An introduction to Riemannian geometry

94 Pages·2006·0.433 MB·English

by  Gudmundsson S.

Lecture Notes
An Introduction
to
Riemannian Geometry
(version 1.240 - 24 January 2006)
Sigmundur Gudmundsson
(Lund University)
The latest version of this document can be obtained at:
http://www.matematik.lu.se/matematiklu/personal/sigma/index.html

1
Preface
These lecture notes grew out of an M.Sc. course on di(cid:11)erential
geometry which I gave at the University of Leeds 1992. Their main
purpose is to introduce the beautiful theory of Riemannian Geometry
a still very active area of mathematical research. This is a subject with
no lack of interesting examples. They are indeed the key to a good
understanding of it and will therefore play a major role throughout
this work. Of special interest are the classical Lie groups allowing
concrete calculations of many of the abstract notions on the menu.
The study of Riemannian Geometry is rather meaningless without
some basic knowledge on Gaussian Geometry that is the di(cid:11)erential
geometry of curves and surfaces in 3-dimensional space. For this we
recommend the excellent textbook: M. P. do Carmo, Di(cid:11)erential ge-
ometry of curves and surfaces, Prentice Hall (1976).
These lecture notes are written for students with a good under-
standing of linear algebra, real analysis of several variables, the clas-
sical theory of ordinary di(cid:11)erential equations and some topology. The
most important results stated in the text are also proved there. Other
smaller ones are left to the reader as exercises, which follow at the end
of each chapter. This format is aimed at students willing to put hard
work into the course.
For further reading we recommend the very interesting textbook: M.
P. do Carmo, Riemannian Geometry, Birkha(cid:127)user (1992).
I am very grateful tomy many students who throughout the years have
contributed to the text by (cid:12)nding numerous typing errors and giving
many useful comments on the presentation.
It is my intention to extend this very incomplete draft version and
include some of the di(cid:11)erential geometry of the Riemannian symmetric
spaces.
Lund University, 15 January 2004
Sigmundur Gudmundsson

Contents
Chapter 1. Introduction 5
Chapter 2. Di(cid:11)erentiable Manifolds 7
Chapter 3. The Tangent Space 19
Chapter 4. The Tangent Bundle 33
Chapter 5. Riemannian Manifolds 43
Chapter 6. The Levi-Civita Connection 55
Chapter 7. Geodesics 63
Chapter 8. The Curvature Tensor 75
Chapter 9. Curvature and Local Geometry 83
3

CHAPTER 1
Introduction
On the 10th of June 1854 Riemann gave his famous "Habilita-
tionsvortrag" in the Colloquium of the Philosophical Faculty at Go(cid:127)tt-
ingen. His talk with the title "U(cid:127)ber die Hypothesen, welche der Ge-
ometrie zu Grunde liegen" is often said to be the most important in
the history of di(cid:11)erential geometry. Gauss, at the age of 76, was in the
audience andis said tohave been very impressed by his former student.
Riemann’s revolutionary ideas generalized the geometry of surfaces
which had been studied earlier by Gauss, Bolyai and Lobachevsky.
Later this lead to an exact de(cid:12)nition of the modern concept of an
abstract Riemannian manifold.
5

CHAPTER 2
Di(cid:11)erentiable Manifolds
The main purpose of this chapter is to introduce the concepts of a
di(cid:11)erentiablemanifold,asubmanifoldandadi(cid:11)erentiablemapbetween
manifolds. By this we generalize notions from the classical theory of
curves and surfaces studied in most introductory courses on di(cid:11)erential
geometry.
For a natural number m let Rm be the m-dimensional real vector
space equipped with the topology induced by the standard Euclidean
metric d on Rm given by
d(x;y) = (x y )2 +:::+(x y )2:
1 1 m m
(cid:0) (cid:0)
For positive natural nupmbers n;r and an open subset U of Rm we
shall by Cr(U;Rn) denote the r-times continuously di(cid:11)erentiable maps
U Rn. By smooth maps U Rn we mean the elements of
! !
1
C1(U;Rn) = Cr(U;Rn):
r=1
\
The set of real analytic maps U Rn will be denoted by C!(U;Rn).
!
For the theory of real analytic maps we recommend the book: S.
G. Krantz and H. R. Parks, A Primer of Real Analytic Functions,
Birkha(cid:127)user (1992).
De(cid:12)nition 2.1. Let (M; ) be a topological Hausdor(cid:11) space with
T
a countable basis. Then M is said to be a topological manifold if
there exists a natural number m and for each point p M an open
2
neighbourhood U of p and a continuous map x : U Rm which is a
!
homeomorphism onto its image x(U) which is an open subset of Rm.
The pair (U;x) is called a chart (or local coordinates) on M. The
natural number m is called the dimension of M. To denote that the
dimension of M is m we write Mm.
Following De(cid:12)nition 2.1 a topological manifold M is locally home-
omorphic to the standard Rm for some natural number m. We shall
now use the charts on M to de(cid:12)ne a di(cid:11)erentiable structure and make
M into a di(cid:11)erentiable manifold.
7
8 2. DIFFERENTIABLE MANIFOLDS
De(cid:12)nition 2.2. LetM beatopologicalmanifold. Then aCr-atlas
for M is a collection of charts
= (U ;x ) (cid:11) I
(cid:11) (cid:11)
A f j 2 g
such that covers the whole of M i.e.
A
M = U
(cid:11)
(cid:11)
[
and for all (cid:11);(cid:12) I the corresponding transition map
2
x(cid:12) (cid:14)x(cid:0)(cid:11)1jx(cid:11)(U(cid:11)\U(cid:12)) : x(cid:11)(U(cid:11) \U(cid:12)) ! Rm
is r-times continuously di(cid:11)erentiable.
A chart (U;x) on M is said to be compatible with a Cr-atlas on
A
M if (U;x) is a Cr-atlas. A Cr-atlas ^ is said to be maximal
A[f g A
if it contains all the charts that are compatible with it. A maximal
atlas ^ on M is also called a Cr-structure on M. The pair (M; ^)
A A
is said to be a Cr-manifold, or a di(cid:11)erentiable manifold of class
Cr, if M is a topological manifold and ^ is a Cr-structure on M. A
A
di(cid:11)erentiable manifold is said to be smooth if its transition maps are
C and real analytic if they are C!.
1
It should be noted that a given Cr-atlas on M determines a
A
unique Cr-structure ^ on M containing . It simply consists of all
A A
charts compatible with . For the standard topological space (Rm; )
A T
we have the trivial C!-atlas
= (Rm;x) x : p p
A f j 7! g
inducing the standard C!-structure ^ on Rm.
A
Example 2.3. Let Sm denote the unit sphere in Rm+1 i.e.
Sm = p Rm+1 p2 + +p2 = 1
f 2 j 1 (cid:1)(cid:1)(cid:1) m+1 g
equippedwiththesubsettopologyinducedbythestandard onRm+1.
T
Let N be the north pole N = (1;0) R Rm and S be the south pole
2 (cid:2)
S = ( 1;0) on Sm, respectively. Put U = Sm N , U = Sm S
N S
(cid:0) (cid:0)f g (cid:0)f g
and de(cid:12)ne xN : UN Rm, xS : US Rm by
! !
1
x : (p ;:::;p ) (p ;:::;p );
N 1 m+1 2 m+1
7! 1 p
1
(cid:0)
1
x : (p ;:::;p ) (p ;:::;p ):
S 1 m+1 2 m+1
7! 1+p
1
Then the transition maps
xS (cid:14)x(cid:0)N1;xN (cid:14)x(cid:0)S1 : Rm (cid:0)f0g ! Rm (cid:0)f0g