A nonimmersibility theorem of Chern-Kuiper type (Q2744399)

The following theorem is proved: Let \(M\) be an \(n\)-dimensional compact Riemannian manifold with Ricci curvature Ric. Assume that the scalar curvature \(R\) satisfies \(\text{Ric}+R\geq 0\) and \(R< n(n-1)\lambda^{-2}\) for some constant \(\lambda>0\). Then no isometric immersion of \(M\) into the Euclidean space \(\mathbb{R}^{n+1}\) is contained in a ball \(B^{n+1}\) of radius \(\lambda\). This result generalizes a result due to \textit{S. Deshmukh} and \textit{M. A. Al-Gwaiz} [Q. J. Math., Oxf. II. Ser. 45, 151-157 (1994; Zbl 0810.53046)].