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Calculus and Differential Geometry: an Introduction to Curvature Donna Dietz Howard Iseri

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Calculus and Differential Geometry: an Introduction to Curvature Donna Dietz Howard Iseri Calculus and Differential Geometry: An Introduction to Curvature Donna Dietz Howard Iseri Department of Mathematics and Computer Information Science, Mansfield University, Mansfield, PA 16933 E-mail address: [email protected] Contents Chapter 1. Angles and Curvature 1 1. Rotation 1 2. Angles 3 3. Rotation 4 4. Definition of Curvature 6 5. Impulse Curvature 8 Chapter 2. Solid Angles and Gauss Curvature 11 1. Total curvature for cone points 11 2. Total curvature for smooth surfaces 13 3. Gauss curvature and impulse curvature 14 4. Gauss-Bonnet Theorem (Exact exerpt from Creative Visualization handout. 15 5. Defining Gauss curvature 16 6. Intrinsic aspects of the Gauss curvature 19 Chapter 3. Intrinsic Curvature 21 1. Parallel vectors 21 Chapter 4. Functions 25 1. Introduction 25 2. Piecewise-Linear Approximations for Functions of One Variable 25 3. Uniform Continuity 27 4. Differentiation in One Variable 29 5. Derivatives and PL Approximations 33 6. Parametrizations of Curves 35 7. Functions of Two Variables 37 8. Differentiability for Functions of Two Variables 37 Chapter 5. The Riemannian Curvature Tensor in Two Dimensions 47 1. Parametrizations 48 Chapter 6. Riemannian Curvature Tensor 53 1. The Riemannian Metric for a Plane 53 2. The Riemannian Metric for Curved Surfaces 56 3. Curvature 60 4. The Inverse of the Metric 62 Chapter 7. Riemannian Curvature Tensor 63 1. Intrinsic Interpretations 63 3 4 CONTENTS Chapter 8. Curvature of 3-Dimensional Spaces 69 1. What we know 69 2. What is the geometry like around a vertex of a cubed 3-manifold? 69 3. A positive curvature example 69 CHAPTER 1 Angles and Curvature 0.1. Overview. As you walk around a closed path (along a simple closed curve on the floor), the direction you are facing will make a net rotation of 2π radians or 360◦. 1. Rotation Imagine a circle drawn on the floor (the radius might be ten feet). You are to walk around the circle once in a counter-clockwise direction. If you are initially facing north, you will soon be facing north-west and then west. We can naturally ◦ π say that the direction in which you are facing has changed by 90 or 2 radians. After that, you will face south, then east, and finally north again. The direction in which you are facing has experienced a rotation of 360◦. We will want to think of this rotation as describing how the direction you are facing has changed as opposed to your change in location as you make an orbit around the circle. For a curve in the plane, we can talk about the rotation of a tangent vector in the same way that we have talked about the rotation of our body as we walk along a curve drawn on the floor. Intuitively at least, we would like to identify these two concepts. That is, what we discover about one should apply equally to the other. Throughout this book, we will use the convention that counter-clockwise rota- tions are positive. For example, if you were to turn 45◦ to the left and then 90◦ to the right, the net rotation would be −45◦. A B C Figure 1. Walk along this path marked on the floor. (Exercise 1) 1 2 1. ANGLES AND CURVATURE 1.1. Exercises. 1. Suppose you are walking around the curve shown in Figure 1 in a counter- clockwise direction. Assume that the curve is smooth (the direction varies smoothly) and that the direction you are facing is the same as that of a tangent vector. How does the direction you face change as you move from the starting point A to the point B? From B to C? From A to C? What is your total (net) rotation for the entire circuit? Figure 2. Walk along this path marked on the floor. (Exercise 2) 2. What would your total rotation be as you walked in the direction indicated around the path shown in Figure 2? Figure 3. Walk along this path marked on the floor. (Exercise 3) 3. What would your total rotation be as you walked in the direction indicated around the path shown in Figure 3? 4. Make a conjecture about the net rotation of a tangent vector moving around a simple closed curve in the plane in a counter-clockwise direction. 5. Make a conjecture about the net rotation of a normal vector moving around a simple closed curve in the plane in a counter-clockwise direction. Does it make a difference whether the normal vector is pointing outward or inwards? Are there other directions that a normal vector can point? 1.2. Overview. Angles are abrupt changes in direction. Total curvature is the net change in direction over some section of a curve or polygonal path. 2. ANGLES 3 2. Angles One of the most important theorems in Euclidean geometry states that the sum of the angles of a triangle is 180◦. Virtually all of the theorems that involve angle measure or parallelism can be proved with this fact. Among these would be that the angle sum of a quadrilateral is 360◦, the angle sum of a pentagon is 540◦, the angle sum of a hexagon is 720◦, and in general, Theorem 1. The angle sum of a (convex) n-gon is (n − 2) · 180◦ 95◦ 95◦ 100◦ 70◦ Figure 4. The turning angles for a quadrilateral. This is all very nice, but the sequence of theorems just mentioned can be restated more simply and intuitively in terms of the turning angle or angle defect. The reason for using the term turning angle should become clear, and angle defect refers to the idea that the turning angle measures how far the angle is from being a straight angle. In Figure 4, a quadrilateral is shown with the turning angles marked. You should imagine yourself walking around the quadrilateral in a counter-clockwise direction. The turning angles then measure the amount you must turn to your left as you start the next edge. In this case, the sum of the turning angles is 360◦. If you imagine yourself walking around any closed path, taking left turns, and coming back to your original position, you must have rotated a full 360◦. This should agree completely with your answers to the exercises in the previous section. It seems reasonable, therefore, that the sum of the turning angles is 360◦ for any polygon. This is in fact true, and Theorem 1 can be restated as Theorem 2. The turning angle sum of a (convex) n-gon is 360◦. It is not necessarily true that Theorem 2 is a better theorem than Theorem 1, but it is certainly simpler and more intuitive. The angle sum theorem is probably more convenient for analyzing geometric figures, but we are wanting to understand curvature, and the turning angle sum theorem sets us off in the right direction. 2.1. Exercises. 6. Theorem 1 states that the angle sum of an n-gon is (n − 2)180◦ or n − 2 times the angle sum of a triangle. Draw a figure illustrating that a convex pentagon has the angle sum of three triangles. Do the same for a hexagon. 4 1. ANGLES AND CURVATURE 7. Suppose the quadrilateral of Figure 4 is drawn on the floor with up in the picture corresponding to north, and you are to walk around it in the counter- clockwise direction. Draw a picture of the face of a compass, and for one of the sides, draw the position of the needle corresponding to the direction you are facing as you walk along it. On the same picture, draw the needle positions corresponding to the other three sides. At each vertex, you would need to pivot as you finish walking along one side of the quadrilateral and start on the next. In your picure, for each vertex, indicate which directions you sweep through as you turn to the left. 3. Rotation Our goal is to formulate definitions in differential geometry. Before we do that for curves in the plane, let us summarize what we have so far. Given an object moving in a counter-clockwise direction around a simple closed curve, a vector tangent to the curve and associated with the object must make a “full” rotation of 2π radians or 360◦. In other words, if we were to think of this tangent vector (of if you wish, a copy of it) as having its tail fixed at the origin, then as the object moves around the curve, the tangent vector will sweep through all possible directions. This rotation of the tangent vector will be predominantly in the counter-clockwise direction, but it may, for example, sweep clockwise for a bit, come back counter-clockwise an equal amount, and then continue on. These clockwise rotations are always countered by an extra counter-clockwise rotation, and the total net result is always 360◦ of counter-clockwise rotation. If the curve is smooth (whatever that means), we can easily describe a tangent vector in terms of a derivative. There are some difficulties at non-smooth parts of a curve. At the corners of a quadrilateral, for example, a derivative will not specify a unique tangent direction. In this case at least, we will be able to find a tangent direction entering the vertex and one leaving. We can and will account for the directions swept through as we pivot from one direction to the other, and we will avoid curves that are “less smooth” than this. In order to motivate the definitions describing rotations in terms of derivatives, we will consider the following.
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