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Projective connections. (English) Zbl 1114.53014
Two approaches (of È. Cartan and of T. Y. Thomas and J. H. C. Whitehead) to the study of projective connections are compared. Spray geometry is reviewed, the Thomas-Whitehead construction and its generalization to sprays are discussed, and afterwards the Cartan theory in the affine case is described. A certain bundle is introduced which realizes explicitly Cartan’s idea of attaching a projective space to each point of a manifold, called the Cartan bundle. This bundle is defined independent of any particular choice of connection, but to any projective class of sprays one can associate a unique Cartan connection on the Cartan bundle, as it is shown. In the course of the discussion a Cartan normal projective connection is derived for a system of second-order ordinary differential equations (extending the results of Cartan from a single equation to many) and the concept of a normal Thomas-Whitehead connection is generalized from affine to general sprays. Finally it is described in detail how to derive the Cartan connection from the generalized Thomas-Whitehead data for any projective equivalence class of sprays.

53B10 Projective connections
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