论文标题
产品歧管和第二类的曲率操作员
Product manifolds and the curvature operator of the second kind
论文作者
论文摘要
我们研究了第二种产品的曲率操作员，并获得了一些最佳的刚度结果。例如，我们证明，$ n $维的非流量的通用封面与$（n+\ frac {n-2} {n} {n}）$ - nonnegative（分别为$（n+\ \ \ \\ frac {n+\ frac {n-2} {n} {n} {n}）$ - 非posive curv curv curv curv curv curv curververements $ \ mathbb {s}^{n-1} \ times \ mathbb {r} $（分别为$ \ mathbb {h}^{n-1} \ times \ times \ times \ mathbb {r} $）直至缩放。我们还证明了$ \ mathbb {s}^{n_1} \ times \ Mathbb {s}^{n_2} $和$ \ mathbb { riemannian流形，以及$ \ mathbb {cp}^{m_1} \ times \ times \ mathbb {cp}^{m_2} $和$ \ mathbb {ch}^{ch}^{m_1} Kähler歧管。我们的方法是尖锐而代数。
We investigate the curvature operator of the second kind on product Riemannian manifolds and obtain some optimal rigidity results. For instance, we prove that the universal cover of an $n$-dimensional non-flat complete locally reducible Riemannian manifold with $(n+\frac{n-2}{n})$-nonnegative (respectively, $(n+\frac{n-2}{n})$-nonpositive) curvature operator of the second kind must be isometric to $\mathbb{S}^{n-1}\times \mathbb{R}$ (respectively, $\mathbb{H}^{n-1}\times \mathbb{R}$) up to scaling. We also prove analogous optimal rigidity results for $\mathbb{S}^{n_1}\times \mathbb{S}^{n_2}$ and $\mathbb{H}^{n_1}\times \mathbb{H}^{n_2}$, $n_1,n_2 \geq 2$, among product Riemannian manifolds, as well as for $\mathbb{CP}^{m_1}\times \mathbb{CP}^{m_2}$ and $\mathbb{CH}^{m_1}\times \mathbb{CH}^{m_2}$, $m_1,m_2\geq 1$, among product Kähler manifolds. Our approach is pointwise and algebraic.
