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Flat Affine, Projective and Conformal Structures on Manifolds: A Historical Perspective

In: Geometry in History

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  • William M. Goldman

    (University of Maryland, Department of Mathematics)

Abstract

This historical survey reports on the theory of locally homogeneous geometric structures as initiated in Ehresmann’s 1936 paper Sur les espaces localement homogènes. Ehresmann, Charles (1905–1979) 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(π 1( Σ), G). Holonomy principle We briefly Representation variety survey recent results in flat affine, projective, and conformal structures, in particular the tameness of developing maps and uniqueness of structures with given holonomy.

Suggested Citation

  • William M. Goldman, 2019. "Flat Affine, Projective and Conformal Structures on Manifolds: A Historical Perspective," Springer Books, in: S. G. Dani & Athanase Papadopoulos (ed.), Geometry in History, chapter 0, pages 515-552, Springer.
    • Handle: RePEc:spr:sprchp:978-3-030-13609-3_14
    DOI: 10.1007/978-3-030-13609-3_14
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