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Flat affine, projective and conformal structures on manifolds: a historical perspective. (English) Zbl 1454.57002
Dani, S. G. (ed.) et al., Geometry in history. Cham: Springer. 515-552 (2019).
Summary: This historical survey reports on the theory of locally homogeneous geometric structures as initiated in C. Ehresmann’s 1936 paper [“Sur les espaces localement homogènes”, Enseign. Math. 35, 317–333 (1936; Zbl 0015.39404)]. Beginning with Euclidean geometry, we describe some highlights of this subject and threads of its evolution. In particular, we discuss the relationship to the subject of discrete subgroups of Lie groups. We emphasize the classification of geometric structures from the point of view of fiber spaces and the later work of Ehresmann on infinitesimal connections. The holonomy principle, first isolated by W. Thurston in the late 1970’s, relates this classification to the representation variety Hom \((\pi_1(\Sigma), G)\). We briefly survey recent results in flat affine, projective, and conformal structures, in particular the tameness of developing maps and uniqueness of structures with given holonomy.
For the entire collection see [Zbl 1426.01005].
MSC:
57-03 History of manifolds and cell complexes
57M05 Fundamental group, presentations, free differential calculus
20-03 History of group theory
Biographic References:
Ehresmann, Charles
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