Page 355 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 29: Geodesics

CONTENTS
Objectives

Introduction
29.1 Geodesics
29.2 The Riemann Curvature Tensor
29.3 The Second Variation of Arc length
29.4 Summary

29.5 Keywords
29.6 Self Assessment
29.7 Review Questions

29.8 Further Readings

Objectives

After studying this unit, you will be able to:
Define Geodesics

Explain the Riemann Curvature Tensor

Discuss the Second Variation of Arc length

Introduction

In particular, we will ignore the Gauss map and the second fundamental form. Thanks to Gauss’
Theorema Egregium, we will still be able to take the Gauss curvature into account. In last unit,
you have studied about local intrinsic Geometry of Surfaces.

29.1 Geodesics

Definition 1. Let (U, g) be a Riemannian surface, and let : I  U be a curve. A vector field along
.
2
i
 is a smooth function Y : I   The covariant derivative of Y  y  along  is the vector field:
i
j
 y  
   Y   i i jk y   k  . i
Note that if Z is any extension of Y, i.e., a any vector field defined on a neighborhood V of the
image   I of  in U, then we have:
 Z    i Z .
  Y     ;i

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