Riemannian Manifolds: An Introduction to Curvature by Lee

[micromass]

• Author: John Lee
• Title: Riemannian Manifolds: An Introduction to Curvature
• Amazon link https://www.amazon.com/dp/0387983228/?tag=pfamazon01-20
• Prerequisities: "Introduction to Smooth Manifolds" by Lee seems like a prereq.
• Level: Grad

Table of Contents:

[LIST]
[*] Preface 
[*] What Is Curvature?
[LIST]
[*] The Euclidean Plane
[*] Surfaces in Space
[*] Curvature in Higher Dimensions
[/LIST]
[*] Review of Tensors, Manifolds, and Vector Bundles 
[LIST]
[*] Tensors on a Vector Space
[*] Manifolds
[*] Vector Bundles
[*] Tensor Bundles and Tensor Fields
[/LIST]
[*] Definitions and Examples of Riemannian Metrics 
[LIST]
[*] Riemannian Metrics
[*] Elementary Constructions Associated with Riemannian Metrics
[*] Generalizations of Riemannian Metrics
[*] The Model Spaces of Riemannian Geometry
[*] Problems
[/LIST]
[*] Connections 
[LIST]
[*] The Problem of Differentiating Vector Fields
[*] Connections
[*] Vector Fields Along Curves
[*] Geodesics
[*] Problems
[/LIST]
[*] Riemannian Geodesics 
[LIST]
[*] The Riemannian Connection
[*] The Exponential Map
[*] Normal Neighborhoods and Normal Coordinates
[*] Geodesics of the Model Spaces
[*] Problems
[/LIST]
[*] Geodesics and Distance 
[LIST]
[*] Lengths and Distances on Riemannian Manifolds
[*] Geodesics and Minimizing Curves
[*] Completeness
[*] Problems
[/LIST]
[*] Curvature
[LIST]
[*] Local Invariants
[*] Flat Manifolds
[*] Symmetries of the Curvature Tensor
[*] Ricci and Scalar Curvatures
[*] Problems
[/LIST]
[*] Riemannian Submanifolds 
[LIST]
[*] Riemannian Submanifolds and the Second Fundamental Form
[*] Hypersurfaces in Euclidean Space
[*] Geometric Interpretation of Curvature in Higher Dimensions
[*] Problems
[/LIST]
[*] The Gauss–Bonnet Theorem 
[LIST]
[*] Some Plane Geometry
[*] The Gauss–Bonnet Formula
[*] The Gauss–Bonnet Theorem
[*] Problems
[/LIST]
[*] Jacobi Fields 
[LIST]
[*] The Jacobi Equation
[*] Computations of Jacobi Fields
[*] Conjugate Points 
[*] The Second Variation Formula
[*] Geodesics Do Not Minimize Past Conjugate Points
[*] Problems
[/LIST]
[*] Curvature and Topology 
[LIST]
[*] Some Comparison Theorems 
[*] Manifolds of Negative Curvature
[*] Manifolds of Positive Curvature
[*] Manifolds of Constant Curvature
[*] Problems
[/LIST]
[*] References 
[*] Index
[/LIST]

 

[Fredrik]

This is an excellent book, one of my favorites. Definitely the best place to learn about connections, parallel transport, geodesics and curvature. This is covered in the first 7 chapters, 129 pages. The rest of the book covers stuff that most physicists probably won't be very interested in.

You should study the basics of manifold theory in "Introduction to smooth manifolds" before you study this one. You don't need to study the entire book, but you should make sure that you understand the terms smooth manifold, tangent space, cotangent space, vector bundle, tensor and tensor field.