An Introduction to Riemannian Geometry Matematikcentrum

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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 abstract notions. on the menu, The study of Riemannian geometry is rather meaningless without. some basic knowledge on Gaussian geometry i e 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. 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 Birkha user 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 No bbelo v the 2nd of February 2018,Sigmundur Gudmundsson.
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. Chapter 7 Geodesics 85,Chapter 8 The Riemann Curvature Tensor 103. Chapter 9 Curvature and Local Geometry 117,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 Go ttingen His talk U ber 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 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,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 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. 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. C U R C r U Rn, 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 G Krantz and H R Parks A Primer of Real An. alytic Functions Birkha user 1992, Definition 2 1 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 2 1 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 2 DIFFERENTIABLE MANIFOLDS. Definition 2 2 Let M be an m dimensional topological manifold. Then a C r atlas on M is a collection, of local charts on M such that A covers the whole of M i e. and for all I the corresponding transition maps,x U U x U U R R.
are r times continuously differentiable i e 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. A differentiable manifold is said to be smooth if its transition maps. are C and real analytic if they are C, Remark 2 3 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 2 4 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 2 5 Let S m denote the unit sphere in Rm 1 i e. 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 Put UN S m N. US S m S and define the homeomorphisms xN UN Rm and. xS US Rm by,xN p1 pm 1 7 p2 pm 1,xS p1 pm 1 7 p2 pm 1. 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, so A UN xN US xS is a C atlas on S m The C manifold. S m A is called the m dimensional standard sphere, Another interesting example of a differentiable manifold is the m. dimensional real projective space RP m, Example 2 6 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. Let RP m be the quotient space Rm 1 0 and,Rm 1 0 RP m.
be the natural projection mapping a point p Rm 1 0 onto the. equivalence class p RP m i e the line,p p Rm 1 R, through the origin generated by p Then equip the set RP m with the. quotient topology T induced by and Tm 1 on Rm 1 This means that. a subset U of RP m is open if and only if its pre image 1 U is open. in Rm 1 For k 1 m 1 we define the open subset Uk of RP m. Uk p RP m pk 6 0,and the local chart xk Uk RP m Rm by. p1 pk 1 pk 1 pm 1,pk pk pk pk, If p q then p q for some R so pl pk ql qk for all l This. shows that the maps xk Uk RP m Rm are all well defined A line. p RP m is represented by a non zero point p Rm 1 so at least one. of its components is non zero This shows that,The corresponding transition maps. l xl Ul Uk xl Ul Uk R R,10 2 DIFFERENTIABLE MANIFOLDS.
are given by,p1 pl 1 pl 1 pm 1 p1 pk 1 pk 1 pm 1,pl pl pl pl pk pk pk pk. so the collection,A Uk xk k 1 m 1, is a C atlas on RP m The real analytic manifold RP m A is called. the m dimensional real projective space, Remark 2 7 The above definition of the real projective space. RP m might seem very abstract But later on we will embed RP m. into the real vector space Sym Rm 1 of symmetric m 1 m 1. matrices For this see Example 3 26, Example 2 8 Let C be the extended complex plane given by. and put C C 0 U0 C and U C 0 Then define the, local charts x0 U0 C and x U C on C by x0 z 7 z and.
x w 7 1 w respectively Then the corresponding transition maps. 0 x0 x C C, are both given by z 7 1 z so A U0 x0 U x is a C atlas on. C The real analytic manifold C A is called the Riemann sphere. For the product of two differentiable manifolds we have the following. important result, Proposition 2 9 Let M1 A 1 and M2 A 2 be two differentiable. manifolds of class C r Let M M1 M2 be the product space with the. product topology Then there exists an atlas A on M turning M A. into a differentiable manifold of class C r and the dimension of M sat. dim M dim M1 dim M2,Proof See Exercise 2 1, The concept of a submanifold of a given differentiable manifold. will play an important role as we go along and we will be especially. interested in the connection between the geometry of a submanifold. and that of its ambient space, Definition 2 10 Let m n be positive integers with m n and. N n A N be a C r manifold A subset M of N is said to be a sub. manifold of N if for each point p M there exists a local chart. 2 DIFFERENTIABLE MANIFOLDS 11,Up xp A N such that p Up and xp Up N Rm Rn m.
xp Up M xp Up Rm 0, The natural number n m is called the codimension of M in N. Proposition 2 11 Let m n be positive integers with m n and. N A N be a C r manifold Let M be a submanifold of N equipped with. the subset topology and Rm Rn m Rm be the natural projection. onto the first factor Then,AM Up M xp Up M p M, is a C r atlas for M Hence the pair M A M is an m dimensional C r. manifold The differentiable structure A M on M is called the induced. structure by A N,Proof See Exercise 2 2, Remark 2 12 Our next aim is to prove Theorem 2 16 which is a. useful tool for the construction of submanifolds of Rm For this we use. the classical inverse mapping theorem stated below Note that if. is a differentiable map defined on an open subset U of Rm then its. differential dFp Rm Rn at the point p U is a linear map given by. the n m matrix,F1 x1 p F1 xm p,Fn x1 p Fn xm p,If R U is a curve in U such that 0 p and 0 v Rm. then the composition F R Rn is a curve in Rn and according. to the chain rule we have,dFp v F s s 0, This is the tangent vector of the curve F at F p Rn.
The above shows that the differential dFp can be seen as. a linear map that maps tangent vectors at p U to tangent. vectors at the image point F p Rn This will later be gen. eralised to the manifold setting, We now state the classical inverse mapping theorem well known. from multivariable analysis,12 2 DIFFERENTIABLE MANIFOLDS. Fact 2 13 Let U be an open subset of Rm and F U Rm be a. C map If p U and the differential, of F at p is invertible then there exist open neighbourhoods Up around p. and Uq around q F p such that F F Up Up Uq is bijective and. the inverse F 1 Uq Up is a C r map The differential dF 1 q of. F 1 at q satisfies,dF 1 q dFp 1, i e it is the inverse of the differential dFp of F at p. Before stating the classical implicit mapping theorem we remind. the reader of the following well known notions, Definition 2 14 Let m n be positive natural numbers U be an.
open subset of Rm and F U Rn be a C r map A point p U is. said to be regular for F if the differential, is of full rank but critical otherwise A point q F U is said to be. a regular value of F if every point in the pre image F 1 q of q is. Remark 2 15 Note that if m n are positive integers with m n. then p U is a regular point for,F F1 Fn U Rn, if and only if the gradients gradF1 gradFn of the coordinate func. tions F1 Fn U R are linearly independent at p or equivalently. the differential dFp of F at p satisfies the following condition. det dFp dFp t 6 0, The next result is a very useful tool for constructing submanifolds. of the classical vector space Rm, Theorem 2 16 The Implicit Mapping Theorem Let m n be pos. itive integers with m n and F U Rn be a C r map from an open. subset U of Rm If q F U is a regular value of F then the pre image. F 1 q of q is an m n dimensional submanifold of Rm of class. Proof Let p be an element of F 1 q and Kp be the kernel of. the differential dFp i e the m n dimensional subspace of Rm given. by Kp v Rm dFp v 0 Let p Rm Rm n be a linear map,2 DIFFERENTIABLE MANIFOLDS 13.
such that p Kp Kp Rm n is bijective p Kp 0 and define the. map Gp U Rn Rm n by,Gp x 7 F x p x, Then the differential dGp p Rm Rm of Gp with respect to the. decompositions Rm Kp Kp and Rm Rn Rm n is given by. hence bijective It now follows from the inverse function theorem that. there exist open neighbourhoods Vp around p and Wp around Gp p. such that G p Gp Vp Vp Wp is bijective the inverse G 1. is C r d G 1 1 1, p Gp p dGp p and d G p y is bijective for all y Wp. Now put U p F 1 q Vp then, so if Rn Rm n Rm n is the natural projection onto the second. factor then the map,x p Gp U p U p q Rm n Wp Rm n, is a local chart on the open neighbourhood U p of p The point q F U. is a regular value so the set,A U p x p p F 1 q,is a C r atlas for F 1 q.
Applying the implicit function theorem we obtain the following. interesting examples of the m dimensional sphere S m and its tangent. bundle T S m as differentiable submanifolds of Rm 1 and R2m 2 respec. An Introduction to Riemannian Geometry The Riemann Curvature Tensor 101 Chapter 9 Curvature and Local Geometry 115 3 CHAPTER 1

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