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Proof of Chern conjecture for flat affine manifolds

Mihail Cocos

TL;DR

The paper proves Chern's conjecture by constructing a global one-parameter deformation $ abla^t$ from a flat affine connection to the Levi-Civita connection of a global metric, and producing a transgression form via a polarized Pfaffian. The curvature $ ext{Ω}^t$ remains skew-symmetric throughout, enabling the definition of $e_t(x)= rac{1}{(2π)^n} ext{Pf}( ext{Ω}^t(x))$ with $e_0=0$ and $e_1$ as the genuine Euler form; a transgression formula shows $e_1$ is exact, implying $oxed{χ(M)= rac{1}{(2π)^n} ext{Pf}( ext{Ω}^{g})}$ integrates to zero. The approach unifies prior partial results and provides a complete proof via a local-to-global deformation and an invariant polarized Pfaffian, with implications for deforming flat connections to globally metric ones. The work also clarifies obstructions in terms of the Euler form for locally metric connections, effectively removing them in the flat affine setting.

Abstract

We prove Chern conjecture, which states that the Euler characteristic vanishes for closed flat affine manifolds. Our key innovation is a deformation argument for the Euler form.

Proof of Chern conjecture for flat affine manifolds

TL;DR

The paper proves Chern's conjecture by constructing a global one-parameter deformation from a flat affine connection to the Levi-Civita connection of a global metric, and producing a transgression form via a polarized Pfaffian. The curvature remains skew-symmetric throughout, enabling the definition of with and as the genuine Euler form; a transgression formula shows is exact, implying integrates to zero. The approach unifies prior partial results and provides a complete proof via a local-to-global deformation and an invariant polarized Pfaffian, with implications for deforming flat connections to globally metric ones. The work also clarifies obstructions in terms of the Euler form for locally metric connections, effectively removing them in the flat affine setting.

Abstract

We prove Chern conjecture, which states that the Euler characteristic vanishes for closed flat affine manifolds. Our key innovation is a deformation argument for the Euler form.

Paper Structure

This paper contains 10 sections, 7 theorems, 53 equations.

Table of Contents

  1. A brief history of the problem. Main result
  2. Construction of cohomology classes $e_t$
  3. The local metrics $h^x$
  4. The global one parameter family of connections $\nabla^t$ and their Euler class
  5. The Transgression Form and the Polarized Pfaffian
  6. Skew-Symmetry of the Time-Derivative Connection
  7. Definition and Invariance of the Polarized Pfaffian
  8. Transgression Formula and the Main Result
  9. Conclusion
  10. The global connection tensor

Key Result

Lemma 2.3

Let $(y^1,\dots,y^m)$ be local affine coordinates on an open subset $U \subset M$, and let $x \in U$ be arbitrary. Let $\Gamma$ denote the Christoffel symbols. Then In particular, $\nabla^t$ depends smoothly on $t$ and $x$, and by definition is a global connection on $TM$.

Theorems & Definitions (18)

  • Remark 2.1: Orientability Assumption
  • Definition 2.2: Global one-parameter family of connections
  • Lemma 2.3: Scaling of Christoffel symbols
  • proof
  • Lemma 2.4
  • proof
  • Remark 2.5
  • Lemma 3.1: Skew-Symmetry of the Time-Derivative Connection One-Form
  • proof
  • Definition 3.2: Polarized Pfaffian
  • ...and 8 more