On Riemannian manifolds admitting a function whose gradient is of constant norm
Let \((M,g)\) be a connected complete Riemannian manifold of nonnegative Ricci curvature and \(f : M \rightarrow \mathbb{R}\) a smooth function with \(|\nabla f |= a>0\), where \(\nabla f\) denotes the gradient vector field of \(f\). The author generalizes the fact that any smooth function \(f : \mathbb{R}^m\rightarrow \mathbb{R}\) with \(|\nabla f|=\) const is an affine function. If the Ricci curvature satisfies Ricci \(m\geq-(m-1),\) \(m=\dim M\), and \(|\nabla f |= 1\), then \(|\Delta f |\leq m-1\) for the Laplacian of \(M\).
scientific article; zbMATH DE number 895929