Page 341 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Sachin Kaushal, Lovely Professional University

Notes Unit 27: Principal Curvatures

CONTENTS
Objectives

Introduction
27.1 Principal Curvatures
27.2 Two Dimensions: Curvature of Surfaces
27.2.1 Gaussian Curvature
27.2.2 Mean Curvature

27.2.3 Second Fundamental Form
27.3 Higher Dimensions: Curvature of Space
27.4 Generalizations

27.5 Summary
27.6 Keywords
27.7 Self Assessment
27.8 Review Question
27.9 Further Reading

Objectives

After studying this unit, you will be able to:
Define Principal curvatures

Discuss the concept of two dimensions: Curvature of surfaces

Explain the Mean curvature

Explain the Higher dimensions: Curvature of space

Introduction

In last unit, you have studied about the meaning and concept of curvature. Gaussian curvature,
sometimes also called total curvature, is an intrinsic property of a space independent of the
coordinate system used to describe it. The Gaussian curvature of a regular surface in  at a point
3
p is formally defined as (S(p)) where S is the shape operator and det denotes the determinant
K(p) = det. This unit will provides you information related to principal curvatures.

27.1 Principal Curvatures

All curves with the same tangent vector will have the same normal curvature, which is the same
as the curvature of the curve obtained by intersecting the surface with the plane containing T
and u.

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