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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 \(M\).

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