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Let $\mathbb{T}^n = \mathbb{C}^n /\Lambda$ be a complex torus of (complex) dimension $n$. If $n=2$, it is a theorem of Berger that the Ricci-flat metrics on $\mathbb{T}^2$ are flat. This follows from the Chern-Gauss-Bonnet theorem and the fact that the Euler characteristic of $\mathbb{T}^2$ is zero.

Are there Ricci-flat metrics on $\mathbb{T}^{n \geq 3}$ that are not flat?

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  • $\begingroup$ I'm guess you are asking about Ricci flat Kähler metrics? $\endgroup$
    – Donu Arapura
    Commented Jul 15, 2023 at 22:50
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    $\begingroup$ The answer is no, see e.g. ams.org/journals/bull/1974-80-01/S0002-9904-1974-13368-9/…, The Calabi construction for compact Ricci flat Riemannian manifolds, Arthur E. Fischer and Joseph A. Wolf, Bull. Amer. Math. Soc. 80 (1974), 92-97. The fact the torus is complex is not relevant. $\endgroup$
    – Igor Belegradek
    Commented Jul 15, 2023 at 22:57
  • $\begingroup$ @IgorBelegradek Thank you, I have heard Berger's result mentioned time and time again, but surprisingly was never made aware of this result of Fischer and Wolf. Thank you! $\endgroup$
    – AmorFati
    Commented Jul 15, 2023 at 23:08
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    $\begingroup$ More generally one can appeal to Cheeger-Gromoll splitting that says that the universal Riemannian cover of a compact manifold $M$ of nonnegative Ricci curvature splits isometrically as the product of a Euclidean space and a compact simply-connected manifold of nonnegative Ricci curvature. If $M$ is diffeomorphic to an $n$-torus, the simply-connected factor is a point and hence $M$ is locally isometric to $\mathbb R^n$. $\endgroup$
    – Igor Belegradek
    Commented Jul 16, 2023 at 14:03

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