Projectively recurrent semi-definite Riemannian manifolds.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1034.53018
author: Yong-Soo Pyo
publication date: 2001-01-01T00:00:00Z
Mathematics Subject Classification ID: 53B30
Mathematics Subject Classification ID: 53C15
Mathematics Subject Classification ID: 53C50
zbMATH DE Number: 1657763
zbMATH Keywords: Semi-definite Riemannian manifolds
zbMATH Keywords: projectively recurrent
zbMATH Keywords: projectively symmetric
MaRDI profile type: Publication
title: Projectively recurrent semi-definite Riemannian manifolds.
published in: Mathematical Sciences Research Hot-Line
review text: Let \(M\) be a \(n\) (\(\geq 3\))-dimensional semi-definite Riemannian manifold with semi-definite Riemann metric \(g\), Riemannian connection \(\nabla\), Riemannian curvature tensor \(R\) and Ricci tensor \(S\). Locally, the projective curvature tensor \(W\) with respect to an orthogonal frame, is given by NEWLINE\[NEWLINEW_{ijkl} = R_{ijkl} - \frac{1}{n-1}\epsilon_i (S_{jk}g_{il} - S_{jl}g_{ik}).NEWLINE\]NEWLINE The manifold \(M\) is called projectively recurrent, if there exists a 1-form \(\alpha\) such that \(\nabla W = \alpha \otimes W\). If \(\nabla W = 0\), the manifold is called projectively symmetric. The author shows, that a projectively symmetric Riemannian manifold of dimension \(n \geq 3\) is locally symmetric and further that a projectively recurrent manifold is of constant curvature or locally symmetric.
reviewed by: Klaus Horneffer
description: scientific article; zbMATH DE number 1657763