Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 Geometry of Surfaces in R3
- 2 Vector Fields
- 3 Sub-Riemannian Structures
- 4 Pontryagin Extremals: Characterization and Local Minimality
- 5 First Integrals and Integrable Systems
- 6 Chronological Calculus
- 7 Lie Groups and Left-Invariant Sub-Riemannian Structures
- 8 Endpoint Map and Exponential Map
- 9 2D Almost-Riemannian Structures
- 10 Nonholonomic Tangent Space
- 11 Regularity of the Sub-Riemannian Distance
- 12 Abnormal Extremals and Second Variation
- 13 Some Model Spaces
- 14 Curves in the Lagrange Grassmannian
- 15 Jacobi Curves
- 16 Riemannian Curvature
- 17 Curvature in 3D Contact Sub-Riemannian Geometry
- 18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
- 19 Asymptotic Expansion of the 3D Contact Exponential Map
- 20 Volumes in Sub-Riemannian Geometry
- 21 The Sub-Riemannian Heat Equation
- Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
- References
- Index
16 - Riemannian Curvature
Published online by Cambridge University Press: 28 October 2019
Book contents
- Frontmatter
- Dedication
- Contents
- Preface
- Introduction
- 1 Geometry of Surfaces in R3
- 2 Vector Fields
- 3 Sub-Riemannian Structures
- 4 Pontryagin Extremals: Characterization and Local Minimality
- 5 First Integrals and Integrable Systems
- 6 Chronological Calculus
- 7 Lie Groups and Left-Invariant Sub-Riemannian Structures
- 8 Endpoint Map and Exponential Map
- 9 2D Almost-Riemannian Structures
- 10 Nonholonomic Tangent Space
- 11 Regularity of the Sub-Riemannian Distance
- 12 Abnormal Extremals and Second Variation
- 13 Some Model Spaces
- 14 Curves in the Lagrange Grassmannian
- 15 Jacobi Curves
- 16 Riemannian Curvature
- 17 Curvature in 3D Contact Sub-Riemannian Geometry
- 18 Integrability of the Sub-Riemannian Geodesic Flow on 3D Lie Groups
- 19 Asymptotic Expansion of the 3D Contact Exponential Map
- 20 Volumes in Sub-Riemannian Geometry
- 21 The Sub-Riemannian Heat Equation
- Appendix Geometry of Parametrized Curves in Lagrangian Grassmannians Igor Zelenko
- References
- Index
Summary
In this chapter we introduce "Ehresmann connections,"with their associated notions of parallel transportand curvature. We then specify these notions in thecase of a Riemannian manifold, where one can find acanonical connection associated with the metricstructure, called Levi–Civita connection. We thenexplain how this connection is related to the theoryof Jacobi curves developed in the previouschapter.
- Type
- Chapter
- Information
- A Comprehensive Introduction to Sub-Riemannian Geometry , pp. 551 - 570Publisher: Cambridge University PressPrint publication year: 2019