论文标题
关于正标曲率指标的均匀类型
On the homotopy type of the space of metrics of positive scalar curvature
论文作者
论文摘要
令$ m^d $为尺寸的简单连接的旋转歧管$ d \ geq 5 $允许riemannian的正标曲率指标。用$ \ Mathcal {r}^+（m^d）$在$ m^d $上的空间。我们表明$ \ MATHCAL {r}^+（m^d）$是同质的，等于$ \ Mathcal {r}^+（s^d）$，其中$ s^d $表示$ d $ d $ dimensional Sphere具有标准平滑结构。 对于简单连接的非旋转歧管$ M^d $，带有$ d \ geq 5 $和$ d \ neq 8 $，我们还显示出类似的结果。在这种情况下，让$ w^d $是非平凡$ s^{d-2} $ - 带有结构组$ so（d-1）$的总空间，超过$ s^2 $。然后，$ \ Mathcal {r}^+（m^d）$是同质的，相当于$ \ Mathcal {r}^+（w^d）$。
Let $M^d$ be a simply connected spin manifold of dimension $d \geq 5$ admitting Riemannian metrics of positive scalar curvature. Denote by $\mathcal{R}^+(M^d)$ the space of such metrics on $M^d$. We show that $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(S^d)$, where $S^d$ denotes the $d$-dimensional sphere with standard smooth structure. We also show a similar result for simply connected non-spin manifolds $M^d$ with $d\geq 5$ and $d\neq 8$. In this case let $W^d$ be the total space of the non-trivial $S^{d-2}$-bundle with structure group $SO(d-1)$ over $S^2$. Then $\mathcal{R}^+(M^d)$ is homotopy equivalent to $\mathcal{R}^+(W^d)$.
