Page 323 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Richa Nandra, Lovely Professional University

Notes Unit 26: Lines of Curvature

CONTENTS
Objectives

Introduction
26.1 Lines of Curvature
26.2 Examples
26.3 Surface Area
26.4 Bernstein’s Theorem

26.5 Theorema Egregium
26.6 Summary
26.7 Keywords

26.8 Self Assessment
26.9 Review Questions
26.10 Further Readings

Objectives

After studying this unit, you will be able to:

Define lines of curvature

Explain the examples of lines of curvature

Describe the surface area and Bernstein's theorem

Introduction

In last unit, you have studied about curvature. In general, there are two important types of
curvature: extrinsic curvature and intrinsic curvature. The extrinsic curvature of curves in two-
and three-space was the first type of curvature to be studied historically, culminating in the
Frenet formulas, which describe a space curve entirely in terms of its "curvature," torsion, and
the initial starting point and direction. This unit will explains the concept of lines of curvature.
26.1 Lines of Curvature

Definition 1. A curve  on a parametric surface X is called a line of curvature if   is a principal
direction.
The following proposition, due to Rodriguez, characterizes lines of curvature as those curves
whose tangents are parallel to the tangent of their spherical image under the Gauss map.

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