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Differentiable Manifolds Differentiable Manifolds Lecture Notes for MATH 4033 (Spring 2018) FREDERICK TSZ-HO FONG Hong Kong University of Science and Technology (Version: January 23, 2018) Contents Preface ix Chapter 1. Regular Surfaces 1 1.1. Local Parametrizations 1 1.2. Level Surfaces 8 1.3. Transition Maps 11 1.4. Maps and Functions from Surfaces 14 1.5. Tangent Planes and Tangent Maps 18 Chapter 2. Abstract Manifolds 23 2.1. Smooth Manifolds 23 2.2. Functions and Maps on Manifolds 32 2.3. Tangent Spaces and Tangent Maps 37 2.4. Inverse Function Theorem 45 2.5. Immersions and Submersions 50 2.6. Submanifolds 56 Chapter 3. Tensors and Differential Forms 61 3.1. Cotangent Spaces 61 3.2. Tangent and Cotangent Bundles 64 3.3. Tensor Products 75 3.4. Wedge Products 82 3.5. Exterior Derivatives 89 Chapter 4. Generalized Stokes’ Theorem 109 4.1. Manifolds with Boundary 109 4.2. Orientability 115 4.3. Integrations of Differential Forms 121 4.4. Generalized Stokes’ Theorem 130 Chapter 5. De Rham Cohomology 139 vii viii Contents 5.1. De Rham Cohomology 140 5.2. Deformation Retracts 146 5.3. Mayer-Vietoris Theorem 151 Appendix A. Geometry of Curves 161 A.1. Curvature and Torsion 161 A.2. Fundamental Theorem of Space Curves 172 A.3. Plane Curves 177 Appendix B. Geometry of Surfaces 181 B.1. First Fundamental Form 181 B.2. Second Fundamental Form 187 B.3. Curvatures 197 B.4. Covariant Derivatives 202 B.5. Theorema Egregium 207 B.6. Geodesics and Minimal Surfaces (work in progress) 211 B.7. Gauss-Bonnet’s Theorem (work in progress) 211 Bibliography 213 Preface This lecture note is written for the course MATH 4033 (Calculus on Manifolds) taught by the author in the Hong Kong University of Science and Technology. The main goal of the course is to introduce advanced undergraduates and first-year graduates the basic language of differentiable manifolds and tensor calculus. The topics covered in the course is essential for further studies on Riemannian geometry, general relativity, string theory, and related fields. The prerequisite of the course is a solid conceptual background of linear algebra and multivariable calculus. The course MATH 4033 covers Chapters 1 to 5 in this lecture note. These chapters are about the analytic, algebraic, and topological aspects of differentiable manifolds. The appendix in this lecture note forms a crush course on differential geometry of curves and surfaces. They are not the essential parts of the course, but is strongly recommended for readers who want to acquire some workable knowledge in differential geometry (such as for the purpose of my UROP) The author would like to thank the following students for their diligent readings of the earlier version of this lecture notes and for pointing out many typographical errors: Chow Ka-Wing, Alex Chan Yan-Long, Aaron Chow Tsz-Kiu, Jimmy Choy Ka-Hei, Toby Cheung Hin-Wa, Poon Wai-Tung, Cheng Chun-Kit, Chu Shek-Kit, Wan Jingbo, and Nicholas Chin Cheng-Hoong. ix Chapter 1 Regular Surfaces “God made solids, but surfaces were the work of the devil.” Wolfgang Pauli A manifold is a space which locally resembles an Euclidean space. Before we learn about manifolds in the next chapter, we first introduce the notion of regular surfaces in R3 which motivates the definition of abstract manifolds and related concepts in the next chapter. 1.1. Local Parametrizations In Multivariable Calculus, we expressed a surface in R3 in two ways, namely using a parametrization F(u, v) or by a level set f (x, y, z) = 0. In this section, let us first focus on the former. In MATH 2023, we used a parametrization F(u, v) to describe a surface in R3 and to calculate various geometric and physical quantities such as surface areas, surface integrals and surface flux. To start the course, we first look into several technical and analytical aspects concerning F(u, v), such as their domains and images, their differentiability, etc. In the past, we can usually cover (or almost cover) a surface by a single parametrization F(u, v). Take the unit sphere as an example. We learned that it can be parametrized with the help of spherical coordinates: F(q, j) = (sin j cos q, sin j sin q, cos j) where 0 < q < 2p and 0 < j < p. This parametrization covers almost every part of the sphere (except the north and south poles, and a half great circle connecting them). In order to cover the whole sphere, we need more parametrizations, such as G(q, j) = (sin j cos q, sin j sin q, cos j) with domain −p < q < p and 0 < j < p. Since the image of either F or G does not cover the whole sphere (although almost), from now on we call them local parametrizations. 1 2 1. Regular Surfaces Definition 1.1 (Local Parametrizations of Class Ck). Consider a subset M ⊂ R3.A function F(u, v) : U!O from an open subset U ⊂ R2 onto an open subset O ⊂ M is called a Ck local parametrization (or a Ck local coordinate chart) of M (where k ≥ 1) if all of the following holds: (1) F : U! R3 is Ck when the codomain is regarded as R3. (2) F : U!O is a homeomorphism, meaning that F : U!O is bijective, and both F and F−1 are continuous. (3) For all (u, v) 2 U, the cross product: ¶F ¶F × 6= 0. ¶u ¶v The coordinates (u, v) are called the local coordinates of M. If F : U! M is of class Ck for any integer k, then F is said to be a C¥ (or smooth) local parametrization. Definition 1.2 (Surfaces of Class Ck). A subset M ⊂ R3 is called a Ck surface, where k 2 N [ f¥g, in R3 if at every point p 2 M, there exists an open subset U ⊂ R2, an open subset O ⊂ M containing p, and a Ck local parametrization F : U!O which satisfies all three conditions stated in Definition 1.1. We say M is a regular surface in R3 if it is a C¥ surface. Figure 1.1. smooth local parametrization To many students (myself included), the definition of regular surfaces looks obnox- ious at the first glance. One way to make sense of it is to look at some examples and understand why each of the three conditions is needed in the definition. The motivation behind condition (1) in the definition is that we are studying differential topology/geometry and so we want the parametrization to be differentiable as many times as we like. Condition (2) rules out surfaces that have self-intersection such as the Klein bottle (see Figure 1.2a). Finally, condition (3) guarantees the existence of a unique tangent plane at every point on M (see Figure 1.2b for a non-example). 1.1. Local Parametrizations 3 (a) Klein Bottle has a self-intersection. (b) F(u, v) = (u3, v3, uv) fails condition (3). Figure 1.2. Examples of non-smooth parametrizations Example 1.3 (Graph of a Function). Consider a smooth function f (u, v) : U! R 2 defined on an open subset U ⊂ R . The graph of f , denoted by G f , is the subset 3 f(u, v, f (u, v)) : (u, v) 2 Ug of R . One can parametrize G f by a global parametrization: F(u, v) = (u, v, f (u, v)). Condition (1) holds because f is given to be smooth. For condition (2), F is clearly one-to-one, and the image of F is the whole graph G f . Regarding it as a map F : U! G f , the inverse map F−1(x, y, z) = (x, y) is clearly continuous. Therefore, F : U! G f is a homeomorphism. To verify condition (3), we compute the cross product: ¶F ¶F ¶ f ¶ f × = − , − , 1 6= 0 ¶u ¶v ¶u ¶v for all (u, v) 2 U. Therefore, F is a smooth local parametrization of G f . Since the image of this single smooth local parametrization covers all of G f , we have proved that G f is a regular surface. Figure 1.3. The graph of any smooth function is a regular surface. 4 1. Regular Surfaces Exercise 1.1. Show that F(u, v) : (0, 2p) × (0, 1) ! R3 defined by: F(u, v) = (sin u, sin 2u, v) satisfies conditions (1) and (3) in Definition 1.1, but not condition (2). [Hint: Try to −1 show F is not continuous by finding a diverging sequence f(un, vn)g such that fF(un, vn)g converges. See Figure 1.4 for reference.] Figure 1.4. Plot of F(u, v) in Exercise 1.1 In Figure 1.3, one can observe that there are two families of curves on the surface. These curves are obtained by varying one of the (u, v)-variables while keeping the other constant. Precisely, they are the curves represented by F(u, v0) and F(u0, v) where ¶F ¶F u0 and v0 are fixed. As such, the partial derivatives ¶u (p) and ¶v (p) give a pair of ¶F ¶F tangent vectors on the surface at point p. Therefore, their cross product ¶u (p) × ¶v (p) is a normal vector to the surface at point p (see Figure 1.5). Here we have abused the ¶F ¶F −1 notations for simplicity: ¶u (p) means ¶u evaluated at (u, v) = F (p). Similarly for ¶F ¶v (p). ¶F ¶F Condition (3) requires that ¶u × ¶v is everywhere non-zero in the domain of F.
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