论文标题
高斯河网不平等超出了非球形猜想
The Gauss-Bonnet inequality beyond aspherical conjecture
论文作者
论文摘要
直到尺寸五，我们可以证明，鉴于任何具有非负标量曲率的封闭的Riemannian歧管，通用覆盖物在所有$ k \ geq 3 $中都消失了同源组$ h_k $，要么是平坦的，要么具有Gauss-bonnet数量（（1.3）定义（1.3）），没有更大的$8π$。在第二种情况下，高斯 - 骨网量的平等产生的平等性覆盖了覆盖分裂为$ 2 $ sphere的Riemannian产品，具有非负分段曲率和欧几里得空间。我们还建立了此结果的主导版本及其在同型$ 2 $ - 总结估计中的应用统一的[BBN10]和[Zhu20]的结果。
Up to dimension five, we can prove that given any closed Riemannian manifold with nonnegative scalar curvature, of which the universal covering has vanishing homology group $H_k$ for all $k\geq 3$, either it is flat or it has Gauss-Bonnet quantity (defined by (1.3)) no greater than $8π$. In the second case, the equality for Gauss-Bonnet quantity yields that the universal covering splits as the Riemannian product of a $2$-sphere with non-negative sectional curvature and the Euclidean space. We also establish a dominated version of this result and its application to homotopical $2$-systole estimate unifies the results from [BBN10] and [Zhu20].
