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 This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions.
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The book is written in a very clear and understandable way, with careful explanation of the main ideas and many remarks and comments, and it includes systematic suggestions for further reading ... It can be warmly recommended to mathematicians (in geometry and global analysis, in particular) as well as to physicists interested in string theory. EMS The first part is a very effective introduction to basic notions and results of modern differential geometry ... This book is highly recommended for people who are interested in the very recent developments of differential geometry and its relationships with present research in theoretical physics. Zentralblatt MATH
About the Author
University lecturer and tutorial fellow at Lincoln College, Oxford Dr D Joyce Lincoln College Oxford OX1 3DR Tel. 01865 279800 Email: dominic.joyce@lincoln.ox.ac.uk
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This review is from: Compact Manifolds with Special Holonomy (Oxford Mathematical Monographs) (Hardcover)
This book is, as the author mentioned in the preface, a marriage of two parts. The first part provided more or less a self-contained introduction to the theory of Riemannian holonomy groups, which usually couldn't be found in differential geometry textbooks. The second part is a research monograph on exceptional holonomy groups, which is the subject that the author is famous at. This book contains lots of topics which are hard to be found in any other books. For example, it contains a proof of the Calabi conjecture, which I've never seen in anywhere else except Yau's original papers. It also has a concise introduction to Calabi-Yau manifolds, which includes lots of topics about CY manifolds that are hard to be found in just a single book. Overall, it's a great introduction to the theory of holonomy groups. And also provides a good start about differential geometric side of the theory of Calabi-Yau manifolds, together with a roughly complete list of further references.
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Inside This Book
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First Sentence:
In this chapter we explain some background necessary for understanding the rest of the book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
crepant resolution, using toric geometry, free antiholomorphic involution, holomorphic volume form, manifold asymptotic, using complex algebraic geometry, complex orbifold, complex symplectic manifold, exceptional holonomy groups, orbifold groups, hypercomplex manifolds, calibrated submanifolds, topological choices, compact associative, isolated quotient singularity, nontrivial finite subgroup, complex symplectic form, hyperkahler structure, asymptotic coordinate system, parallel spinors, local moduli space, calibrated geometry, hyperkahler manifolds, quaternionic manifolds, principal subbundle Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, String Theory, Ricci-flat Kahler, Proof Define, Asymptotically Locally Euclidean, Berger's Theorem, Kondrakov Theorem, Lefschetz Hyperplane Theorem, Sobolev Embedding Theorem, Applying Lemma, Chow's Theorem, Ilf Ilc, Proof Choose, Therefore Theorem New!
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Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
In this chapter we explain some background necessary for understanding the rest of the book. Read the first page Key Phrases - Statistically Improbable Phrases (SIPs): (learn more)
crepant resolution, using toric geometry, free antiholomorphic involution, holomorphic volume form, manifold asymptotic, using complex algebraic geometry, complex orbifold, complex symplectic manifold, exceptional holonomy groups, orbifold groups, hypercomplex manifolds, calibrated submanifolds, topological choices, compact associative, isolated quotient singularity, nontrivial finite subgroup, complex symplectic form, hyperkahler structure, asymptotic coordinate system, parallel spinors, local moduli space, calibrated geometry, hyperkahler manifolds, quaternionic manifolds, principal subbundle Key Phrases - Capitalized Phrases (CAPs): (learn more)
Proof Let, String Theory, Ricci-flat Kahler, Proof Define, Asymptotically Locally Euclidean, Berger's Theorem, Kondrakov Theorem, Lefschetz Hyperplane Theorem, Sobolev Embedding Theorem, Applying Lemma, Chow's Theorem, Ilf Ilc, Proof Choose, Therefore Theorem New!
Books on Related Topics | Concordance | Text Stats Browse Sample Pages:
Front Cover | Table of Contents | First Pages | Index | Back Cover | Surprise Me!
Citations (learn more)
This book cites 31
books:
See all 31 books this book cites
5
books
cite this book:
- Essays on Mirror Manifolds (Mathematical Physics) by Shing-Tung Yau on 4 pages
- New Trends in Algebraic Geometry (London Mathematical Society Lecture Note Series) by K. Hulek in Back Matter (1), Back Matter (2), and Back Matter (3)
- Differential Geometry (Proceedings of Symposia in Pure Mathematics ; V. 54 Part 1, 2, 3) (Pt.1-3) by Robert Greene in Back Matter (1), and Back Matter (2)
- Surveys in Differential Geometry, Vol. 6: Essays on Einstein Manifolds by Claude LeBrun in Back Matter (1), and Back Matter (2)
- Mathematical Aspects of String Theory (Advanced Series in Mathematical Physics) by S. T. Yau in Back Matter (1), and Back Matter (2)
See all 31 books this book cites
- Advances in Analysis and Geometry: New Developments Using Clifford Algebras (Trends in Mathematics) by Tao Qian on page 207
- Fourier-Mukai Transforms in Algebraic Geometry (Oxford Mathematical Monographs) by Daniel Huybrechts in Front Matter
- Geometric and Topological Methods for Quantum Field Theory (Lecture Notes in Physics) by Hernan Ocampo on page 177
- Calabi-Yau Manifolds and Related Geometries by Mark Gross in Back Matter
- String Theory and M-Theory: A Modern Introduction by Katrin Becker in Back Matter
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