双曲平面上Hilbert非浸入定理的推广
Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane
摘要：
我们讨论了希尔伯特著名定理的推广，即双曲平面不完全等距浸入欧氏三维空间。我们表明，这个问题是非常自然地表示为对称空间的反射子流形的某些同构的存在性问题。因此，我们得出结论，只有其他（非紧）的情况下，这一定理可以推广的问题是等距浸入与平坦的正常丛的双曲空间$H^n$到一个欧几里得空间$E^{n+k}$，$n \geq 2$，和拉格朗日等距浸入的问题$H^n$到$\cc^n$，$n \geq 2$。此外，这些问题有自然的紧对应物，对于紧的情况下，我们证明了定理实际上推广：局部嵌入存在，但完全浸入不存在。
英文摘要：
We discuss generalizations of the well-known theorem of Hilbert that there is no complete isometric immersion of the hyperbolic plane into Euclidean 3-space. We show that this problem is expressed very naturally as the question of the existence of certain homotheties of reflective submanifolds of a symmetric space. As such, we conclude that the only other (non-compact) cases to which this theorem could generalize are the problem of isometric immersions with flat normal bundle of the hyperbolic space $H^n$ into a Euclidean space $E^{n+k}$, $n \geq 2$, and the problem of Lagrangian isometric immersions of $H^n$ into $\cc^n$, $n \geq 2$. Moreover, there are natural compact counterparts to these problems, and for the compact cases we prove that the theorem does in fact generalize: local embeddings exist, but complete immersions do not.
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