Page 347 - DMTH402_COMPLEX_ANALYSIS_AND_DIFFERENTIAL_GEOMETRY

Complex Analysis and Differential Geometry Sachin Kaushal, Lovely Professional University

Notes Unit 28: Local Intrinsic Geometry of Surfaces

CONTENTS
Objectives

Introduction
28.1 Riemannian Surfaces
28.2 Lie Derivative
28.3 Covariant Differentiation
28.4 Summary

28.5 Keywords
28.6 Self Assessment
28.7 Review Questions

28.8 Further Readings

Objectives

After studying this unit, you will be able to:
Define Riemannian Surfaces

Discuss Lie Derivative

Explain the Concept of Covariant Differentiation

Introduction

In this unit, we change our point of view, and study intrinsic geometry, in which the starting
point is the first fundamental form. Thus, given a parametric surface, we will ignore all
information which cannot be recovered from the first fundamental form and its derivatives
only.

28.1 Riemannian Surfaces

Definition 1. Let U   be open. A Riemannian metric on U is a smooth function g : U   2 2 .
2


A Riemannian surface patch is an open set U equipped with a Riemannian metric.
The tangent space of U at u  U is  . The Riemannian metric g defines an inner-product on each
2
tangent space by:
g(Y,Z) = g y z,
i j
ij
where y and z are the components of Y and Z with respect to the standard basis of  . We will
2
i
j
2
write Y  g(Y,Y), and omit the subscript g when it is not ambiguous.
g

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