Compact Manifolds with Special Holonomy [Hardcover]

The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and K?hler geometry. Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperk?hler manifolds). These are constructed and studied using complex algebraic geometry. The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated. The first known examples of these manifolds were discovered by the author in 1993-5. This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.

1. Background material
2. Introduction to connections, curvature and holonomy groups
3. Riemannian holonomy groups
4. K?hler manifolds
5. The Calabi conjecture
6. Calabi-Yau manifolds
7. Hyperk?hler manifolds
8. Asymptotically locally Euclidean metrics with holonomy SU (m)
9. QALE metrics with holonomy SU(m) and Sp(m)
10. Introduction to the exceptional holonomy groups
11. Construction of compactG[2-manifolds
12. Examples of compact 7-manifolds with holonomyG[2
13. Construction of compact Spin(7)-manifolds
14. Examples of compact 8-manifolds with holonomy Spin(7)
15. A second construction of compact 8-manifolds with holonomy Spin(7)
References
Index