Page 251 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 21: New Spherical Indicatrices and
their Characterizations

CONTENTS

Objectives
Introduction
21.1 Preliminaries
21.2 Tangent Bishop Spherical Images of a Regular Curve
21.3 M Bishop Spherical Images of a Regular Curve
1
21.4 M Bishop Spherical Images of a Regular Curve
2
21.5 Summary
21.6 Keywords

21.7 Self Assessment
21.8 Review Questions
21.9 Further Readings

Objectives

After studying this unit, you will be able to:
Define Preliminaries

Explain Tangent Bishop Spherical Images

Identify M and M Bishop
 1 2
Introduction

In the existing literature, it can be seen that, most of classical differential geometry topics have
been extended to Lorentzian manifolds. In this process, generally, researchers used standard
moving Frenet-Serret frame. Using transformation matrix among derivative vectors and frame
vectors, some of kinematical models were adapted to this special moving frame. Researchers
aimed to have an alternative frame for curves and other applications. Bishop frame, which is
also called alternative or parallel frame of the curves, was introduced by L.R. Bishop in 1975 by
means of parallel vector fields.
Spherical images of a regular curve in the Euclidean space are obtained by means of Frenet-
Serret frame vector fields, so this classical topic is a well-known concept in differential geometry
of the curves. In the light of the existing literature, this unit aims to determine new spherical
images of regular curves using Bishop frame vector fields. We shall call such curves, respectively,
Tangent, M and M Bishop spherical images of regular curves. Considering classical methods,
1
2
we investigated relations among Frenet-Serret invariants of spherical images in terms of Bishop
invariants. Additionally, two examples of Bishop spherical indicatrices are presented.

244 LOVELY PROFESSIONAL UNIVERSITY