eBook ISBN:  9781470403232 
Product Code:  MEMO/154/730.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 
eBook ISBN:  9781470403232 
Product Code:  MEMO/154/730.E 
List Price:  $57.00 
MAA Member Price:  $51.30 
AMS Member Price:  $34.20 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 154; 2001; 122 ppMSC: Primary 57; Secondary 53
An affine manifold is a manifold with torsionfree flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with a holonomy group consisting of affine transformations fixing a common fixed point. We decompose a closed radiant affine \(3\)manifold into radiant \(2\)convex affine manifolds and radiant concave affine \(3\)manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective \(n\)manifolds developed earlier. Then we decompose a \(2\)convex radiant affine manifold into convex radiant affine manifolds and concavecone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of a holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine \(3\)manifold admits a total crosssection, confirming a conjecture of Carrière, and hence every closed radiant affine \(3\)manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a Euler characteristic zero surface. In Appendix C, Thierry Barbot and the author show the nonexistence of certain radiant affine \(3\)manifolds and that compact radiant affine \(3\)manifolds with nonempty totally geodesic boundary admit total crosssections, which are key results for the main part of the paper.
ReadershipGraduate students and research mathematicians interested in manifolds and cell complexes, and differential geometry.

Table of Contents

Chapters

0. Introduction

1. Preliminary

2. ($n$  1)convexity: previous results

3. Radiant vector fields, generalized affine suspensions, and the radial completeness

4. Threedimensional radiant affine manifolds and concave affine manifolds

5. The decomposition along totally geodesic surfaces

6. 2convex radiant affine manifolds

7. The claim and the rooms

8. The radiant tetrahedron case

9. The radiant trihedron case

10. Obtaining concavecone affine manifolds

11. Concavecone radiant affine 3manifolds and radiant concave affine 3manifolds

12. The nonexistence of pseudocrescentcones


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An affine manifold is a manifold with torsionfree flat affine connection. A geometric topologist's definition of an affine manifold is a manifold with an atlas of charts to the affine space with affine transition functions; a radiant affine manifold is an affine manifold with a holonomy group consisting of affine transformations fixing a common fixed point. We decompose a closed radiant affine \(3\)manifold into radiant \(2\)convex affine manifolds and radiant concave affine \(3\)manifolds along mutually disjoint totally geodesic tori or Klein bottles using the convex and concave decomposition of real projective \(n\)manifolds developed earlier. Then we decompose a \(2\)convex radiant affine manifold into convex radiant affine manifolds and concavecone affine manifolds. To do this, we will obtain certain nice geometric objects in the Kuiper completion of a holonomy cover. The equivariance and local finiteness property of the collection of such objects will show that their union covers a compact submanifold of codimension zero, the complement of which is convex. Finally, using the results of Barbot, we will show that a closed radiant affine \(3\)manifold admits a total crosssection, confirming a conjecture of Carrière, and hence every closed radiant affine \(3\)manifold is homeomorphic to a Seifert fibered space with trivial Euler number, or a virtual bundle over a circle with fiber homeomorphic to a Euler characteristic zero surface. In Appendix C, Thierry Barbot and the author show the nonexistence of certain radiant affine \(3\)manifolds and that compact radiant affine \(3\)manifolds with nonempty totally geodesic boundary admit total crosssections, which are key results for the main part of the paper.
Graduate students and research mathematicians interested in manifolds and cell complexes, and differential geometry.

Chapters

0. Introduction

1. Preliminary

2. ($n$  1)convexity: previous results

3. Radiant vector fields, generalized affine suspensions, and the radial completeness

4. Threedimensional radiant affine manifolds and concave affine manifolds

5. The decomposition along totally geodesic surfaces

6. 2convex radiant affine manifolds

7. The claim and the rooms

8. The radiant tetrahedron case

9. The radiant trihedron case

10. Obtaining concavecone affine manifolds

11. Concavecone radiant affine 3manifolds and radiant concave affine 3manifolds

12. The nonexistence of pseudocrescentcones