论文标题
贝蒂数字和第二类的曲率运算符
Betti numbers and the curvature operator of the second kind
论文作者
论文摘要
我们表明，带有$ \ frac {n+2} {2} $的紧凑型，$ n $ dimensional riemannian歧管 - 第二类的非负曲率操作员是有理同源球或平坦的。更普遍地，我们获得了$ p $ -th betti号码的消失，但前提是第二类的曲率运算符为$ c（p，n）$ - 阳性。随着$ P $的增加，我们的曲率条件变得较弱。对于$ p = \ frac {n} {2} $，我们有$ c（p，n）= \ frac {3n} {2} {2} \ frac {n+2} {n+4} $，$ 5 \ leq p \ leq p \ leq p \ leq \ frac {n} n} $ c（第二类，负RICCI曲率。
We show that compact, $n$-dimensional Riemannian manifolds with $\frac{n+2}{2}$-nonnegative curvature operators of the second kind are either rational homology spheres or flat. More generally, we obtain vanishing of the $p$-th Betti number provided that the curvature operator of the second kind is $C(p,n)$-positive. Our curvature conditions become weaker as $p$ increases. For $p=\frac{n}{2}$ we have $C(p,n)= \frac{3n}{2} \frac{n+2}{n+4} $, and for $5 \leq p \leq \frac{n}{2}$ we exhibit a $C(p,n)$-positive algebraic curvature operator of the second kind with negative Ricci curvatures.
