- Title Pages
- Preface
- 1 Smooth manifolds
- 2 Matrices and Lie groups
- 3 Introduction to vector bundles
- 4 Algebra of vector bundles
- 5 Maps and vector bundles
- 6 Vector bundles with ℂ<sup>n</sup> as fiber
- 7 Metrics on vector bundles
- 8 Geodesics
- 9 Properties of geodesics
- 10 Principal bundles
- 11 Covariant derivatives and connections
- 12 Covariant derivatives, connections and curvature
- 13 Flat connections and holonomy
- 14 Curvature polynomials and characteristic classes
- 15 Covariant derivatives and metrics
- 16 The Riemann curvature tensor
- 17 Complex manifolds
- 18 Holomorphic submanifolds, holomorphic sections and curvature
- 19 The Hodge star
- List of lemmas, propositions, corollaries and theorems
- List of symbols
- Index
Properties of geodesics
Properties of geodesics
- Chapter:
- (p.96) 9 Properties of geodesics
- Source:
- Differential Geometry
- Author(s):
Clifford Henry Taubes
- Publisher:
- Oxford University Press
This chapter examines the properties of geodesics. It discusses the maximal extension of a geodesic; the exponential map; Gaussian coordinates; and the proof of the geodesic theorem.
Keywords: exponential map, Gaussian coordinates, geodesic theorem
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- Title Pages
- Preface
- 1 Smooth manifolds
- 2 Matrices and Lie groups
- 3 Introduction to vector bundles
- 4 Algebra of vector bundles
- 5 Maps and vector bundles
- 6 Vector bundles with ℂ<sup>n</sup> as fiber
- 7 Metrics on vector bundles
- 8 Geodesics
- 9 Properties of geodesics
- 10 Principal bundles
- 11 Covariant derivatives and connections
- 12 Covariant derivatives, connections and curvature
- 13 Flat connections and holonomy
- 14 Curvature polynomials and characteristic classes
- 15 Covariant derivatives and metrics
- 16 The Riemann curvature tensor
- 17 Complex manifolds
- 18 Holomorphic submanifolds, holomorphic sections and curvature
- 19 The Hodge star
- List of lemmas, propositions, corollaries and theorems
- List of symbols
- Index
