Immersions equivariant for a given Killing vector. II (Q1821367)

[For part I, cf. J. Lond. Math. Soc., II. Ser. 29, 323-330 (1984; Zbl 0544.53040).] This paper studies isometric immersions of complete Riemannian manifolds \(M^ n\) into \({\mathbb{E}}^ N\) or \({\mathbb{H}}^ N\) which are equivariant with respect to the 1 parameter group of isometries associated to some fixed non-trivial Killing field K on \(M^ n\). Non-existence results are proved in the following three cases: A) The target is \({\mathbb{E}}^ N\) and the sectional curvature \(K_{M^ n}\) is bounded from above by \(-a\cdot r^{-2}\), \(a>0\), where r stands for the distance to some base point. B) The target is \({\mathbb{H}}^ N\) and \(K_{M^ n}\leq -c<-1\). C) The target is \({\mathbb{H}}^ N\) and \(M^ n\) is an irreducible symmetric space with Ricci curvature \(<-(n-1)\). The technique is to derive from the curvature hypothesis on \(M^ n\) a lower bound on the growth of \(| K|\) along a suitably chosen geodesic. This bound contradicts the growth of actual Killing fields in the target space.