An Introduction to Riemannian Geometry - Matematikcentrum

1 Lecture Notes in Mathematics An Introduction to Riemannian Geometry Sigmundur Gudmundsson (Lund University). (version - 27 September 2018). The latest version of this document can be found at 1. Preface These lecture notes grew out of an course on differential 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.

2 The study of Riemannian Geometry is rather meaningless without some basic knowledge on Gaussian Geometry the Geometry of curves and surfaces in 3-dimensional Euclidean space. For this we recommend the following text: M. P. do Carmo, Differential Geometry 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 classical theory of ordinary differential equations and some topology. The most important results stated in the text are also proven there. Others are left to the reader as exercises, which follow at the end of each chapter.

3 This format is aimed at students willing to put hard work into the course. For further reading we recommend the excellent standard text: M. P. do Carmo, Riemannian Geometry , Birkh auser (1992). I am very grateful to my enthusiastic students and many other readers who have, throughout the years, contributed to the text by giving numerous valuable comments on the presentation. Norra N obbel ov the 2nd of February 2018. Sigmundur Gudmundsson Contents Chapter 1. Introduction 5. Chapter 2. Differentiable Manifolds 7. Chapter 3. The Tangent Space 23. Chapter 4. The Tangent Bundle 39. Chapter 5. Riemannian Manifolds 57. Chapter 6. The Levi-Civita Connection 73.

4 Chapter 7. Geodesics 85. Chapter 8. The Riemann Curvature Tensor 103. Chapter 9. Curvature and Local Geometry 117. 3. CHAPTER 1. Introduction On the 10th of June 1854 Georg Friedrich Bernhard Riemann (1826- 1866) gave his famous Habilitationsvortrag in the Colloquium of the Philosophical Faculty at G ottingen. His talk Uber die Hypothesen, welche der Geometrie zu Grunde liegen is often said to be the most important in the history of differential Geometry . Johann Carl Friedrich Gauss (1777-1855) was in the audience, at the age of 77, and is said to have been very impressed by his former student. Riemann's revolutionary ideas generalised the Geometry of surfaces which had earlier been initiated by Gauss.

5 Later this lead to an exact definition of the modern concept of an abstract Riemannian manifold. The development of the 20th century has turned Riemannian ge- ometry into one of the most important parts of modern mathematics. For an excellent survey on this vast field we recommend the following work written by one of the main actors: Marcel Berger, A Panoramic View of Riemannian Geometry , Springer (2003). 5. CHAPTER 2. Differentiable Manifolds In this chapter we introduce the important notion of a differentiable manifold. This generalises curves and surfaces in R3 studied in classi- cal differential Geometry . Our manifolds are modelled on the classical differentiable structure on the vector spaces Rm via compatible local charts.

6 We give many examples of differentiable manifolds, study their submanifolds and differentiable maps between them. Let Rm be the standard m-dimensional real vector space equipped with the topology induced by the Euclidean metric d on Rm given by p d(x, y) = (x1 y1 )2 + .. + (xm ym )2 . For a natural number r and an open subset U of Rm we will by C r (U, Rn ) denote the r-times continuously differentiable maps from U to Rn . By smooth maps U Rn we mean the elements of . \. n C (U, R ) = C r (U, Rn ). r=0. The set of real analytic maps from U to Rn will be denoted by C (U, Rn ). For the theory of real analytic maps we recommend the important text: S.

7 G. Krantz and H. R. Parks, A Primer of Real An- alytic Functions, Birkh auser (1992). Definition Let (M, T ) be a topological Hausdorff space with a countable basis. Then M is called a topological manifold if there exists an m Z+ such that for each point p M we have an open neighbourhood U of p, an open subset V of Rm and a homeomorphism x : U V . The pair (U, x) is called a local chart (or local coordi- nates) on M . The integer m is called the dimension of M . To denote that the dimension of M is m we write M m . According to Definition an m-dimensional topological manifold M m is locally homeomorphic to the standard Rm . We will now intro- duce a differentiable structure on M via its local charts and turn it into a differentiable manifold.

8 7. 8 2. DIFFERENTIABLE MANIFOLDS. Definition Let M be an m-dimensional topological manifold. Then a C r -atlas on M is a collection A = {(U , x )| I}. of local charts on M such that A covers the whole of M [. M= U .. and for all , I the corresponding transition maps x x 1 m |x (U U ) : x (U U ) R R. m are r-times continuously differentiable of class C r . A local chart (U, x) on M is said to be compatible with a C r -atlas A if the union A {(U, x)} is a C r -atlas. A C r -atlas A is said to be maximal if it contains all the local charts that are compatible with it. A maximal atlas A on M is also called a C r -structure on M . The pair (M, A) is said to be a C r -manifold, or a differentiable manifold of class C r , if M is a topological manifold and A is a C r -structure on M.]

9 A differentiable manifold is said to be smooth if its transition maps are C and real analytic if they are C . Remark It should be noted that a given C r -atlas A on a topological manifold M determines a unique C r -structure A on M. containing A. It simply consists of all local charts compatible with A. Example For the standard topological space (Rm , Tm ) we have the trivial C -atlas A = {(Rm , x)| x : p 7 p}. inducing the standard C -structure A on Rm . Example Let S m denote the unit sphere in Rm+1 S m = {p Rm+1 | p21 + + p2m+1 = 1}. equipped with the subset topology T induced by the standard Tm+1. on Rm+1 . Let N be the north pole N = (1, 0) R Rm and S be the south pole S = ( 1, 0) on S m , respectively.

10 Put UN = S m \ {N }, US = S m \ {S} and define the homeomorphisms xN : UN Rm and xS : US Rm by 1. xN : (p1 , .. , pm+1 ) 7 (p2 , .. , pm+1 ), 1 p1. 1. xS : (p1 , .. , pm+1 ) 7 (p2 , .. , pm+1 ). 1 + p1. 2. DIFFERENTIABLE MANIFOLDS 9. Then the transition maps xS x 1 1 m m N , xN xS : R \ {0} R \ {0}. are both given by x x 7 , |x|2. so A = {(UN , xN ), (US , xS )} is a C -atlas on S m . The C -manifold is called the m-dimensional standard sphere. (S m , A). Another interesting example of a differentiable manifold is the m- dimensional real projective space RP m . Example On the set Rm+1 \ {0} we define the equivalence relation by p q if and only if there exists a R such that p = q.