Abstract We study isometric embeddings of C 2 Riemannian manifolds in the Euclidean space and we establish that the Holder space C 1 , 1 2 is critical in a suitable sense: in particular we prove that for α > 1 2 the Levi-Civita connection of any isometric immersion is induced by the Euclidean connection, whereas for any α 1 2 we construct C 1 , α isometric embeddings of portions of the standard 2-dimensional sphere for which such property fails.

中文翻译：

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C1，极冠的等距嵌入

摘要 我们研究了 C 2 黎曼流形在欧几里得空间中的等距嵌入，并且我们确定 Holder 空间 C 1 , 1 2 在合适的意义上是关键的：特别是我们证明对于 α > 1 2 任何等距浸入是由欧几里得连接引起的，而对于任何 α 1 2 ，我们构造 C 1 ，α 等距嵌入标准二维球体的部分，对于这种特性失败。