such manifolds are equivariantly diffeomorphic to S⁵, S³×S², the Wu manifold W=SU(3)/SO(3), or specific biquotients Nₘ,ₙˡ; for positive curvature, only S⁵ with a linear action is possible, while W and other Nₘ,ₙˡ fail due to fixed point set dimension or isotropy obstructions.

ABSTRACT

We classify compact simply-connected 5-dimensional manifolds which admit a metric of nonnegative curvature with a connected non-abelian group acting by isometries. We show that they are diffeomorphic to either S^5, S^3 x S^2, the nontrivial S^3-bundle over S^2 or the Wu-manifold, SU(3)/SO(3). This result is a consequence of our equivariant classification of all SO(3) and SU(2)-actions on compact simply-connected 5-manifolds. In the case of positive curvature we obtain a partial classification.

Motivation & Objective

• To classify 5-dimensional simply-connected manifolds that admit an effective action by a connected non-abelian Lie group (SO(3) or SU(2)) and support an invariant metric of nonnegative curvature.
• To determine which of these manifolds can admit a metric of positive curvature under the same symmetry assumption.
• To provide a complete classification of such manifolds up to equivariant diffeomorphism, using orbit space geometry and isotropy data.
• To analyze the role of fixed point sets and isotropy groups in obstructing positive curvature, particularly via Frankel’s Lemma and Gauss-Bonnet arguments.
• To construct and characterize a family of biquotients Nₘ,ₙˡ as model spaces for non-abelian group actions on 5-manifolds with nonnegative curvature.

Proposed method

• Classify all simply-connected 5-manifolds admitting a non-abelian Lie group action by analyzing their diffeomorphism types: S⁵, S³×S², the nontrivial S³-bundle over S², connected sums of S³×S², and connected sums of copies of the Wu manifold W and Brieskorn manifold B.
• Construct the biquotient spaces Nₘ,ₙˡ = (SU(2) × S³)/S¹ via a free S¹-action defined by (p, (z,w)) ↦ (p xˡ, (xᵐ z, xⁿ w)), with x ∈ S¹ ⊂ SU(2), and show they are diffeomorphic to S³×S² if m+n even, to S³ ×̃ S² otherwise.
• Define an SO(3) or SU(2) action on Nₘ,ₙˡ by left multiplication on the SU(2) factor, and compute isotropy types: Zₘ, Zₙ, Zgcd(m,n), or SO(2), with ineffective kernel Z₂ when gcd(m,n) even.
• Apply O’Neill’s formula to show that the standard product metric on S³×S³ induces a nonnegatively curved invariant metric on Nₘ,ₙˡ.
• Use orbit space geometry: the quotient M/G is a 3-dimensional orbifold with boundary, and its curvature properties (nonnegative or positive) are inherited via O’Neill’s formula.
• Apply the Soul Theorem and Gauss-Bonnet to constrain the number of isolated fixed points: ≤3 for nonnegative curvature, ≤2 for positive curvature, based on angle sums in the orbit space polygon.

Experimental results

Research questions

• RQ1 Which simply-connected 5-manifolds admit an effective action by SO(3) or SU(2) and support an invariant metric of nonnegative curvature?
• RQ2 Which of these manifolds can support a metric of positive curvature under the same symmetry condition?
• RQ3 What is the structure of the orbit space M/G for such actions, and how does its curvature constrain the possible manifolds?
• RQ4 How do isotropy groups and fixed point sets obstruct positive curvature, particularly via Frankel’s Lemma?
• RQ5 Which biquotients Nₘ,ₙˡ are candidates for admitting positive curvature, and what conditions on m, n, l are necessary?

Key findings

• The only simply-connected 5-manifolds admitting an effective SO(3) or SU(2) action and an invariant metric of nonnegative curvature are S⁵, S³×S², the Wu manifold W=SU(3)/SO(3), and the biquotients Nₘ,ₙˡ.
• For positive curvature, only S⁵ with a linear SO(3) or SU(2) action admits such a metric; all other candidates, including W and Nₘ,ₙˡ with gcd(m,n)≥3, are ruled out.
• The action on W has exactly three isolated fixed points, which violates the upper bound of two fixed points allowed in positively curved manifolds with such symmetry.
• The biquotient Nₘ,ₙˡ with gcd(m,n)≥3 has a fixed point set of dimension 3 for the principal isotropy group Zgcd(m,n), violating Frankel’s Lemma in positively curved manifolds.
• The manifold N₀,₀¹ has a fixed point set that is the disjoint union of two 3-spheres, which also violates the dimension sum constraint in Frankel’s Lemma for positive curvature.
• For fixed m,n with gcd(m,n)=1 or 2, only three of the Nₘ,ₙˡ are candidates for positive curvature, as shown by equivariance and isotropy analysis.