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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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##### References:
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