Addendum to the paper: Sphere theorem by means of the ratio of mean curvature functions. (Q2731120)

The proof of theorem B in [Glasg. Math. J. 42, 91--95 (2000; Zbl 0957.53026)] does not apply directly by assuming that \(\phi\) is an immersion. By changing the argument of it, the authors get the following:NEWLINENEWLINELet \(N^{n+1}\) be either the Euclidean \(\mathbb {R}^{n+1}\), the hyperbolic space \(\mathbb{H}^{n+1}\) or the open half-sphere \(\mathbb{S}_{+}^{n+1}\) and \(\Phi\): \(M^{n} \rightarrow N^{n+1}\) an isometric immersion of a compact oriented \(n\)-dimensional manifold without boundary \(M^{n}\) and let \(H_{k}\) denote the \(k\)-th mean curvature function of \(M^{n}\), where \(H_{0}\) is defined to be 1. \(H_{1}\) and \(H_{n}\) are the mean curvature and Gauss-Kronecker curvature respectively. If the ratio \(\frac{H_{k}}{H_{l}}\) is constant for some \(k,l=1,2, \dots, n\), \(k>1\) and \(H_{l}\) does not vanish on \(M^{n}\), then \(\phi (M^{n})\) is a geodesic hypersphere.