Isometric immersions of RCD spaces
Isometric immersions of RCD spaces
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DOI：
10.4171/cmh/519
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发表时间：
2020-05
影响因子：
0.9
通讯作者：
Shouhei Honda
中科院分区：
文献类型：
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作者：
Shouhei Honda
证明了如果RCD空间在欧氏空间中有正则等距浸入，则该浸入是局部双Lipschitz嵌入映射。这一结果使我们证明了：如果紧致非塌陷RCD空间通过特征映射在欧氏空间中有等距浸入，则该特征映射是球面的局部双Lipschitz嵌入映射，从而将Takahashi在子流形理论中的一个基本定理推广到非光滑情形.这些结果的应用包括一个拓扑球定理和拓扑有限性定理，这是新的，甚至封闭的黎曼流形。
We prove that if an RCD space has a regular isometric immersion in a Euclidean space, then the immersion is a locally bi-Lipschitz embedding map. This result leads us to prove that if a compact non-collapsed RCD space has an isometric immersion in a Euclidean space via an eigenmap, then the eigenmap is a locally bi-Lipschitz embedding map to a sphere, which generalizes a fundamental theorem of Takahashi in submanifold theory to a non-smooth setting. Applications of these results include a topological sphere theorem and topological finiteness theorems, which are new even for closed Riemannian manifolds.