Isometric \(C^1\)-immersions for pairs of Riemannian metrics.
In this work the authors deal with the following problem: let \(h_{1}\) be the standard Euclidean metric on \(\mathbb{R}^{q}\) with canonical coordinates \((x^{1},x^{2},\dots,x^{q})\) and \(h_{2}=\sum^{q}_{i=1}\lambda^{2}_{i}dx_ i^ 2\), where it is assumed that \(0<\lambda_{1}<\lambda_{2}<\dots<\lambda_{q}\). Let us consider an \(n\)-dimensional \(C^{\infty}\) manifold \(M^{n}\) endowed with two \(C^{\infty}\)-Riemannian metrics \(g_{1}\) and \(g_{2}\), then the question is to find out when there exists a \(C^{1}\)-immersion \(f: M^{n}\rightarrow\mathbb{R}^{q}\) such that \[ f^{*}(h_{1})=g_{1}\quad \text{and} \quad f^{*}(h_{2})=g_{2}. \] This is an extension of the classical isometric immersion problem solved by \textit{J. Nash} in 1954 [Ann. Math. (2) 60, 383--396 (1954; Zbl 0058.37703)]. He proved that if an \(n\)-dimensional manifold \((M^{n},g)\) admits a \(C^{1}\)-immersion (resp. embedding) into \(\mathbb{R}^{q}\), where \(q\geq n+2\), then it admits a \(C^{1}\)-isometric immersion (resp. embedding) into \(\mathbb{R}^{q}\). By refining Nash's techniques, \textit{N. H. Kuiper} improved this result by showing that it is true even when \(q\geq n+1\) [Nederl. Akad. Wet., Proc., Ser. A 58, 545--556, 683--689 (1955; Zbl 0067.39601)]. It is clear that not every pair \((g_{1},g_{2})\) of metrics on \(M^{n}\) can be induced from a pair of metrics \((h_{1},h_{2})\) on \(\mathbb{R}^{q}\) by a single map. A necessary condition for that is \[ \lambda_{1}^{2}g_{1}\leq g_{2}\leq \lambda_{q}^{2}g_{1}. \] In this paper, the authors obtain a generalization of Nash's result to the case where the manifolds come with pairs of Riemannian metrics, by showing a certain family of metrics \(h_{2}\) for which a condition similar to the above one solves the problem. They first explain the basic ideas involved in Nash's proof and then make a neat nontrivial adaptation of Nash's techniques to the new context. More concretely, they prove the following result: Let \(a\) and \(b\) be two positive numbers, \(a<b\), such that \(\lambda_{i}<a\) for \(i\leq 3n+2\) and \(\lambda_{i}>b\) for \(i\geq q-3n-1\), and none of the \(\lambda_{i}\) lie between \(a\) and \(b\). If \(g_{1}\) and \(g_{2}\) are two \(C^{\infty}\)-Riemannian metrics on \(M^{n}\) satisfying the relation \(a^{2}g_{1}<g_{2}<b^{2}g_{1}\), then there exists a \(C^{1}\)-immersion \(f: M^{n}\rightarrow\mathbb{R}^{q}\) such that \[ f^{*}(h_{1})=g_{1}\quad \text{and} \quad f^{*}(h_{2})=g_{2}. \] As a corollary, they obtain conditions on the positive and negative ranks of \(h_{2}-h_{1}\) under which a Riemannian manifold \((M^{n},g)\) admits a \(C^{1}\)-immersion \(f: M^{n}\rightarrow\mathbb{R}^{q}\) such that \[ f^{*}(h_{1})=g=f^{*}(h_{2}). \]
scientific article; zbMATH DE number 1925764