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

William M. Goldman ()
Additional contact information
William M. Goldman: University of Maryland, Department of Mathematics

Chapter Chapter 14 in Geometry in History, 2019, pp 515-552 from Springer

Abstract: 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.

Keywords: Locally homogeneous geometric structures; Projective connection; Affine connection; Real projective space; Fundamental group; Covering space; 57M05 (Low-dimensional topology); 20H10 (Fuchsian groups and their generalizations) (search for similar items in EconPapers)
Date: 2019
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-030-13609-3_14

Ordering information: This item can be ordered from
http://www.springer.com/9783030136093

DOI: 10.1007/978-3-030-13609-3_14

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().