标题： 关于霍普夫猜想和正的第二中间里奇曲率 显示英文标题

标题： On Hopf's conjecture and positive second intermediate Ricci curvature

摘要： 霍普夫猜想认为，具有正截面曲率的偶维闭黎曼流形具有正欧拉示性数。 该猜想的结论在放松正截面曲率假设的任何方式下均已知不成立，包括转换为正第二中间里奇曲率。 在这里，我们证明，如果一个具有正第二中间里奇曲率的流形的维度能被四整除，并且具有至少十阶的环面对称性，则它具有正欧拉示性数。 一个关键的新工具是将第一作者的四周期性定理非平凡地扩展到上同调的周期性不延伸到零度的情况。 显示英文摘要

摘要： Hopf conjectured that even-dimensional closed Riemannian manifolds with positive sectional curvature have positive Euler characteristic. The conclusion of the conjecture is known to fail if the positive sectional curvature assumption is relaxed in any number of ways, including to positive second intermediate Ricci curvature. Here we prove that if a manifold with positive second intermediate Ricci curvature has dimension divisible by four and torus symmetry of rank at least ten, then it has positive Euler characteristic. A crucial new tool is a non-trivial extension of the first author's Four Periodicity Theorem to situations where the periodicity of the cohomology does not extend all the way down to degree zero.