- #1
- 22,183
- 3,322
- 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:
Code:
[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]
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