Surfaces: (almost) everything you wanted to know about them lectures by Anatole Katok notes taken and TeXed by Vaughn Climenhaga Contents Chapter 1. Various Ways of representing Surfaces 7 1.1. Lecture 2: Wednesday, Aug. 29 7 a. Equations for surfaces and local coordinates 7 b. Other ways of introducing local coordinates 8 c. Metrics on surfaces 9 1.2. Lecture 3: Friday, Aug. 31 10 a. Remarks concerning the problem set 10 b. Parametric representations of curves 11 c. Other means of representation 12 d. Regularity conditions for parametric surfaces 13 1.3. Lecture 4: Wednesday, Sept. 5 14 a. Review of metric spaces and topology 14 b. Isometries 16 1.4. Lecture 5: Friday, Sept. 7 17 a. Issues relating to a lecture by A. Kirillov 17 b. Isometries of the Euclidean plane 18 c. Isometries of the sphere and the projective plane 20 1.5. Lecture 6: Monday, Sept. 10 20 a. Area of a spherical triangle 20 b. Isometries of the sphere 21 c. Spaces with lots of isometries 23 1.6. Lecture 7: Wednesday, Sept. 12 24 a. Symmetric spaces 24 b. Remarks concerning direct products 25 c. Topology and combinatorial structure on surfaces 26 Chapter 2. Combinatorial Structure and Topological Classification of Surfaces 31 2.1. Lecture 8: Friday, Sept. 14 31 a. Triangulation 31 b. Euler Characteristic 34 2.2. Lecture 9: Monday, Sept. 17 36 a. Continuation of the proof of Theorem 1 36 b. Calculation of Euler characteristic 42 2.3. Lecture 10: Wednesday, Sept. 19 43 a. From triangulations to maps 43 3 4 CONTENTS b. Examples 46 2.4. Lecture 11: Friday, Sept. 21 48 a. Euler characteristic of planar models 48 b. Attaching handles 49 c. Orientability 51 d. Inverted handles and M¨obius caps 52 2.5. Lecture 12: Monday, Sept. 24 53 a. Non-orientable surfaces and M¨obius caps 53 b. Calculation of Euler characteristic 54 c. Covering non-orientable surfaces 55 d. Classification of orientable surfaces 57 2.6. Lecture 13: Wednesday, Sept. 26 58 a. Proof of the classification theorem 58 b. Non-orientable surfaces: Classification and models 61 2.7. Lecture 14: Friday, Sept. 28 62 a. Chain complexes and Betti numbers 62 b. Homology of surfaces 64 c. A second interpretation of Euler characteristic 66 d. Interpretation of the Betti numbers 67 e. Torsion in the first homology and non-orientable surfaces 69 2.8. Lecture 15: Monday, Oct. 1 69 a. Alternate method for deriving interpretation of Betti numbers 69 Chapter 3. Differentiable (Smooth) Structure on Surfaces. 71 3.1. Continuation of lecture 15: Monday, Oct. 1 and lecture 16: Monday, Oct. 8 71 a. Charts and atlases 71 b. First examples of atlases 73 3.2. Lectures 17: Wednesday, Oct. 10 and 18: Friday October 12 75 a. Differentiable manifolds 75 b. Diffeomorphisms 77 c. More examples of charts and atlases 78 3.3. Lecture 19: Monday, Oct. 15 80 a. Embedded surfaces 80 b. Gluing surfaces 81 c. Quotient spaces 81 d. Removing singularities 82 e. Riemann surfaces 84 3.4. Lecture 20: Wednesday, Oct. 17 85 a. More on Riemann surfaces 85 b. Conformal property of holomorphic functions and invariance of angles on Riemann surfaces 87 c. Differentiable functions on real surfaces 88 3.5. Lecture 21: Friday, Oct. 19 90 a. More about smooth functions on surfaces 90 CONTENTS 5 b. The third incarnation of Euler characteristics 94 3.6. Lecture 22: Monday, Oct. 22 96 a. Functions with degenerate critical points 96 b. Degree of a circle map 98 c. Zeroes of a vector field and their indices 100 3.7. Lecture 23: Wednesday, Oct. 24 100 a. More on degrees 100 b. More on indices 102 c. Tangent vectors, tangent spaces, and the tangent bundle 103 Chapter 4. Riemannian Metrics on Surfaces 107 4.1. Lecture 24: Friday, Oct. 26 107 a. Definition of a Riemannian metric 107 b. Partitions of unity 110 4.2. Lecture 25: Monday, Oct. 29 111 a. Existence of partitions of unity 111 b. Global properties from local and infinitesimal 114 c. Lengths, angles, and areas 115 4.3. Lecture 26: Wednesday, Oct. 31 117 a. Geometry via a Riemannian metric 117 b. Differential equations 118 c. Geodesics 118 4.4. Lecture 27: Friday, Nov. 2 120 a. First glance at curvature 120 b. The hyperbolic plane: two conformal models 122 c. Geodesics and distances on H2 125 4.5. Lecture 28: Monday, Nov. 5 127 a. Detailed discussion of geodesics and isometries in the upper half-plane model 127 b. The cross-ratio 131 4.6. Lectures 29: Wednesday, Nov. 7 and 30: Friday, Nov. 9 134 a. Three approaches to hyperbolic geometry 134 b. Characterization of isometries 134 c. Classification of isometries 138 d. Geometric interpretation of isometries 143 4.7. Lecture 31: Monday, Nov. 12 146 a. Area of triangles in different geometries 146 b. Area and angular defect in hyperbolic geometry 147 4.8. Lecture 32: Wednesday, Nov. 14 151 a. Hyperbolic metrics on surfaces of higher genus 151 b. Curvature, area, and Euler characteristic 155 4.9. Lecture 33: Friday, Nov. 16 158 a. Geodesic polar coordinates 158 b. Curvature as an error term in the circle length formula 159 c. The Gauss-Bonnet Theorem 160 6 CONTENTS d. Comparison with traditional approach 162 Chapter 5. Smooth and Combinatorial Structure revisited 165 5.1. Lecture 34: Monday, Nov. 26 165 a. More on indices 165 b. The Fundamental Theorem of Algebra 167 5.2. Lecture 35: Wednesday, Nov. 28 168 a. Jordan Curve Theorem 168 b. Another interpretation of genus 171 5.3. Lecture 36: Friday, Nov. 30 172 a. A remark on tubular neighbourhoods 172 b. Jordan Curve Theorem 173 c. Poincar´e-Hopf formula 174 5.4. Lecture 37: Monday, Dec. 3 175 a. Proof of the Poincar´e-Hopf Index Formula 175 b. The ubiquitous Euler characteristic 178 CHAPTER 1 Various Ways of representing Surfaces 1.1. Lecture 2: Wednesday, Aug. 29 a. Equations for surfaces and local coordinates. Consider the problem of writing an equation for the torus; that is, finding a function F : R3 R such that the torus is the solution set to F (x,y,z) = 0. (We will consider→ parametric representations later on). Because the torus is a surface of revolution, we begin with the equation for a circle in the x z plane with radius 1 and centre at (2, 0): − S1 = (x, z) R2 : (x 2)2 + z2 = 1 { ∈ − } To obtain the surface of revolution, we introduce y into the equation by making the substitution x x2 + y2, and obtain 7→ T2 = (x,y,z) Rp3 : ( x2 + y2 2)2 + z2 1 = 0 { ∈ − − } At first glance, then, setting F (x,y,zp ) = ( x2 + y2 2)2 + z2 1 gives our desired solution. However, this suffers from the− defect that−F is not p 1 along the z-axis; we can overcome this fairly easily with a little algebra. ExpandingC the equation, isolating the square root, and squaring both sides, we obtain x2 + y2 + 4 4 x2 + y2 + z2 1 = 0 − − px2 + y2 + z2 +3 = 4 x2 + y2 (x2 + y2 + z2 + 3)2 16(x2 + y2) = 0 − p It can be checked that this new choice of F does not introduce any extraneous points to the solution set, and now F is 1 on all of R3. C This argument cannot be applied literally to the homework problem N3 (see optional exercise below) but the general method can be used; namely one should start from the intersection of a sphere with two handles with a horizontal plane which may look e.g. as the union of three circles and build the rest of the surface symmetrically above and below. Exercise 1. Prove that a sphere with m 2 handles cannot be repre- sented as a surface of revolution. ≥ What good is all this? What benefit do we gain from representing the torus, or any other surface, by an equation? To answer this, we first back- track a bit and discuss graphs of functions. In particular, given a function 7 8 1. VARIOUSWAYS OFREPRESENTINGSURFACES f : R2 R, the graph of f is → graph f = (x,y,z) R3 : z = f(x,y) { ∈ } Assuming f is ‘nice’, its graph is a ‘nice’ surface sitting in R3. Of course, most surfaces cannot be represented globally as the graph of such a function; the sphere, for instance, contains two points on the z-axis, and hence we require at least two functions to describe it in this manner. In fact, more than two functions are required if we adopt this approach. The sphere is given as the solution set of x2 + y2 + z2 = 1, so we can write it as the union of the graphs of f1 and f2, where f (x,y) = 1 x2 y2 1 − − 2 2 f2(x,y) = p 1 x y − − − The graph of f1 is the northern hemisphere,p the graph of f2 the southern. However, we run into problems at the equator z = 0, where the derivatives of f1 and f2 become infinite; we permit ourselves to use only functions with bounded derivatives. By using graphs with x or y as the dependent variable, we can cover the eastern and western hemispheres, as it were, but find that we require six graphs to deal with the entire sphere. This approach has wide validity; if we have any surface S given as the zero set of a function F : R3 R, we say that a point on S is regular if the gradient of F is nonzero→ at that point.