- #1
- 22,183
- 3,324
- 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:
Code:
[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]
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