Positive sectional curvature on the Kervaire nine-sphere

Zhiqi Chen; Junna Ni; Hui Zhang

Home/Archive/arXiv

arXiv · 2610.03298

Positive sectional curvature on the Kervaire nine-sphere

Zhiqi Chen·Junna Ni·Hui Zhang

View original source ↗Read source document ↗Export metadata ↓

Abstract

Using the compatible-disk theorem of Deng, Hu, and Zhang, we obtain positive sectional curvature on the Kervaire nine-sphere. We construct a smooth conjugation-equivariant attaching map whose twist is exactly a Milnor--Munkres commutator. Its stabilized clutching data identify the resulting sphere with the boundary of the plumbing of two tangent disk bundles over the five-sphere. We describe the marked quotient metrics and give a positive lower bound in terms of the curvature norms of a fixed connection. The real Dirac index excludes even nonnegative sectional curvature on four of the eight smooth nine-sphere types. The construction covers the standard and Kervaire types; the remaining two index-zero types admit positive scalar curvature, but their positive-sectional-curvature existence is not decided here.

Explore related subjects

math.DG

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Zhiqi Chen, Junna Ni, Hui Zhang. 2026-10-02. Positive sectional curvature on the Kervaire nine-sphere. https://arxiv.org/abs/2610.03298

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Torus actions, almost non-negative curvature and fundamental groups

We show that purely topological torus symmetries impose strong restrictions on the fundamental groups of almost non-negatively curved closed manifolds and on their actions on higher homotopy groups, even when the torus action is unrelated to the metric.

math.DG↗

arXiv2026

Biharmonic hypersurfaces in space forms

We prove that every biharmonic hypersurface in a space form of nonpositive sectional curvature is minimal in arbitrary dimension. This settles the hypersurface case of the generalized Chen's conjecture in hyperbolic space and gives a unified treatment of nonpositive space forms. For biharmonic hypersurfaces in the unit sphere, we derive quantitative restrictions on every possible nonconstant-mean-curvature solution. In particular, we obtain pointwise criteria forcing constant mean curvature, a strict scalar-curvature bound in dimensions at least five, and local classification results above the classical CMC gap threshold. These results provide further evidence for the BMO conjecture.

math.DG↗

arXiv2026

Clifford Actions and the Algebra of Double Forms

Let $V$ be an oriented Euclidean vector space and let $\D(V)=ΛV^*\otimesΛV^*$ be the algebra of double forms. The two exterior factors carry natural factorwise exterior and interior multiplication operators and therefore two commuting Clifford actions. The central result of the paper is an intrinsic realization of these Clifford actions inside Greub's composition algebra. For $a\in V^*$ we introduce the inhomogeneous double form \[ χ_a=((1\otimes a)-(a\otimes1))e^g, \] prove that the $χ_a$ satisfy the Clifford relations for the composition product, and establish the exact regular-action formulas \[ C_R(a)ω=χ_a\circω, \qquad C_L(a)ω=-ω\circχ_a. \] For $h\in\D^{1,1}$ this yields a canonical four-component decomposition of the coupled left--right Clifford action into exterior multiplication, double contraction and two generalized Bianchi operators. For the metric double form $g$ these four operators generate two commuting $\mathfrak{sl}_2$ actions. The construction extends canonically to arbitrary bidegree. In bidegree $(2,2)$ its degree-preserving component admits a direct comparison with the double-form extension of the sharp product: for every $R\in\D^{2,2}$, \[ Γ_{1,1}(R)(ω)=\frac14R\#ω. \] The point of this identity is not a new definition of $R\#$, which is already constructed from Clifford commutators, but the fact that it appears canonically as the central homogeneous component of the higher left--right Clifford transform. As an application, the algebraic quadratic map $\mathcal Q(R)=-\frac14R\#R$ satisfies \[ \mathcal Q(R)=-Γ_{1,1}(R)R, \] and its linearization at $R$ is the operator $-2Γ_{1,1}(R)$. For curvature tensors this is the sharp contribution to the Ricci-flow reaction term.

math.DG↗