What Makes Riemannian Geometry by Do Carmo Essential for Grad Students?

[micromass]

• Author: Manfredo Do Carmo
• Title: Riemannian Geometry
• Amazon link https://www.amazon.com/dp/0817634908/?tag=pfamazon01-20
• Prerequisities: Basic differential geometry, topology, calculus 3, linear algebra
• Level: Grad

Table of Contents:

[LIST]
[*] Preface
[*] How to use this book
[*] Differentiable Manifolds
[LIST]
[*] Introduction
[*] Differentiable manifolds; tangent space
[*] Immersions and embeddings; examples
[*] Other examples of manifolds. Orientation
[*] Vector fields; brackets. Topology of manifolds
[/LIST]
[*] Riemannian Metrics
[LIST]
[*] Introduction
[*] Riemannian Metrics
[/LIST]
[*] Affine Connections; Riemannian Connections
[LIST]
[*] Introduction
[*] Affine connections
[*] Riemannian connections
[/LIST]
[*] Geodesics; Convex Neighborhoods
[LIST]
[*] Introduction
[*] The geodesic flow
[*] Minimizing properties of geodesics
[*] Convex neighborhoods
[/LIST]
[*] Curvature
[LIST]
[*] Introduction
[*] Curvature
[*] Sectional curvature
[*] Ricci curvature and scalar curvature
[*] Tensors on Riemannian manifolds
[/LIST]
[*] Jacobi Fields
[LIST]
[*] Introduction
[*] The Jacobi equation
[*] Conjugate points
[/LIST]
[*] Isometric Immersions
[LIST]
[*] Introduction
[*] The second fundamental form
[*] The fundamental equations
[/LIST]
[*] Complete Manifolds; Hopf-Rinow and Hadamard Theorems
[LIST]
[*] Introduction
[*] Complete manifolds; Hopf-Rinow Theorem
[*] The Theorem of Hadamard
[/LIST]
[*] Spaces of Constant Curvature
[LIST]
[*] Introduction
[*] Theorem of Cartan on the determination of the metric by means of the curvature
[*] Hyperbolic space
[*] Space forms
[*] Isometries of the hyperbolic space; Theorem of Liouville
[/LIST]
[*] Variations of Energy
[LIST]
[*] Introduction
[*] Formulas for the first and variations of energy
[*] The theorems of Bonnet-Myers and of Synge-Weinstein
[/LIST]
[*] The Rauch comparison theorem
[LIST]
[*] Introduction
[*] The theorem of Rauch
[*] Applications of the Index Lemma to immersions
[*] Focal points and an extension of Rauch's Theorem
[/LIST]
[*] The Morse Index Theorem
[LIST]
[*] Introduction
[*] The Index Theorem
[/LIST]
[*] The Fundamental Group of Manifolds of Negative Curvature
[LIST]
[*] Introduction
[*] Existence of closed geodesics
[/LIST]
[*] The Sphere Theorem
[LIST]
[*] Introduction
[*] The cut locus
[*] The estimate of the injectivity radius
[*] The Sphere Theorem
[*] Some further developments
[/LIST]
[*] References
[*] Index
[/LIST]

 

[WannabeNewton]

This book is not as rigorous as Lee's book on the same subject and doesn't have many diagrams but it has a very nice motivation for each chapter, covers more topics, and has problems that are pretty much mini lessons in and of themselves (but beware don't look at the hints whatever you do because they are basically the solutions xD). If you are already well acquainted with a lot of smooth manifold theory then just use Lee's book on the same subject and maybe use this book for the problems. It has a noticeably leisurely tone.

 

[micromass]

This is an excellent book on Riemannian Geometry. It is very similar to Lee's masterpiece, but most leisurely. If you went through the previous book by Do Carmo: "Differential Geometry of Curves and Surfaces", then you should have no problem with this book. I do wish that Do Carmo used the language of differential forms more.