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1 Geometry and Topology Geom/Top RTG 2017 The Rotation Index of Regular Curves Author: Prof. Caine Notre Dame Name: Directions: These notes are a supplemental resource to the first lecture by Prof. Arlo Caine at the 2017 Geometry Topology RTG Workshop at Notre Dame. Some exercises and references are included for further reading and investigation. 1 Geometry and Topology Welcome to the 2017 Geometry and Topology RTG Workshop at the University of Notre Dame. RTG stands for Research Tutorial Group, the name of the National Science Foundation grant program which is funding this workshop as part of the broader efforts of the mathematicians at Notre Dame to recruit and develop new experts in these mathematical subjects. The word geometry of course is quite old, deriving from the latin geometria which stems from the Greek words g¯e and metria, meaning earth and measure, respectively. Inspired by ancient efforts to measure, describe, and construct objects in our physical world, geometry has grown over the millenia into a vast subject which is one of the three pillars of modern mathematics: analysis, algebra, and geometry. As humanity turned its attention to the measurement of the relative positions of stars and planets, geometry expanded from the study of shapes on the earth to trigonometry for the celestial sphere. The ancient theories of ratio and proportion grew through our artistic endeavors into projective geometry which explains our visual perspective on the world around us and, ultimately, allows us to create the stunning visual illusions we see in images and film today. With the development of the Calculus and the concept of coordinate systems, came analytic geometry and later vector calculus, central tools in our physical description of the motion of objects. This theory of dynamics sent people people to the moon and is responsible for our landing of robotic explorers on the surface of Mars. Along the way, we discovered the utility of studying shapes not just in 1, 2, and 3 dimensions, but in higher dimensions as well. The mathematics of geometry enabled mathematicians to \see" and understand shapes in higher dimensions, such as the very fabric of spacetime (which we imagine is 4 dimensional) or, more abstractly, the spaces of solutions to systems of equations (which can have just about any dimension depending on the complexity and number of variables in those equations). Today, the modern subject of geometry includes the vast subjects of differential geometry and algebraic geometry. Differential geometry, which includes dynamics and the mechanics of motion and measurement, uses the subjects of analysis and linear algebra to study shapes in all dimensions. Algebraic geometry, which includes projective geometry, uses the tools of abstract algebra to study the shapes of solution sets to systems of polynomial equations in many variables which arise all over mathematics and applications, in subjects ranging from number theory to robotics. Through these lectures and problem sessions we will dip our toes in the waters of differential geometry and its interface with another mathematical subject, topology. Relatively speaking, topology is a much younger subject, entering the mathematical scene in the late 1800s, early 1900s. The latin roots of the word mean the study of place or position but this phrase has a different meaning than you might imagine. In geometry, we measure length, angles, areas, and volumes, we locate features of shape and describe relative positions of them. The focus of geometry is often local (think of the tangent line to a point on a curve or the concavity of the graph of a function) and the study of geometric objects is quantitative in nature (e.g., compute the volume of this solid). Topology, by comparison, studies global features of a space and is more qualitative in nature. It is not specifically the features of a space near any one position that concerns this subject, but how the points of a space neighbor one another and are connected to form the whole that matter. With topology, one can determine what is possible and impossible within certain constraints. Can one shape be continuously deformed into another? Must every continuous function from [0; 1] to itself have a fixed point? This subject too has physical applications. For example, with topology you can prove that no matter how you heat a circular metal ring, at each instance there is at least one pair of diametrically opposite points with the same temperature. First courses in topology often concern the axiomatic development of what is referred to as point- set topology, the collection of abstract ideas and theorems which allow us to define and study what is mathematically possible and impossible to do in certain kinds of spaces. Further courses push on to algebraic or differential topology, the subjects which combine these tools with abstract algebra and analysis to compute characteristic features of spaces. These features help mathematicians distinguish spaces from each other, determining what is possible in one realm and not in another. In fact, some mathematicians use topology as a lens through which to view mathematics itself. There are ways of making spaces out of mathematical theories and using topology to classify those theories, determining what is possible to determine with those theories and what is not. In this sense, topology has evolved over the last century into a kind of qualitative theory of mathematics itself. 2 Oriented Regular Curves For this first introduction to some concepts from differential geometry and topology we will begin with some interesting results about curves in the plane. Drawing a coordinate system, we can regard this plane 2 as R , the set of ordered pairs of real numbers. When most people think of a curve in the plane, they think of a shape drawn with a single (possibly lengthy) stroke of a pen or pencil on a flat surface. Following this 2 intuition, we could try to define a plane curve to be the image of a continuous function X : I ! R from 2 2 an interval I in R into R . Thinking of the elements of the interval I as times, we can imagine X(t) in R as the point the tip of the pen is over at time t. The function X then describes the motion of the pen tip with time. 2 Exercise 2.1. Using the vector space operations on R , we can write the function X(t) = (1 − t; 2t) from 2 [0; 1] into R in the form X(t) = (1; 0) + t(−1; 2) and see that the image of X is the line segment from 2 2 (1; 0) to (0; 2). Suppose (a1; b1) and (a2; b2) are points in R . Find a function X from [0; 1] to R which describes drawing the line segment from (a1; b1) to (a2; b2) in one second. 2 Exercise 2.2. The function Y (t) = (sin(t); cos(t)) from [0; 2π] to R describes a point traveling around 2 2 2 the unit circle f(x; y) 2 R j x + y = 1g clockwise starting at (0; 1). Sketch the image of the function 2 X(t) = (sin(t) + t; cos(t)) from [0; 2π] to R . Hint: Use vector operations to write X(t) = Y (t) + Z(t) for some simple function Z(t). It turns out that requiring only continuity of the function X in our definition of curve allows strange examples of sets in the plane that most people would not refer to as curves. A rather mind-blowing fact that one can prove with real analysis is that there exists a function X from the one dimensional interval 2 [0; 1] in R into the plane R which is continuous and whose image is the two-dimensional square [0; 1]×[0; 1]! While we might be okay with referring to the perimeter of a square as a curve, albeit a one made of straight segments joined at sharp corners, it seems a little ridiculous to refer to a two-dimensional figure as a curve. Here, Calculus comes to the rescue. If we require the function X to be more than continuous and in fact differentiable at each time t, then the image will have a tangent line at each point X(t) such that the derivative X0(t) is not the zero vector. Thus, we would have a better chance of thinking of the image as being one dimensional. 2 Definition 2.1. An oriented regular curve in R is a twice continuously differentiable function X from an 2 0 interval I into R such that X (t) 6= ~0 for each t 2 I. The locus of the curve is the image of this function in 2 R . If I = [a; b], then X(a) is called the initial point of the curve and X(b) is called the final point of the curve. Such a curve is said to be closed if the initial point and final point are the same and the velocity vectors X0(a) and X0(b) agree. 2 Remark 2.1. There are various definitions of oriented regular curves in R that differ from the one above. In fact, there are a variety of names as well (e.g., regular parametric curves of class C2). Some require other orders of differentiability at every point while others may allow the derivative to be zero on a subset of the interval of a certain size. With our definition, there is no way to realize the perimeter of a square as the locus of an oriented plane curve because any parametrization of such a figure would either have zero derivative when it reached a corner or fail to be differentiable there.
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