Page 369 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 30: Geodesic Curvature and Christoffel Symbols

CONTENTS
Objectives

Introduction
30.1 Geodesic Curvature and the Christoffel Symbols
30.2 Principal Curvatures, Gaussian Curvature, Mean Curvature
30.3 The Gauss Map and its Derivative dN
30.4 The Dupin Indicatrix

30.5 Clairaut’s Theorem
30.6 Gauss–Bonnet theorem
30.6.1 Statement of the Theorem

30.6.2 Interpretation and Significance
30.6.3 Special Cases
30.6.4 Combinatorial Analog
30.6.5 Generalizations
30.7 The Theorema Egregium of Gauss, the Equations of Codazzi-Mainardi, and Bonnet’s
Theorem
30.8 Lines of Curvature, Geodesic Torsion, Asymptotic Lines
30.9 Summary

30.10 Keywords
30.11 Self Assessment
30.12 Review Questions
30.13 Further Readings

Objectives

After studying this unit, you will be able to:
Explain the Gauss Map and its Derivative dN

Define the Dupin Indicatrix

Describe the theorema Egregium of Gauss, the Equations of Codazzi-Mainardi, and Bonnet’s

Theorem
Define Lines of Curvature, Geodesic Torsion, Asymptotic Lines


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