Results related to generalizations of Hilbert's non-immersibility theorem for the hyperbolic plane

الأصل · EN

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 Hⁿ into a Euclidean space Eⁿ⁺ᵏ, n ≥ 2, and the problem of Lagrangian isometric immersions of Hⁿ into ⁿ, n ≥ 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.