We show that the extended principal bundle of a Cartan geometry of type $(A(m,\mathbb{R}),GL(m,\mathbb{R}))$, endowed with its extended connection $\hat{\omega }$, is isomorphic to the principal $A(m,\mathbb{R})$-bundle of affine frames endowed with the affine connection as defined in classical Kobayashi-Nomizu volume I.

Then we classify the local holonomy groups of the Cartan geometry canonically associated to a Riemannian manifold. It follows that if the holonomy group of the Cartan geometry canonically associated to a Riemannian manifold is compact then the Riemannian manifold is locally a product of cones.

中文翻译：

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锥体和嘉当几何

我们证明了类型 Cartan 几何的扩展主丛 $(\mathrm{一种}(米,\mathbb{电阻}),G升(米,\mathbb{电阻}))$, 赋予其扩展连接 $\widehat{\omega }$, 与主同构 $\mathrm{一种}(米,\mathbb{电阻})$- 一组仿射框架，赋予经典 Kobayashi-Nomizu 卷 I 中定义的仿射连接。

然后我们对与黎曼流形规范相关的嘉当几何的局部完整群进行分类。因此，如果与黎曼流形规范相关的嘉当几何的完整群是紧的，则黎曼流形局部是锥的乘积。