Riemannian Geometry in an Orthogonal Frame: Lectures on Differential Geometry by Elie Cartan – A Classic from the Sorbon

Introduction

Dive into the profound world of differential geometry with Riemannian Geometry in an Orthogonal Frame, a classic text derived from the legendary lectures delivered by Élie Cartan at the Sorbonne in 1926-27. This hardcover edition, published by World Scientific Publishing Company, brings to English readers a seminal work that has shaped modern mathematical thought. For Indian students and researchers in pure mathematics, this book is not merely a historical artifact but a living source of deep geometric insight, offering a unique approach to Riemannian manifolds through the powerful lens of orthogonal frames and exterior forms.

Book Overview

Originally transcribed from Cartan’s lectures by Sergei P. Finikov and translated into Russian in 1960, this volume has now been masterfully rendered into English by Vladislav V. Goldberg, a Distinguished Professor of Mathematics. The text is distinguished by its early introduction of exterior differential forms and its systematic use of orthogonal frames to explore the intrinsic geometry of Riemannian spaces. Cartan’s course addresses a wide array of problems, from Euclidean and non-Euclidean geometries to variational problems on geodesics, all while laying foundational concepts like intrinsic normal differentiation and Gaussian torsion of submanifolds. It is an indispensable resource for anyone seeking a rigorous, frame-theoretic understanding of Riemannian geometry.

Key Highlights

• Pioneering Approach: Introduces exterior forms at the outset, a method that revolutionizes the study of Riemannian manifolds.
• Orthogonal Frame Method: Employs moving orthogonal frames extensively, providing a clear geometric and computational framework.
• Original Insights: Features Cartan’s innovative concepts such as intrinsic normal differentiation and Gaussian torsion for submanifolds.
• Historical Significance: Based on the actual 1926-27 Sorbonne lectures of one of the 20th century’s greatest geometers.
• Authoritative Translation: Translated by Vladislav V. Goldberg, ensuring mathematical precision and readability for modern audiences.

Inside the Book

The text unfolds with a systematic development of Riemannian geometry using Cartan’s method of moving frames. It begins with the algebra of exterior forms and moves into the geometry of curves and surfaces in Euclidean space. The core of the book explores Riemannian manifolds, introducing the structural equations of Cartan, curvature forms, and the concept of torsion. A significant portion is devoted to submanifold theory, where Cartan’s techniques illuminate the geometry of embeddings, including the normal bundle and its affine connection. Variational principles and geodesic theory are treated with elegance, demonstrating Cartan’s mastery of combining analytic and geometric reasoning.

Key Topics

• Exterior differential forms and their geometric applications
• Orthogonal moving frames and the structural equations
• Riemannian metrics and curvature tensors
• Intrinsic normal differentiation and Gaussian torsion
• Submanifolds in Euclidean spaces and spaces of constant curvature
• Affine connections in normal fiber bundles
• Variational problems and geodesic flows

Reader Benefits

This book offers a transformative reading experience for advanced students and professionals. By working through Cartan’s lectures, readers will develop a deep, intuitive grasp of curvature and geometry that standard textbooks often miss. The orthogonal frame method equips you with powerful computational tools for tackling complex problems in differential geometry, general relativity, and geometric analysis. The historical context enriches your appreciation of how modern geometry evolved, while the clear translation ensures that Cartan’s original brilliance remains accessible.

Learning Outcomes

• Master the use of exterior forms in Riemannian geometry.
• Understand and apply Cartan’s structural equations to compute curvature.
• Analyze the geometry of submanifolds using intrinsic normal differentiation.
• Solve variational problems on geodesics with frame-theoretic methods.
• Gain proficiency in moving frame techniques for research in geometry and physics.

Who Should Read

This book is ideal for postgraduate students and researchers in pure mathematics, especially those specializing in differential geometry, topology, and geometric analysis. It is also highly valuable for theoretical physicists working in general relativity and gauge theories, where Cartan’s methods are increasingly applied. Indian students preparing for competitive research fellowships or advanced coursework will find this a challenging yet rewarding companion. Professors and educators seeking a classic text to deepen their own understanding or to use in advanced seminars will treasure this edition.

About the Author

Élie Cartan (1869–1951) was one of the most influential mathematicians of the 20th century. A Frenchman, he made foundational contributions to Lie groups, differential geometry, and theoretical physics. His work on moving frames and exterior differential systems revolutionized geometric thinking. Cartan’s lectures at the Sorbonne, captured in this volume, represent the pinnacle of his teaching, blending deep conceptual insight with rigorous calculation.

About the Publisher

World Scientific Publishing Company is a leading international academic publisher with a strong presence in India. Renowned for its high-quality monographs and textbooks in science, technology, and mathematics, World Scientific brings this classic work to English readers in a durable hardcover format, ensuring it remains a permanent resource for libraries and personal collections.

Conclusion

Riemannian Geometry in an Orthogonal Frame is more than a book—it is a masterclass from one of history’s greatest geometers. For Indian students and scholars committed to understanding the deepest structures of space, this volume offers an unparalleled journey into the heart of Riemannian geometry. Add this essential hardcover to your library and engage with the lectures that continue to inspire generations of mathematicians.