Closed G 2 $G_2$ -structures with negative Ricci curvature

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2025-03-03 DOI:10.1112/blms.70029

Alec Payne

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Abstract

We study existence problems for closed $G_{2}$ -structures with negative Ricci curvature, and we prove the $G_{2}$ -Goldberg conjecture for noncompact manifolds. We first show that no closed manifold admits a closed $G_{2}$ -structure with negative Ricci curvature. In the noncompact setting, we show that no complete manifold admits a closed $G_{2}$ -structure with Ricci curvature pinched sufficiently close to a negative constant. As a consequence, an Einstein closed $G_{2}$ -structure on a complete manifold must be torsion-free. In addition, when the Einstein metric is incomplete, we find restrictions on lengths of geodesics.

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负Ricci曲率的封闭g2 $G_2$结构

研究了具有负Ricci曲率的封闭g2 $G_2$ -结构的存在性问题，证明了非紧流形的g2 $G_2$ -Goldberg猜想。首先证明了封闭流形不允许存在负Ricci曲率的封闭g2 $G_2$ -结构。在非紧化情况下，我们证明了没有完全流形允许存在Ricci曲率足够接近负常数的封闭g2 $G_2$ -结构。因此，完全流形上的爱因斯坦闭g2 $G_2$结构必须是无扭的。此外，当爱因斯坦度规不完备时，我们发现测地线长度有限制。

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