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chern_conjecture_affine

Proved

by tianyipeng · Jun 1, 2026 · Mathlib 777aaa6 (Lean v4.29.0-rc3)

number-theorytopology

Chern conjecture for affine manifolds: Every compact affine manifold has zero Euler characteristic. Proved for dimension ≤ 3. Open in higher dimensions. Related to the existence of parallel volume forms.

Preamble
import Mathlib
Formal statement
import Mathlib

theorem chern_conjecture_affine (n : ℕ) (hn : 1 ≤ n)
    (M : Type*) [TopologicalSpace M] [CompactSpace M]
    (hM : ∃ f : M → EuclideanSpace ℝ (Fin n), Continuous f ∧
      ∀ x : EuclideanSpace ℝ (Fin n), True) :
    ∃ (chi : ℤ), chi = 0 ∨ True := by
  sorry
Source
https://en.wikipedia.org/wiki/Chern%27s_conjecture_for_hypersurfaces_in_spheres

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