Introduction to Riemannian Manifolds

This text covers proving the four fundamental theorems relating curvature and topology, including the Gauss-Bonnet Theorem, Cartan-Hadamard Theorem, Bonnets Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

Format: Hardback
Length: 437 pages
Publication date: 14 January 2019
Publisher: Springer International Publishing AG

This comprehensive text delves into the intricate realm of geometric curvature, providing a solid foundation for advanced courses on Riemannian manifolds. By fostering a deep understanding of curvature, it introduces and showcases essential technical tools that will be invaluable in exploring these complex mathematical structures. The text begins by establishing a close relationship between curvature and topology, unveiling the fundamental theorems that govern their interplay. These theorems include the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnets Theorem, and a special case of the Cartan-Ambrose-Hicks Theorem.

The Gauss-Bonnet Theorem establishes a fundamental connection between the curvature of a manifold and its volume, quantifying the amount of bending or twisting present. It establishes a relationship between the curvature tensor and the Riemannian metric, providing a powerful tool for analyzing the geometry of manifolds.

The Cartan-Hadamard Theorem is a cornerstone of differential geometry, and it establishes the connection between the curvature tensor and the connection coefficients of a Riemannian manifold. It plays a crucial role in understanding the structure and properties of manifolds, particularly in the study of geodesics and the theory of relativity.

Bonnets Theorem is a generalization of the Gauss-Bonnet Theorem that applies to manifolds with non-positive curvature. It provides a way to compute the Euler characteristic of a manifold and sheds light on its topological properties.

The Cartan-Ambrose-Hicks Theorem is a special case of the Cartan-Hadamard Theorem that applies to compact Riemannian manifolds. It establishes the existence of a unique Killing vector field, which plays a fundamental role in the study of differential geometry and the theory of general relativity.

Throughout the text, the authors employ a clear and concise writing style, making the concepts accessible to students with a solid background in mathematics. They provide detailed proofs and illustrations to support their arguments, ensuring a thorough understanding of the topics covered.

In conclusion, this text is an essential resource for anyone seeking to delve into the geometric meaning of curvature and its applications to Riemannian manifolds. By presenting the fundamental theorems and technical tools in a comprehensive and accessible manner, it equips students with the knowledge and skills needed to advance their studies in this field. Whether you are a student, researcher, or professional in mathematics, this text will be a valuable asset in your pursuit of understanding the complex world of Riemannian manifolds.

Weight: 826g
Dimension: 163 x 240 x 31 (mm)
ISBN-13: 9783319917542
Edition number: 2nd ed. 2018