Page 301 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Sachin Kaushal, Lovely Professional University

Notes Unit 24: Two Fundamental Form

CONTENTS
Objectives

Introduction
24.1 Surfaces
24.2 The First Fundamental Form
24.3 The Second Fundamental Form
24.4 Examples

24.5 Summary
24.6 Keywords
24.7 Self Assessment

24.8 Review Questions
24.9 Further Readings

Objectives

After studying this unit, you will be able to:
Define surfaces

Explain the first fundamental form

Describe the second fundamental form

Discuss some example related to fundamental forms

Introduction

In last unit, you have studied about development surfaces. In mathematics, specifically in topology,
a surface is a two-dimensional topological manifold. The most familiar examples are those that
arise as the boundaries of solid objects in ordinary three-dimensional Euclidean space R - for
3
example, the surface of a ball. There are surfaces, such as the Klein bottle, that cannot be embedded
in three-dimensional Euclidean space without introducing singularities or self-intersections.
24.1 Surfaces

Definition 1. A parametric surface patch is a smooth mapping:

X :U   3 ,
where U   is open, and the Jacobian dX is non-singular.
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