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seminar:2007:006

第06回

  • 講演者:David Brander 氏 (神戸大学)
    • 題目:Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane
    • 日時:平成19年10月24日(水)16:30-17:30

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. 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 complex Euclidean space, $C^n$, $n \geq 2$. 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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seminar/2007/006.txt · 最終更新: 2017/11/17 15:11 (外部編集)