by K Yang (Arkansas State)

This book is an introduction to the theory of almost complex homogeneous spaces and certain closely related class of spaces, so called partial G-flag manifolds. Submanifolds, in particular holomorphic curves, are also treated using the theory of moving frames and the structure theory of compact lie groups. The exposition is reasonably self-contained and this book is strongly recommended as a text for beginning graduate students.

• Structures of Compact Lie Groups:
  • Definitions and Examples
  • Lie Algebras — Basic Results
  • Orthogonal and Unitary Representations
  • Maximal Tori and Stiefel Diagrams
  • Weyl Groups, Dynkin Diagrams and the Classification
• Almost Complex Homogeneous Spaces:
  • Homogeneous Spaces
  • Partial G-Flag Manifolds and Invariant Metrics
  • Invariant Complex Structures
  • The Maurer-Cartan Form
  • Integrability Condition
  • The Horizontal Distribution
• Complex Submanifolds:
  • Pseudocomplex Maps
  • Moving Frames
  • U(n)-Flag Manifolds
  • SO(2n,R)-Flag Manifolds
• Examples:
  • Holomorphic Curves: Holomorphic Curves in F_1,2,3(C³)
  • Horizontal Curves U(n)/T
  • Holomorphic Curves in CP^m
  • Horizontal Curves in Sp(n)/T
• Algebraic Curves:
  • Singular Metrics and the Gauss-Bonnet-Chem Theorem
  • Project Curves and Their Associated Curves
  • Horizontal Curves and the Frenet Frames
  • Plücker Formulae for Projective Curves
  • The Symplectic Plücker Formulae
• Exterior Differential Systems:
  • Exterior Algebra, Completely Integrable Systems and the Cauchy Characteristics
  • Cartan-Kähler Theory
  • Prolongation