Hypersurfaces with two distinct principal curvatures in a real space form (Q646613)

Building on numerous rigidity results for a hypersurface \(M^{n}\) in a real space form \(M^{n+1}(c)\) with constant mean or scalar curvature, the authors of this paper prove several rigidity results for such hypersurfaces under the assumption that \(\rho^{2}\), the square of the length of the trace free part of the second fundamental form of \(M^{n}\), is constant. Through a series of calculations and using a previous result due to \textit{T. Otsuki} [Am. J. Math. 92, 145--173 (1970; Zbl 0196.25102)] on the integrability of the distribution of the space of principal vectors under the assumption of constant multiplicity on the principal curvatures, the authors prove two main results, one of which is the following theorem: Let \(M^{n}\) be an (\(n \geq 3\))-dimensional complete connected and oriented hypersurface in a real space form \(M^{n+1}(c)\). If \(\rho^{2}\) is constant and \(M^{n}\) has two distinct principle curvatures, each of which has constant multiplicity greater than one, then \(M^{n}\) is isometric to one of the Riemannian products (1) \(\mathbb{R}^{k} \times S^{n-k}(a)\) for \(c=0\), (2) \(S^{k}(a) \times S^{n-k}(\sqrt{1-a^{2}})\) for \(c=1\), or (3) \(H^{k}(\tanh^{2}\varrho - 1) \times S^{n-k}(\coth^{2}\varrho -1)\) for \(c=-1\) (where \(1 < k < n-1\)).