On Riemannian manifolds admitting a function whose gradient is of constant norm
type: Item
instance of: scholarly article
title: On Riemannian manifolds admitting a function whose gradient is of constant norm
zbMATH Open document ID: 0881.53035
author: Takashi Sakai
published in: Kodai Mathematical Journal
publication date: 1998-01-13T00:00:00Z
review text: 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 \(M\).
reviewed by: Nicolae A. Soare
Mathematics Subject Classification ID: 53C21
zbMATH DE Number: 895929
zbMATH Keywords: Ricci curvature
zbMATH Keywords: gradient vector field
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2996/kmj/1138043545
OpenAlex ID: W4241130797
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Wikidata QID: Q115224576
DOI: 10.2996/KMJ/1138043545
description: scientific article; zbMATH DE number 895929