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

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Geometry in History

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. 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). 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.

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Acknowledgements

The author gratefully acknowledges research support from NSF Grants DMS1065965, DMS1406281, DMS1065965 as well as the Research Network in the Mathematical Sciences DMS1107367 (GEAR).

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  1. Department of Mathematics, University of Maryland, College Park, MD, USA

    William M. Goldman

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  1. University of Mumbai & Department of Atomic Energy, Centre for Excellence in Basic Sciences, Mumbai, Maharashtra, India

    S. G. Dani

  2. Institut de Recherche Mathématique, University of Strasbourg, CNRS, Strasbourg Cedex, France

    Athanase Papadopoulos

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Goldman, W.M. (2019). Flat Affine, Projective and Conformal Structures on Manifolds: A Historical Perspective. In: Dani, S.G., Papadopoulos, A. (eds) Geometry in History. Springer, Cham. https://doi.org/10.1007/978-3-030-13609-3_14

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