We present a rigidity theorem for the action of the mapping class group \(\pi _0({\mathrm{Diff}}(M))\) on the space \(\mathcal {R}^+(M)\) of metrics of positive scalar curvature for high dimensional manifolds M. This result is applicable to a great number of cases, for example to simply connected 6-manifolds and high dimensional spheres. Our proof is fairly direct, using results from parametrised Morse theory, the 2-index theorem and computations on certain metrics on the sphere. We also give a non-triviality criterion and a classification of the action for simply connected 7-dimensional \({\mathrm{Spin}}\)-manifolds.

中文翻译：

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映射类组对正标量曲率度量的作用

我们提出了映射类群\(\pi _0({\mathrm{Diff}}(M))\)在空间\(\mathcal {R}^+(M)\)上的作用的刚性定理高维流形M的正标量曲率度量。这个结果适用于很多情况，例如简单连接的 6 流形和高维球体。我们的证明相当直接，使用参数化莫尔斯理论的结果、2-指数定理和球体上某些度量的计算。我们还给出了简单连接的 7 维\({\mathrm{Spin}}\) -流形的非平凡标准和动作分类。