Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

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J. Korean Math. Soc. 2003; 40(2): 319-340

Printed March 1, 2003

Copyright © The Korean Mathematical Society.

Geometric structures on low-dimensional manifolds

Suhyoung Choi

Seoul National University

Abstract

Geometric structures on $3$-manifolds are often projectively flat structures. Projectively flat structures on $3$-manifolds are given by atlases of charts to ${\bf R} P^3$ with projective transition maps. Equivalently, they are given by projectively flat torsion-free connections. We study the question of putting projective structures on 3-manifolds. This is done by triangulating a given $3$-manifold, and then reducing the question to a $2$-dimensional classical projective geometry problem produced by the Haken diagram of the $3$-manifold. Next, we show that the $2$-dimensional problem can be reduced to solving a system of homogeneous equations that are in product forms of scalar triple products of vectors. Finally, we will compute the deformation spaces of projective structures on a small class of $3$-orbifolds.

Keywords: projectively flat structure on 3-manifolds, real algebraic set, homogeneous polynomial equation

MSC numbers: Primary 57M50

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