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Isometric immersions of Riemannian manifolds. (English) Zbl 0612.53032
Élie Cartan et les mathématiques d’aujourd’hui, The mathematical heritage of Élie Cartan, Semin. Lyon 1984, Astérisque, No.Hors Sér. 1985, 129-133 (1985).
[For the entire collection see Zbl 0573.00010.]
The paper describes two theorems which are proved in the author’s book ”Partial differential relations” (to appear). The isometric immersion theorem states that every n-dimensional Riemannian \(C^{\infty}\) manifold V admits an isometric immersion into the Euclidean space \({\mathbb{R}}^ q\) for \(q=p(n)+2n+3\) with \(p(n)=n(n+1)/2\). Furthermore, the author formulates two conjectures: 1. If \(q<p(n)\), then generically the immersions \(V^ n\to {\mathbb{R}}^ q\) are rigid. 2. If \(q>p(n)\), then (for every fixed \(V^ n)\) there exists an open dense subset of immersions \(f: V^ n\to {\mathbb{R}}^ q\) on which the map \(f\mapsto f^*g_ 0\) \((g_ 0\) the usual metric \({\mathbb{R}}^ q)\) is a submersion into the space of all Riemannian metrics of \(V^ n\).
Reviewer: H.Reckziegel

53C40 Global submanifolds
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces