The projective connections of T. Y. Thomas and J. H. C. Whitehead applied to invariant connections
type: Item
instance of: scholarly article
title: The projective connections of T. Y. Thomas and J. H. C. Whitehead applied to invariant connections
zbMATH Open document ID: 0833.53023
DOI: 10.1016/0926-2245(95)92848-Y
author: Craig W. Roberts
published in: Differential Geometry and its Applications
publication date: 1995-10-17T00:00:00Z
review text: The purpose of the paper is to give a global perspective of some work of T. Y. Thomas and J. H. C. Whitehead on projective connections. On a principal \(\mathbb{R}\)-bundle of volume elements \(\varepsilon (M)\) over the base manifold \(M\) a covariant derivative can be defined, and the linear connection corresponding to this covariant derivative will be called a Thomas-Whitehead projective connection. This will be applied to the study of those projective equivalence classes on \(M\) which contains an invariant torsion-free connection. It is shown that, if \(G\) is a Lie group acting on \(M\) admitting a \(G\)-invariant torsion-free linear connection, then every \(G\)-invariant projective equivalence class on \(M\) contains a \(G\)-invariant torsion-free linear connection. The fundamental theorem of the paper says: To every projective equivalence class on \(M\) there exists a unique Thomas-Whitehead connection \(\nabla\) such that: 1) \(\nabla \Psi_M=0\) for the canonical positive odd scalar density \(\Psi_M\) on \(\varepsilon (M)\), 2) \(\nabla\) is Ricci flat, 3) \(\nabla\) induces the given projective equivalence class on \(M\).
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 53C30
Mathematics Subject Classification ID: 53B10
zbMATH DE Number: 806422
zbMATH Keywords: bundle of volume elements
zbMATH Keywords: Thomas-Whitehead projective connection
Wikidata QID: Q115362537
reviewed by: Lajos Tamássy
MaRDI profile type: Publication
cites work: Q5658761
cites work: Q5728083
cites work: Q5731354
cites work: Projective Structures on Reductive Homogeneous Spaces
cites work: Q3881472
cites work: Q3664070
cites work: The representation of projective spaces
Recommended article: Invariant projectively flat connections and its applications
Recommended article: Relating Thomas-Whitehead projective connections by a gauge transformation
Recommended article: On invariant projectively flat affine connections
Recommended article: Q3776989
Recommended article: Examples of Thomas-Whitehead projective connections
Recommended article: PROJECTIVE STRUCTURES AND -CONNECTIONS
Recommended article: Invariant connections for conformal and projective changes
Recommended article: The T. Y. Thomas construction of projectively related manifolds
Recommended article: Q3772859
Recommended article: Invariant connections on homogeneous spaces
description: scientific article; zbMATH DE number 806422