The Ledger curvature conditions and D'Atri geometry
type: Item
instance of: scholarly article
title: The Ledger curvature conditions and D'Atri geometry
zbMATH Open document ID: 0944.53025
DOI: 10.1016/S0926-2245(99)00026-1
author: Paul Tod
author: Henrik Pedersen
published in: Differential Geometry and its Applications
publication date: 2000-02-08T00:00:00Z
review text: The main result of this paper is the following Theorem. Let \((M,g)\) be a Riemannian 3-manifold which is ``cyclic Ricci-parallel'', i.e., it satisfies the identity \((\nabla_X\text{Ric})(X,X)=0\). Then \((M,g)\) is a D'Atri space (i.e., a space with volume-preserving local geodesic symmetries) and, in particular, it is locally homogeneous. The reviewer [Rend. Semin. Mat., Torino, Fasc. Spec., 131-158 (1983; Zbl 0631.53033)] classified all 3-dimensional D'Atri spaces using the above identity (called the 3rd Ledger condition \(L_3)\) and the next natural identity of higher order (the Ledger condition \(L_5)\), which are both necessary conditions for a Riemannian manifold to be a D'Atri space. Hence, the present authors proved that the condition \(L_5\) follows from the condition \(L_3\) in dimension 3. On the other hand, they give a 5-dimensional example satisfying the condition \(L_3\) but not the condition \(L_5\).
Mathematics Subject Classification ID: 53C25
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53C22
zbMATH DE Number: 1367712
zbMATH Keywords: cyclic Ricci-parallel
zbMATH Keywords: D'Atri space
zbMATH Keywords: Ledger condition
Wikidata QID: Q115337413
reviewed by: Oldřich Kowalski
MaRDI profile type: Publication
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full work available at URL: s0926-2245(99)00026-1
OpenAlex ID: W1994301774
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description: scientific article; zbMATH DE number 1367712