Differential Geometry of Varieties with Degenerate Gauss Maps by Maks A. Akivis – Advanced Mathematics Textbook on Proje

Introduction

This book offers a rigorous and insightful exploration into the differential geometry of varieties whose Gauss maps are degenerate. Authored by Maks A. Akivis, a distinguished mathematician, this volume bridges classical projective differential geometry with modern algebraic geometry. It is an essential reference for researchers and graduate students seeking to understand the deeper structures of submanifolds with singularities in their Gauss mappings. The text systematically develops the theory using moving frames, exterior differential forms, and tensor calculus, making it a valuable addition to any serious mathematics library in India.

Book Overview

Differential Geometry of Varieties with Degenerate Gauss Maps is a comprehensive monograph that delves into the geometry of varieties where the Gauss map has a lower rank than expected. The authors employ a blend of projective differential geometry and classical algebraic geometry to uncover the intrinsic structure of such varieties. The work identifies singular points and focal images, leading to a classification of these degenerate varieties. Published by Springer in a durable hardcover edition, this book is ideal for advanced study and research in geometry.

Key Highlights

• Systematic use of the method of moving frames and exterior differential forms for analyzing degenerate Gauss maps.
• In-depth study of singular points, focal images, and classification of varieties with degenerate Gauss maps.
• Unique integration of projective differential geometry with classical algebraic geometry techniques.
• Rigorous theoretical framework suitable for advanced graduate courses and research seminars.
• Authored by a leading expert in differential geometry, ensuring authoritative content.

Inside the Book

The book is structured to first introduce the foundational methods of differential geometry, including moving frames and tensor calculus. It then applies these tools to investigate the geometry of varieties with degenerate Gauss maps. Key chapters cover the determination of singular points, focal images, and the construction of a classification scheme. The final sections present concrete applications of the theory to specific problems, demonstrating the power of the combined projective and algebraic approaches.

Key Topics

• Degenerate Gauss maps and their geometric significance
• Moving frames and exterior differential forms in differential geometry
• Tensor methods for studying submanifolds
• Singular points and focal varieties
• Classification theory of varieties with degenerate Gauss maps
• Projective differential geometry and its algebraic connections

Reader Benefits

Readers will gain a deep understanding of how degenerate Gauss maps shape the geometry of varieties. The book equips them with advanced analytical tools like moving frames and exterior forms, which are applicable to broader problems in differential geometry. It also provides a clear classification framework that can be used for future research. Indian students and researchers will find the systematic exposition helpful for mastering complex geometric concepts and for preparing competitive examinations or research publications.

Learning Outcomes

• Ability to apply moving frames and exterior differential forms to study degenerate Gauss maps.
• Proficiency in identifying and analyzing singular points and focal images of varieties.
• Understanding of the classification of varieties with degenerate Gauss maps.
• Skill in integrating projective differential geometry with algebraic geometry methods.
• Capacity to conduct independent research in advanced differential geometry.

Who Should Read

This book is intended for graduate students in mathematics, particularly those specializing in differential geometry, algebraic geometry, or projective geometry. It is also an excellent resource for researchers and faculty members who wish to deepen their knowledge of degenerate Gauss maps. Professionals in related fields such as theoretical physics or computer-aided geometric design may also benefit from the geometric insights presented.

About the Author

Maks A. Akivis is a renowned mathematician known for his significant contributions to differential geometry and the theory of webs. His research has focused on the geometry of submanifolds, projective differential geometry, and the application of moving frames. With decades of teaching and research experience, Akivis brings clarity and depth to this complex subject, making it accessible to advanced learners.

About the Publisher

Springer is a globally respected academic publisher known for producing high-quality scientific and mathematical literature. With a legacy of excellence, Springer ensures that this hardcover edition meets the highest standards of accuracy and durability, making it a reliable resource for students and researchers in India and around the world.

Conclusion

Differential Geometry of Varieties with Degenerate Gauss Maps is a masterful work that fills a crucial gap in the literature. Its unique combination of projective differential geometry and algebraic geometry, along with systematic classification, makes it indispensable for anyone serious about advanced geometry. Whether you are a graduate student or a seasoned researcher, this book will enrich your understanding and inspire further exploration. Order your hardcover copy from Bookshops.in today and add this classic to your collection.