The Hopf Conjectures

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

Let $(M,g)$ be a smooth connected closed Riemannian manifold of dimension $2m$, where $m\geq1$ and closed means compact without boundary. For a two-dimensional linear subspace $\sigma\subset T_pM$, let $K_g(\sigma)$ be its sectional curvature. Write $\chi(M)=\sum_{j=0}^{2m}(-1)^j\dim_{\mathbb Q}H_j(M;\mathbb Q)$ for the Euler characteristic.

Statement recorded in the supplied source.

Do the following two assertions hold for every such manifold? \[ \left(K_g(\sigma)>0\text{ for every }p,\sigma\right)\Longrightarrow\chi(M)>0, \] \[ \left(K_g(\sigma)<0\text{ for every }p,\sigma\right)\Longrightarrow(-1)^m\chi(M)>0. \] These are respectively the positive- and negative-sectional-curvature Hopf sign questions. [S2]

Short English statement

Must positive sectional curvature force positive Euler characteristic, and must negative sectional curvature force the dimension-dependent alternating sign?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1210 (GEO-019). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Oliver Knill, Integral geometric Hopf conjectures, arXiv:2001.01398v1 (2020), abstract and Section 1.1. https://arxiv.org/pdf/2001.01398.