Ricci curvature of affine connections
type: Item
instance of: scholarly article
title: Ricci curvature of affine connections
zbMATH Open document ID: 1167.53028
author: Osamu Kobayashi
published in: Tôhoku Mathematical Journal. Second Series
publication date: 2008-11-27T00:00:00Z
review text: For an affine connection on a smooth manifold \(M\), one can define the Ricci curvature as a contraction of the Riemann curvature tensor. The Ricci curvature can be interpreted as a bilinear form on the tangent bundle \(TM\), which is symmetric provided that the connection preserves some volume form on \(M\). The aim of this article is to construct examples of affine connections in low dimensions for which the Ricci curvature is definite. After reviewing some basic facts about Ricci curvature, the author constructs an affine connection on the torus \(T^n\) for \(n\geq 2\) which is projectively flat, and for which the Ricci curvature is negative definite, divergence free, and has constant determinant. Next, it is shown that any parallelizable manifold of dimension \(n\geq 3\) admits an affine connection with positive definite Ricci curvature. The last part is devoted to the case of dimension two. The author constructs an affine connection with negative definite Ricci curvature on the Klein bottle. Finally it is shown that the method does not extend to the sphere \(S^2\).
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53B15
zbMATH DE Number: 5374742
zbMATH Keywords: affine connection
zbMATH Keywords: Ricci curvature
zbMATH Keywords: projectively
zbMATH Keywords: Klein bottle
reviewed by: Andreas Cap
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1223057733
OpenAlex ID: W2091999658
cites work: A variational problem for affine connections
cites work: Metrics of negative Ricci curvature
DOI: 10.2748/TMJ/1223057733
description: scientific article; zbMATH DE number 5374742