On Riemannian manifolds admitting a function whose gradient is of constant norm. II (Q1282074)

Let \((M,g)\) be a complete, connected, smooth Riemannian manifold of dimension \(m\) admitting a smooth function \(f: M\to\mathbb{R}\) whose gradient \(\nabla f\) is of constant norm. Let \(\varphi_t\), \(t\in\mathbb{R}\), be the flow generated by \(\nabla f\). Then, the map \(\Phi: \mathbb{R}\times Z\to M\) defined by \(\Phi(t,z)= \varphi_t(z)\) is a diffeomorphism, where \(Z= f^{-1}(0)\). Assume that the Ricci tensor field Ric of \((M,g)\) satisfies \[ \text{Ric}(\nabla f,\nabla f)\geq -(m- 1)\delta,\tag{1} \] where \(\delta\) is a nonnegative constant. In a previous paper, the author showed that if \(\delta= 0\), then \(\Phi^{-1}\) is an isometry between \((M,g)\) and the Riemannian product \(\mathbb{R}\times Z\), where \(Z\) is endowed with the totally geodesic induced Riemannian metric; if \(\delta= 1\) and \(|\Delta f|\equiv m-1\), where \(\Delta\) denotes the Laplacian, then \(\Phi^{-1}: M\to \mathbb{R}\times Z\) is an isometry, where \(\mathbb{R}\times Z\) is endowed with the warped product metric with warping function \(\psi(t)= \exp(\pm t)\) and \(Z\) is endowed with the induced metric [see part I, Kodai Math. J. 19, 39-51 (1996; Zbl 0881.53035)]. In this paper, the author continues the study of the metrical structure of complete Riemannian manifolds admitting such a real smooth function. More precisely, suppose that Ric satisfies \(\text{Ric}\geq -(m-1)k\) and (1), where \(k>1\), \(0<\delta< 1\). He proves that for any \(R> 0\) and \(\varepsilon> 0\), \(\Phi^{-1}: M\to \mathbb{R}\times Z\) is roughly an \(\varepsilon\)-Hausdorff approximation when restricted to a distance ball \(B_R(p; M)\) centered at \(p\in Z\), if \(\delta> 0\) is sufficiently small and \(Z\) is endowed with a ditance which is close to the induced metric in some sense. The proof is based on recent powerful methods of \textit{J. Cheeger} and \textit{T. H. Colding} [see Ann. Math., II. Ser. 144, 189-237 (1996; Zbl 0865.53037); J. Differ. Geom. 46, 406-480 (1997; Zbl 0902.53034)], and the preprint ``On the structure of spaces with Ricci curvature bounded below. II'']. An application of this result is pointed out. The author also discusses what happens if \(\text{Ric}(\nabla f,\nabla f)\geq -(m-1)- \delta\) \((0<\delta< 1)\) and \(|\Delta f|\) is close to \(m-1\).