×

zbMATH — the first resource for mathematics

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

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