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Notes on conformal differential geometry. (English) Zbl 0911.53020
Slovák, Jan (ed.), Proceedings of the 15th Winter School on geometry and physics, Srní, Czech Republic, January 14–21, 1995. Palermo: Circolo Matemàtico di Palermo, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 43, 57-76 (1996).
This survey paper presents lecture notes from a series of four lectures addressed to a wide audience and it offers an introduction to several topics in conformal differential geometry. In particular, a very nice and gentle introduction to the conformal Riemannian structures themselves, flat or curved, is presented in the beginning. Then the behavior of the covariant derivatives under the rescaling of the metrics is described. This leads to Penrose’s local twistor transport which is introduced in the approach credited to T. Y. Thomas. The resulting object is called the tractor connection and is equivalent to Cartan’s concept of connections on generalized spaces [see also T. N. Bailey, M. G. Eastwood and R. A. Gover, Rocky Mt. J. Math. 24, 1191-1217 (1994; Zbl 0828.53012)].
The third lecture builds explicit links to the representation theory and describes the invariant operators on the homogeneous vector bundles over the round sphere (the homogeneous model for conformally flat Riemannian spaces) in terms of the homomorphisms of Verma modules. The last lecture is then mainly devoted to the discussion of a geometric version of the Jantzen-Zuckermann translation procedure well known in representation theory. At the very end, some open questions are listed.
The whole survey reads nicely and many further links to the literature are given.
For the entire collection see [Zbl 0884.00042].
Reviewer: Jan Slovák (Brno)

MSC:
53C20 Global Riemannian geometry, including pinching
53A30 Conformal differential geometry (MSC2010)
53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry
53C30 Differential geometry of homogeneous manifolds
53C05 Connections, general theory
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