Riemannian Holonomy Groups and Calibrated Geometry

by Dominic D. Joyce

Overview

This graduate level text covers an exciting and active area of research at the crossroads of several different fields in Mathematics and Physics. In Mathematics it involves Differential Geometry, Complex Algebraic Geometry, Symplectic Geometry, and in Physics String Theory and Mirror Symmetry. Drawing extensively on the author's previous work, the text explains the advanced mathematics involved simply and clearly to both mathematicians and physicists. Starting with the basic geometry of connections, curvature, complex and Kähler structures suitable for beginning graduate students, the text covers seminal results such as Yau's proof of the Calabi Conjecture, and takes the reader all the way to the frontiers of current research in calibrated geometry, giving many open problems.

Product Details

ISBN-13:	9780199215591
Publisher:	Oxford University Press
Publication date:	05/03/2007
Series:	Oxford Graduate Texts in Mathematics , #12
Pages:	320
Product dimensions:	9.00(w) x 6.10(h) x 0.70(d)

About the Author

Dominic Joyce came up to Oxford University in 1986 to read Mathematics. He held an EPSRC Advanced Research Fellowship from 2001-2006, was recently promoted to professor, and now leads a research group in Homological Mirror Symmetry. His main research areas so far have been compact manifolds with the exceptional holonomy groups G_2 and Spin(7), and special Lagrangian submanifolds, a kind of calibrated submanifold. He is married, with two daughters.

Table of Contents

Preface1. Background material2. Introduction to connections, curvature and holonomy groups3. Riemannian holonomy groups4. Calibrated geometry5. Kähler manifolds6. The Calabi Conjecture7. Calabi-Yau manifolds8. Special Lagrangian geometry9. Mirror Symmetry and the SYZ Conjecture10. Hyperkähler and quaternionic Kähler manifolds11. The exceptional holonomy groups12. Associative, coassociative and Cayley submanifoldsReferencesIndex