
eBook ISBN: | 978-1-4704-0323-2 |
Product Code: | MEMO/154/730.E |
List Price: | $57.00 |
MAA Member Price: | $51.30 |
AMS Member Price: | $34.20 |

eBook ISBN: | 978-1-4704-0323-2 |
Product Code: | MEMO/154/730.E |
List Price: | $57.00 |
MAA Member Price: | $51.30 |
AMS Member Price: | $34.20 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 154; 2001; 122 ppMSC: Primary 57; Secondary 53
An affine manifold is a manifold with torsion-free 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 concave-cone 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 cross-section, 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 cross-sections, 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.
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Table of Contents
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Chapters
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0. Introduction
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1. Preliminary
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2. ($n$ - 1)-convexity: previous results
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3. Radiant vector fields, generalized affine suspensions, and the radial completeness
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4. Three-dimensional radiant affine manifolds and concave affine manifolds
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5. The decomposition along totally geodesic surfaces
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6. 2-convex radiant affine manifolds
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7. The claim and the rooms
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8. The radiant tetrahedron case
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9. The radiant trihedron case
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10. Obtaining concave-cone affine manifolds
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11. Concave-cone radiant affine 3-manifolds and radiant concave affine 3-manifolds
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12. The nonexistence of pseudo-crescent-cones
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An affine manifold is a manifold with torsion-free 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 concave-cone 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 cross-section, 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 cross-sections, 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. Three-dimensional radiant affine manifolds and concave affine manifolds
-
5. The decomposition along totally geodesic surfaces
-
6. 2-convex radiant affine manifolds
-
7. The claim and the rooms
-
8. The radiant tetrahedron case
-
9. The radiant trihedron case
-
10. Obtaining concave-cone affine manifolds
-
11. Concave-cone radiant affine 3-manifolds and radiant concave affine 3-manifolds
-
12. The nonexistence of pseudo-crescent-cones