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Richa Nandra, Lovely Professional University Unit 31: Joachimsthal's Notations

Unit 31: Joachimsthal's Notations Notes

CONTENTS
Objectives

Introduction
31.1 Geodesic Lines, Local Gauss-Bonnet Theorem
31.2 Covariant Derivative, Parallel Transport, Geodesics Revisited
31.3 Joachimsthal Theorem and Notation
31.4 Tissot’s Theorem

31.5 Summary
31.6 Keywords
31.7 Self Assessment

31.8 Review Questions
31.9 Further Readings

Objectives

After studying this unit, you will be able to:
Define Geodesic Lines, Local Gauss-Bonnet Theorem

Discuss Covariant Derivative, Parallel Transport, Geodesics Revisited

Describe Joachimsthal's Notations

Introduction

In this unit you will go through, Bonnet’s theorem about the existence of a surface patch with
prescribed first and second fundamental form. This will require a discussion of the Theorema
Egregium and of the Codazzi-Mainardi compatibility equations. We will take a Joachimsthal's
Notations

31.1 Geodesic Lines, Local Gauss-Bonnet Theorem

Geodesics play a very important role in surface theory and in dynamics. One of the main reasons
why geodesics are so important is that they generalize to curved surfaces the notion of “shortest
path” between two points in the plane.
More precisely, given a surface X, given any two points p = X(u , v ) and q = X(u , v ) on X, let us
1
1
0
0
look at all the regular curves C on X defined on some open interval I such that p = C(t ) and
0
q = C(t ) for some t , t  I.
1
1
0
It can be shown that in order for such a curve C to minimize the length l (pq) of the curve
C
segment from p to q, we must have k (t) = 0 along [t , t ], where k (t) is the geodesic curvature at
g
g
0
1
X(u(t), v(t)).
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