1 Riemannian Metrics.- 2 Curvature.- 3 Examples.- 4 Hypersurfaces.- 5 Geodesics and Distance.- 6 Sectional Curvature Comparison I.- 7 The Bochner Technique.- 8 Symmetric Spaces and Holonomy.- 9 Ricci Curvature Comparison.- 10 Convergence.- 11 Sectional Curvature Comparison II.- A de Rham Cohomology.- A.1 Elementary Properties.- A.2 Integration of Forms.- A.3 ?ech Cohomology.- A.4 de Rham Cohomology.- A.5 Poincaré Duality.- A.6 Degree Theory.- A.7 Further Study.- B Principal Bundles.- B.1 Cartan Formalism.- B.2 The Frame Bundle.- B.3 Construction of the Frame Bundle.- B.4 Construction of Tensor Bundles.- B.5 Tensors.- B.6 The Connection on the Frame Bundle.- B.7 Covariant Differentiation of Tensors.- B.8 Principal Bundles in General.- B.9 Further Study.- C Spinors.- C.1 Spin Structures.- C.2 Spinor Bundles.- C.3 The Weitzenböck Formula for Spinors.- C.4 The Square of a Spinor.- C.5 Further Study.- References.

Intended for a one year course, this volume serves as a single source, introducing students to the important techniques and theorems, while also containing enough background on advanced topics to appeal to those students wishing to specialise in Riemannian geometry. Instead of variational techniques, the author uses a unique approach, emphasising distance functions and special co-ordinate systems. He also uses standard calculus with some techniques from differential equations to provide a more elementary route. Many chapters contain material typically found in specialised texts, never before published in a single source. This is one of the few works to combine both the geometric parts of Riemannian geometry and the analytic aspects of the theory, while also presenting the most up-to-date research - including sections on convergence and compactness of families of manifolds. Thus, this book will appeal to readers with a knowledge of standard manifold theory, including such topics as tensors and Stokes theorem. Various exercises are scattered throughout the text, helping motivate readers to deepen their understanding of the subject.