Chern's Conjecture (Affine Geometry)

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.

An affine structure on a smooth $n$-manifold $M$ is an atlas of coordinate charts to open subsets of $\mathbb R^n$ whose transitions are restrictions of maps $x\mapsto Ax+b$ with $A\in\operatorname{GL}_n(\mathbb R)$ and $b\in\mathbb R^n$. Equivalently, $TM$ has a flat torsion-free connection. For a closed manifold, let $\chi(M)=\sum_{j=0}^{n}(-1)^j\dim H^j(M;\mathbb Q)$; closed means compact without boundary.

Statement recorded in the supplied source.

For every closed smooth manifold $M$ admitting an affine structure, must \[ \chi(M)=0? \] Neither completeness of the affine connection nor orientability of $M$ is assumed. [S2]

Short English statement

Must every compact manifold without boundary that has affine coordinate changes have zero Euler characteristic?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 1375 (GEOM-016). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 M. Cocos, On Polarized Pfaffians and the Euler Characteristic of Flat Affine Manifolds (2025), arXiv:1504.04852v12, Section 1 and Remark 2.1. https://arxiv.org/abs/1504.04852v12.