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Introduction to Manifolds

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Introduction to Manifolds Introduction to Manifolds Roland van der Veen 2018 2 Contents 1 Introduction 5 1.1 Overview . 5 2 How to solve equations? 7 2.1 Linear algebra . 9 2.2 Derivative . 10 2.3 Intermediate and mean value theorems . 13 2.4 Implicit function theorem . 15 3 Is there a fundamental theorem of calculus in higher dimensions? 23 3.1 Elementary Riemann integration . 24 3.2 k-vectors and k-covectors . 26 3.3 (co)-vector fields and integration . 33 3.3.1 Integration . 34 3.4 More on cubes and their boundary . 37 3.5 Exterior derivative . 38 3.6 The fundamental theorem of calculus (Stokes Theorem) . 40 3.7 Fundamental theorem of calculus: Poincar´elemma . 42 4 Geometry through the dot product 45 4.1 Vector spaces with a scalar product . 45 4.2 Riemannian geometry . 47 5 What if there is no good choice of coordinates? 51 5.1 Atlasses and manifolds . 51 5.2 Examples of manifolds . 55 5.3 Analytic continuation . 56 5.4 Bump functions and partitions of unity . 58 5.5 Vector bundles . 59 5.6 The fundamental theorem of calculus on manifolds . 62 5.7 Geometry on manifolds . 63 3 4 CONTENTS Chapter 1 Introduction 1.1 Overview The goal of these notes is to explore the notions of differentiation and integration in a setting where there are no preferred coordinates. Manifolds provide such a setting. This is not calculus. We made an attempt to prove everything we say so that no black boxes have to be accepted on faith. This self-sufficiency is one of the great strengths of mathematics. Sometimes mathematics texts start by giving answers neglecting to properly state the questions they were meant to answer. We will try motivate concepts and illustrate definitions. In turn the reader is asked to at least try some of the exercises. Doing exercises (and possibly failing some) is an integral part of mathematics. At the beginning of the Dutch national masters program in mathematics there is a one day 'intensive reminder on manifolds' course that consists the following topics: 1. definition of manifolds*, 2. tangent and cotangent bundles*, 3. vector bundles*, 4. differential forms and exterior derivative*, 5. flows and Lie derivative, 6. Cartan calculus, 7. integration and Stokes theorem*. 8. Frobenius theorem Since this is an introductory course we only treat the topics marked *. To illustrate our techniques we will touch upon some concepts in Riemannian, complex and symplectic geometry. More systematically these lecture notes consist of five chapters the first of which is this intro- duction. In chapter two we start by studying non-linear systems of equations by approximating them by linear ones, leading to the implicit function theorem. Basically it says that the solution 5 6 CHAPTER 1. INTRODUCTION set looks like the graph of a function in good cases. Along the way we develop a suitably general notion of derivative. The corresponding notion of integration is developed in the next chapter. This is more involved as it requires defining new kinds of objects as the natural integrands (covector fields, differential forms). Differntiation and integration are connected by a generalization of the fundamental theorem of calculus (Stokes theorem) and the Poincare lemma. In chapter four we briefly explore how our techniques are useful in setting up various kinds of geometry. In the final chapter we will show how to lift all the theory we developed so far to the context of manifolds. Basically a manifold is just several pieces of Rn linked by coordinate transformations. Given that we set up our theory in a way that makes coordinate tranformations easy to deal with, most local aspects of the theory are no different from the way they are in Rn. Most of the material is standard and can be found in references such as Calculus on manifolds by M. Spivak or Introduction to smooth manifolds. However the proofs presented here are simplified and streamlined significantly. This especially goes for the proof of the implicit function theorem and the change of variables theorem for integrals and the Poincar´elemma. I tried to motivate the use of exterior calculus more than usual, while limiting its algebraic preliminaries. f Throughout the text I try to write functions as A 3 a 7−! a2 + a + 1 2 B, instead of f : A ! B defined by f(a) = a2 + a + 1. Acknowledgement. Much of this material is presented in a way inspired by the work of my former master student Jules Jacobs. I would also like to thank Kevin van Helden for his helpful comments, exercises and excellent teaching assistance over the years. Chapter 2 How to solve equations? Postponing the formal definition until chapter 5, manifolds often arise as solution sets to equations. In this preliminary chapter we explore under what conditions a system of n real equations in k + n variables can be solved. Naively one may hope that each equation can be used to determine a variable so that in the end k variables are left undetermined and all others are functions of those. For example consider the two systems of two equations on the left and on the right (k = 1; n = 2): x + y + z = 0 sin(x + y) − log(1 − z) = 0 (2.1) 1 −x + y + z = 0 ey − = 0 (2.2) 1 + x − z The system on the left is linear and easy to solve, we get x = 0 and y = −z. The system on the Figure 2.1: Solutions to the two systems. The yellow surface is the solution to the first equation, blue the second. The positive x; y; z axes are drawn in red, green, blue respectively. 7 8 CHAPTER 2. HOW TO SOLVE EQUATIONS? Figure 2.2: Some random level sets. right is hard to solve explicitly but looks very similar near (0; 0; 0) since sin(x + y) =∼ x + y and log(1 − z) =∼ −z near zero. We will be able to show that just like in the linear situation a curve of solutions passes through the origin. The key point is that the derivative of the complicated looking functions at the origin is precisely the linear function shown on the left. We will look at equations involving only differentiable functions. This means that locally they can be approximated well by linear functions. The goal of the chapter is to prove the implicit function theorem. Basically it says that the linear approximation decides whether or not a system of equations is solvable locally. This is illustrated in the figures above. Later in the course solutions to equations will be an important source of examples of manifolds. Even the solution set to single equation in three unknowns can take many forms. See for example figure 2.2 where we generated random polynomial equations and plotted the solution set. Exercises Exercise 1. (Three holes) Give a single equation in three unknowns such that the solution set is a bounded subset of R3, looks smooth and two-dimensional everywhere and has a hole. Harder: Can you increase the number of holes to three? 2.1. LINEAR ALGEBRA 9 2.1 Linear algebra The basis for our investigation of equations is the linear case. Linear equations can neatly be summarized in terms of a single matrix equation Av = b. Here v is a vector in Rk+n, and b 2 Rn and A is an n×(k +n) matrix. In case b = 0 we call the equation homogeneous and the solution set is some linear subspace S = fv 2 Rk+njAv = 0g, the kernel of the map defined by A. In general, given a single solution p 2 Rk+n such that Ap = b the entire solution set fv 2 Rk+njAv = bg is the affine linear subspace S + p = fs + pjs 2 Sg. In discussing the qualitative properties of linear equations it is more convenient to think in terms of linear maps. Most of this material should be familiar from linear algebra courses but we give a few pointers here to establish notation and emphasize the important points. With some irony, the first rule of linear algebra is YOU DO NOT PICK A BASIS, the second rule of linear algebra is YOU DO NOT PICK A BASIS. In this section W; V will always be real vector spaces of finite dimensions m and n. Of course W is isomorphic to Rm and choosing such an isomorphism b : Rm ! W means choosing a basis. m Usually we write ei for the standard basis of R and abbreviate b(ei) = bi. We may then write P i vectors w 2 W as w = i w bi. However we do not want to pin ourselves down on a specific basis since that makes it harder to switch between various interpretations of W as a space of directions, complex numbers or linear transformations, holomorphic functions and so on. A relevant example is the set of all linear maps from V to W is denoted L(V; W ), it is a vector space in its own right. If we would set V = Rn and W = Rm then ' 2 L(V; W ) could be described i P i by a matrix 'j defined by 'ej = i 'jei. However the matrix might look easier with respect to another basis so we prefer to keep the V; W abstract and describe ' using bases c : Rn ! V and m −1 i b : R ! W . With respect to these bases the matrix of ' 2 L(V; W ) is defined to be (b ◦ ' ◦ c)j. P i So 'cj = i 'jbi. An important special case of the previous is the dual space V ∗ = L(V; R).
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