Affine 3-Manifolds That Fiber by Circles.

摘要

Throughout the paper, M is a closed connected 3-manifold with a free S(sup 1) action such that chi(M/S(sup 1))<0. The authors suppose M has an affine structure and prove Theorem. The affine holonomy of the circle fiber is a homothety. The authors mention that in all known examples this homothety is nontrivial and the affine structure is derived from a projective structure on the base surface M/S(sup 1). On Section 1 they will review the definitions of affine structure and holonomy and the theory for closed manifolds of low dimension. They include the currently known examples of affine structures on M and discuss the possibility that these are the only ones. In Section 2 they will give an argument of Smillie to show that their theorem implies the recent result of Carriere, Dal'bo, and Meigniez that M has no parallel volume form. In Section 3 they take t(epsilon) pi(sub 1)M to be the circle fiber class and they show that the affine holonomy alpha(t) cannot be a nonzero translation. In Section 4 they complete the proof of the theorem by showing that the linear holonomy lambta(t) must be a scalar. This uses the foliation theory developed in Thurston's thesis together with ideas of Smillie on affine manifolds with diagonal holonomy.