A Concise Introduction to Modern Riemannian Geometry

Author: Cao Jianguo
Publisher:
Publish Date: 2006-01-01
Features: This book is a concise textbook on modern Riemann geometry, divided into two parts. The first part, consisting of Chapters 1 to 4, introduces the fundamentals of Riemann geometry, including various forms of comparison theorems, Calabi-Yau volume estimates, discussions on Zheng Shaoyuan's large diameter theorem and the Cheeger finite theorem. The content is novel and clearly presented, especially the proofs of comparison theorems using methods of ordinary differential inequalities, which differ from classical variational methods. The new proofs and discussions are easy to understand and straightforward. The second part of the book, covering Chapters 5 to 7, discusses the latest achievements in three major modern Riemann geometry research fields: geodesic flows, negatively curved manifolds, and positively curved manifolds. Many new research results, such as the new proof of the Cheeger-Gromoll soul conjecture, are among the first to appear in geometry textbooks both in China and abroad. This book is suitable for scholars engaged in research related to Riemann geometry and can also serve as a textbook and reference for advanced undergraduate and graduate students. ... The book is divided into two parts. The first part introduces the fundamentals of Riemann geometry, including various forms of comparison theorems, discussions on Zheng Shaoyuan's large diameter theorem and the Cheeger finite theorem. The second part discusses the latest achievements in the three major modern Riemann geometry research fields: geodesic flows, negatively curved manifolds, and positively curved manifolds.