This book expounds on the recent developments in applications of holomorphic functions in the theory of hypercomplex and anti-Hermitian manifolds as well as in the geometry of bundles. It provides detailed information about holomorphic functions in algebras and discusses some of the areas in geometry with applications. The book proves the existence of a one-to-one correspondence between hyper-complex anti-Kähler manifolds and anti-Hermitian manifolds with holomorphic metrics, and also a deformed lifting to bundles. Researchers and students of geometry, algebra, topology and physics may find the book useful as a self-study guide.
Author(s): Arif Salimov
Publisher: Birkhauser
Year: 2023
Language: English
Pages: 122
City: Singapore
Preface
Reference
Contents
About the Author
1 Holomorphic Manifolds Over Algebras
1.1 Some Basic Concepts of Algebra
1.2 Holomorphic Functions
1.3 Algebraic Π-Structures on Manifolds
1.4 Integrable Structures and Holomorphic Manifolds
1.5 Pure Tensors
1.6 Realizations of Holomorphic Tensors
1.7 Pure Connections
1.8 Torsion Tensors of Pure Connections
1.9 Holomorphic Connections
1.10 Holomorphic Curvature Tensor
2 Anti-Hermitian Geometry
2.1 Equivalence of Holomorphic and Anti-Kähler Conditions
2.2 Kähler-Norden Manifolds
2.3 Hessian-Norden Structures
2.4 Twin Norden Metric Connections
2.5 Norden-Walker Manifolds
2.6 Kähler-Norden-Walker Manifolds
2.7 On Curvatures of Norden-Walker Metrics
2.8 Isotropic Anti-Kähler Manifolds
2.9 Quasi-Kähler-Norden-Walker Manifolds
2.10 The Goldberg Conjecture
3 Problems of Lifts
3.1 Tensor Bundles
3.2 Complete Lifts of (1,1)-Tensor Fields
3.3 Holomorphic Cross-Sections
3.4 Dual-Holomorphic Functions and Tangent Bundles of Order 1
3.5 Deformed Complete Lifts of Vector Fields
3.6 Deformed Complete Lifts of Tensor Fields of Type (1,1)
3.7 Deformed Complete Lifts of 1-Forms
3.8 Deformed Complete Lifts of Riemannian Metrics
3.9 Deformed Complete Lifts of Connections
3.10 Holomorphic Metrics in the Tangent Bundle of Order 2
3.11 Deformed Lifts of Vector Fields in the Tangent Bundle of Order 2
3.12 Deformed Complete and Intermediate Lifts of 1-Forms in the Tangent Bundle of Order 2
3.13 Problems of Lifts in Symplectic Geometry
3.14 Anti-Kähler Manifolds and Musical Isomorphisms
References
Index
