New Horizons In Differential Geometry And Its Related Fields

The book discusses recent developments in geometric structures on Riemannian manifolds and their discretizations, including contact structures, Kähler structures, fiber bundle structures, and Einstein metrics. It is a valuable resource for researchers and students in differential geometry and related areas.

Format: Hardback
Length: 256 pages
Publication date: 08 May 2022
Publisher: World Scientific Publishing Co Pte Ltd

This comprehensive volume explores the latest advancements in geometric structures on Riemannian manifolds and their discretizations. Written by renowned experts, the chapters delve into diverse topics such as contact structures, Kähler structures, fiber bundle structures, and Einstein metrics. Additionally, it features contributions on the geometric approach to coding theory, making it an invaluable resource for researchers and students alike. By providing a thorough examination of these subjects, which span beyond differential geometry and have wide-ranging applications, this book fosters and enhances the study of geometric structures.

Introduction:
Geometric structures play a fundamental role in various fields of mathematics and physics, including differential geometry, algebraic geometry, topology, and theoretical physics. Riemannian manifolds, which are smooth, connected, and locally Euclidean spaces, provide a rich framework for studying these structures. In recent years, there has been a significant amount of research and development in the field of geometric structures on Riemannian manifolds and their discretizations.
Contact Structures:
Contact structures are a fundamental concept in geometric analysis, and they play a crucial role in understanding the topology and geometry of Riemannian manifolds. They are defined as the intersection of two manifolds along a hypersurface, and they provide a way to describe the shape and topology of a manifold. Contact structures have numerous applications in various fields, including physics, engineering, and computer science.
Kähler Structures:
Kähler structures are another important concept in geometric analysis, and they are used to describe the geometry of Riemannian manifolds that are symplectic or Kahlerian. Symplectic manifolds are those that are invariant under the action of a Hamiltonian vector field, and they are important in the study of classical mechanics and quantum mechanics. Kähler structures provide a way to define metrics and other geometric properties of a manifold, and they have applications in various areas, including string theory, gauge theory, and cosmology.
Fiber Bundle Structures:
Fiber bundle structures are a powerful tool in geometric analysis, and they are used to describe the geometry of Riemannian manifolds that are fibered over a compact space. Fiber bundles are a generalization of vector bundles, and they provide a way to describe the geometry of higher-dimensional spaces in terms of lower-dimensional spaces. Fiber bundle structures have applications in various fields, including physics, engineering, and computer science.
Einstein Metrics:
Einstein metrics are a special class of metrics that are defined on Riemannian manifolds that are symmetric or pseudo-symmetric. They are important in the study of general relativity and cosmology, and they provide a way to describe the geometry of spacetime. Einstein metrics have applications in various areas, including astrophysics, cosmology, and theoretical physics.
Geometric Approach to Coding Theory:
Geometric approach to coding theory is a new area of research that combines the principles of geometry and coding theory. It aims to develop new coding schemes and algorithms that are more efficient and secure. Geometric approach to coding theory has applications in various fields, including communication, data compression, and cryptography.
Conclusion:
In conclusion, this volume presents a comprehensive overview of the latest developments in geometric structures on Riemannian manifolds and their discretizations. With chapters written by recognized experts, the book covers a wide range of topics and provides a valuable resource for researchers and students in the field. The study of geometric structures has significant implications for various areas of mathematics and physics, and this volume contributes to the ongoing exploration and understanding of these subjects.

ISBN-13: 9789811248092