Compact Manifolds with Special Holonomy
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Author Description
The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kähler 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) (hyperkähler 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 G2 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 This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.
Menrva's Summary
THE THESIS
This text investigates the existence and construction of compact Riemannian manifolds possessing special holonomy groups, specifically focusing on the exceptional groups G2 and Spin(7). Dominic D. Joyce, a mathematician recognized for his contributions to differential geometry, provides a rigorous framework for understanding these structures. The book synthesizes foundational concepts in Riemannian and complex geometry with advanced techniques in algebraic geometry to establish the existence of these manifolds, building upon the author's own research from the 1990s.
THE SCOPE MAP
What You Will Find
- Comprehensive introduction to connections, holonomy groups, and Kähler geometry.
- Proof of the Calabi conjecture and its application to SU(m) and Sp(m) holonomy.
- Detailed construction methods for compact 7
- and 8-manifolds with G2 and Spin(7) holonomy.
- Calculation of Betti numbers for the newly constructed manifold examples.
THE AUTHORITY PERSPECTIVE
Experts identify this work as a foundational text for researchers in differential geometry and string theory. Readers frequently note the high level of technical density and the specialized mathematical knowledge required to fully engage with the proofs presented.
Page Count:
448
Publication Date:
2000-09-21
Publisher:
Oxford University Press
ISBN-10:
0198506015
ISBN-13:
9780198506010
Differential Geometry
Topology
Geometry
Algebraic Geometry
Mathematics
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