Geometría Riemann

Lecture Notes in Mathematics

An Introduction to Riemannian Geometry
Sigmundur Gudmundsson
(Lund University)

(version 1.271 - 12 August 2010)

The latest version of this document can be found at http://www.matematik.lu.se/matematiklu/personal/sigma/

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Preface
These lecture notes grew out of an M.Sc. 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 abstractnotions on the menu. The study of Riemannian geometry is rather meaningless without some basic knowledge on Gaussian geometry that i.e. the geometry of curves and surfaces in 3-dimensional space. For this I recommend the excellent textbook: M. P. do Carmo, Differential geometry of curves and surfaces, Prentice Hall (1976). These lecture notes are written for students with a good understanding oflinear 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 proved there. Other 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 I recommend the veryinteresting textbook: M. P. do Carmo, Riemannian Geometry, Birkh¨user (1992). a I am very grateful to my many enthusiastic students who throughout the years have contributed to the text by finding numerous typing errors and giving many useful comments on the presentation. Norra N¨bbel¨v, 17 February 2008 o o Sigmundur Gudmundsson

Contents
Chapter 1. Introduction Chapter 2. Differentiable ManifoldsChapter 3. The Tangent Space Chapter 4. The Tangent Bundle Chapter 5. Riemannian Manifolds Chapter 6. The Levi-Civita Connection Chapter 7. Geodesics Chapter 8. The Riemann Curvature Tensor Chapter 9. Curvature and Local Geometry 5 7 21 35 47 59 69 83 95

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CHAPTER 1

Introduction
On the 10th of June 1854 Georg Friedrich Bernhard Riemann (18261866) gave his famous ”Habilitationsvortrag” inthe Colloquium of the ¨ Philosophical Faculty at G¨ttingen. His talk ”Uber die Hypothesen, o 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), at the age of 76, was in the audience and is said to have been very impressed by his former student. Riemann’s revolutionary ideas generalized thegeometry of surfaces which had been studied earlier by Gauss, Bolyai and Lobachevsky. Later this lead to an exact definition of the modern concept of an abstract Riemannian manifold.

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CHAPTER 2

Differentiable Manifolds
In this chapter we introduce the important notion of a differentiable manifold. This generalizes curves and surfaces in R3 studied in classical differential geometry. Ourmanifolds are modelled on the standard differentiable structure on the vector spaces Rm via compatible local charts. We give many examples, study their submanifolds and differentiable maps between manifolds. 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) = For positive natural numbers n,r and an open subset U of Rm we shall 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
∞

(x1 − y1 )2 + . . . + (xm − ym )2 .

C (U, R ) =
r=1

∞

n

C r (U, Rn ).

The set of real analytic maps from U to Rn will be denoted by C ω (U, Rn ). For the theory of real analytic maps we recommend the book: S....