[Paper Review] Positively curved metrics on symmetric spaces with large symmetry rank

Lee Kennard|arXiv (Cornell University)|Sep 20, 2012

Geometric Analysis and Curvature Flows12 references3 citations

TL;DR

This paper establishes a cohomological obstruction to the existence of positively curved Riemannian metrics on symmetric spaces with large symmetry rank, showing that such manifolds must have rational cohomology 4-periodic up to a certain degree. The key result is that if a simply connected, positively curved manifold has symmetry rank at least $2\log_2 n + \frac{c}{2} - 1$, its rational cohomology is 4-periodic up to degree $c$, which severely restricts the possible symmetric space types that can admit such metrics—excluding most rank-two or higher symmetric spaces unless they are products of spheres and rank-one or low-rank Grassmannians.

ABSTRACT

We prove an obstruction at the level of rational cohomology in small degrees to the existence of positively curved metrics with large symmetry rank. The symmetry rank bound is logarithmic in the dimension of the manifold. As an application, we provide evidence for a generalized conjecture of Hopf that says that no symmetric space of rank at least two admits a metric with positive curvature.

Motivation & Objective

• To investigate whether symmetric spaces of rank ≥2 can admit positively curved metrics, extending Hopf's conjecture.
• To establish a topological obstruction based on rational cohomology in small degrees for manifolds with large symmetry rank.
• To provide evidence for the generalized Hopf conjecture by restricting the possible rational cohomology types of positively curved symmetric spaces.
• To analyze the implications of high symmetry rank on Betti numbers and rational ellipticity in positively curved manifolds.
• To connect the results to the Bott and Chern conjectures via symmetry rank constraints.

Proposed method

• Define 4-periodic rational cohomology up to degree $c$ as a key topological condition: existence of a class $x \in H^4(M;\mathbb{Q})$ such that multiplication by $x$ induces isomorphisms in degrees $0 < i < c-4$.
• Use the symmetry rank condition $\geq 2\log_2 n + \frac{c}{2} - 1$ to derive a cohomological obstruction via equivariant cohomology and spectral sequence techniques.
• Apply the Künneth theorem and dimension counting arguments to compare Betti numbers in products of manifolds under periodicity assumptions.
• Use classification results for symmetric spaces and their rational cohomology rings to constrain possible types of symmetric spaces satisfying the periodicity condition.
• Leverage injectivity and surjectivity of multiplication maps in cohomology to prove that irreducible factors beyond the first must be spheres.
• Compare the results with known examples like $\mathbb{C}P^n$, $\mathbb{H}P^n$, and Grassmannians to show that only specific products can satisfy the cohomological condition.

Experimental results

Research questions

• RQ1 Can a symmetric space of rank ≥2 admit a positively curved metric under high symmetry rank?
• RQ2 What rational cohomology conditions must a positively curved manifold satisfy if its symmetry rank is at least $2\log_2 n + \frac{c}{2} - 1$?
• RQ3 Which symmetric spaces can have rational cohomology 4-periodic up to degree $c$ under such symmetry rank constraints?
• RQ4 Does the 4-periodicity condition on rational cohomology rule out products like $S^k \times S^{n-k}$ with $1 < k < 16$?
• RQ5 To what extent does high symmetry rank imply vanishing of odd Betti numbers in positively curved manifolds?

Key findings

• If a closed, simply connected, positively curved $n$-manifold has symmetry rank at least $2\log_2 n + \frac{c}{2} - 1$, then its rational cohomology is 4-periodic up to degree $c$.
• For $c \geq 16$, any symmetric space $N$ with the rational cohomology of such a manifold must be a product of spheres and either a rank-one symmetric space or a Grassmannian $SO(p+q)/SO(p)\times SO(q)$ with $p \in \{2,3\}$.
• The rational cohomology of $S^k \times S^{n-k}$ with $1 < k < 16$ cannot satisfy the 4-periodicity condition required under high symmetry rank, so such products cannot admit positively curved metrics with large symmetry rank.
• If $n \equiv 0 \mod 4$, then the rational cohomology of $M$ in degrees less than $c$ agrees with that of $S^n$, $\mathbb{C}P^{n/2}$, or $\mathbb{H}P^{n/4}$, implying $b_{2i+1}(M) = 0$ for $2i+1 < c$.
• The Betti numbers $b_4(M)$ and $b_8(M)$ are at most 1 for such manifolds, which rules out connected sums of $\mathbb{C}P^n$, $\mathbb{H}P^n$, or $\mathrm{Ca}P^2$ under high symmetry rank.
• The fundamental group of a positively curved manifold with symmetry rank $2\log_2(4n+1)$ is isomorphic to that of a rational homology sphere, supporting modified versions of the Chern conjecture.