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Sasakian Geometry$
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Charles Boyer and Krzysztof Galicki

Print publication date: 2007

Print ISBN-13: 9780198564959

Published to Oxford Scholarship Online: January 2008

DOI: 10.1093/acprof:oso/9780198564959.001.0001

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Sasakian Geometry in Dimensions 3 and 5

Sasakian Geometry in Dimensions 3 and 5

Chapter:
(p.329) Chapter 10 Sasakian Geometry in Dimensions 3 and 5
Source:
Sasakian Geometry
Author(s):

Charles P. Boyer

Krzysztof Galicki

Publisher:
Oxford University Press
DOI:10.1093/acprof:oso/9780198564959.003.0011

This chapter describes Sasakian geometry in low dimensions. In dimension three there is a complete classification; dimension five is large enough to be interesting, yet small enough to hope for some partial classification. We concentrate on the simply connected case, as there we can rely on the Smale-Barden classification. In terms of Sasakian structures, the main focus is on the case of positive Sasakian structures. The chapter describes several remarkable theorems of Kollfiar showing how positivity severely restricts the topology, in particular the torsion of a manifold which admits a positive Sasakian structure. The method heavily depends on the algebraic geometry of log del Pezzo surfaces with cyclic Du Val singularities.

Keywords:   Smale-Barden manifolds, Sasakian 3-manifolds, Sasakian 5-manifolds, log del Pezzo surfaces, Du Val singularities, positive Sasakian structures, Seifert bundles, Sasakian-Seifert structures

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