J. Korean Math. Soc. 2003; 40(2): 319-340
Printed March 1, 2003
Copyright © The Korean Mathematical Society.
Suhyoung Choi
Seoul National University
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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