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The projective connections of T. Y. Thomas and J. H. C. Whitehead applied to invariant connections. (English) Zbl 0833.53023
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 \(M\).

53C05 Connections, general theory
53C30 Differential geometry of homogeneous manifolds
53B10 Projective connections
Full Text: DOI
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