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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

##### MSC:
 53C40 Global submanifolds 53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related $$n$$-spaces
##### Keywords:
rigidity; space of metrics; isometric immersion