On compositions of isometric immersions
The background of this paper is the problem of local and global isometric immersions with low codimension of the \(n\)-dimensional sphere \(S^ n\) into Euclidean space \(\mathbb{R}^{n+p}\); the case \(p = 2\) and \(n \geq 4\) was considered since 1965 [\textit{B. O'Neill}, Duke Math. J. 32, 149-159 (1965; Zbl 0134.395); \textit{J. Erbacher}, Nagoya Math. J. 45, 139-165 (1972; Zbl 0211.537); \textit{W. Henke}, Math. Ann. 219, 261-276 (1976; Zbl 0303.53024); \textit{L. Whitt}, J. Differ. Geom. 14, 295-302 (1979; Zbl 0427.53027)] and \textit{J. D. Moore}]. In the first section of the present paper, the authors give several conditions on isometric immersions \(f: M^ n \to Q_ c^{n+1}\) and \(g: M^ n \to Q_ c^{n+p}\), which imply, that \(g\) is a composition \(g = h\circ f\) with \(h: U \subset Q_ c^{n+1} \to Q_ c^{n+p}\) (\(M^ n\) is an \(n\)-dimensional Riemannian manifold and \(Q^ N_ c\) are complete, simply connected Riemannian manifolds of constant sectional curvature c). Section 2 gives a contribution to the question of realization of an \(n\)- dimensional Riemannian manifold \(M^ n\) as a hypersurface in two space forms of different sectional curvature. From a result of do Carmo-Dajczer it follows, that for \(n \geq 4\quad M^ n\) has to be conformally flat; the authors show, that for \(n = 3\) this is not necessary. Further results, concerning the composition of pseudo-umbilical isometric immersions complete this interesting paper.
scientific article; zbMATH DE number 147454