Exploring Curvature

Frontmatter

1. The Evolution of Geometry

Abstract

Geometry is concerned primarily with the spatial properties of objects and with the endless stream of abstract generalizations to which these properties give rise. Modem geometry is an extremely active field of research by pure and applied mathematicians, and it also has significant applications in physics and engineering. In the present book, we will explore in a physical manner the geometrical properties of curves and surfaces, and will discuss in detail concepts that are essential to modem geometry.

James Casey

2. Basic Operations

Abstract

We start out by performing some geometrical operations that can be easily done on objects located in ordinary three-dimensional space. You will need a ruler and a piece of string (or a tape measure) for measuring lengths, and a protractor for measuring angles.

James Casey

3. Intersecting with a Closed Ball

Abstract

In order to examine the local geometry of a figure, we will need to focus attention on the collection of points that lie close to any given point. We will dissect the figure mentally, as it were. To be clear about what this means, it is best to employ some concepts from set theory.

James Casey

4. Mappings

Abstract

We now discuss the concept of a mapping (or function). The usefulness of this idea for the mathematical sciences can hardly be exaggerated.

James Casey

5. Preserving Closeness: Continuous Mappings

Abstract

In discussing the operations described in Chapter 2, we paid particular attention to whether or not closeness between points of a figure was changed by the operations. It is worth looking into this idea more carefully.

James Casey

6. Keeping Track of Magnitude, Direction and Sense: Vectors

Abstract

In addition to possessing magnitude, many familiar physical quantities, such as wind velocity and force, involve direction in an essential way. These quantities can be represented mathematically by vectors. It turns out that vectors are also very useful in geometry, and you will have ample opportunity to see this when we study curves and surfaces. In the present chapter, the basic properties of vectors are described.

James Casey

7. Curves

Abstract

In preceding chapters, we have discussed some special types of curves, such as the ellipse, the circle, and the straight line. But, these are only a few of the infinite variety of curves that can be imagined. The path of a bird flying through the air, the instantaneous shape of a swinging chain held at its top, the outlines of petals, the forms of arches and suspension bridges-all provide physical examples of curves. We wish to describe such general curves mathematically and to explore their essential properties. To do so, we will utilize the concepts and theoretical tools that have been introduced in earlier chapters. Our aim is to proceed from intuitive notions about curves to a clear, abstract definition. This process - the clarification of ideas - is really one of the most important activities of the mathematician.

James Casey

8. Arc Length

Abstract

Despite the fundamental importance of the concept of length in Euclidean geometry, the Greek geometers did not know how to define the length of a general curve. In other words, the Greek mathematicians were unable to put into mathematical language an idea that every ancient rope-stretcher and tailor must have known! To understand the nature of the difficulty, let us start with an experiment.

James Casey

9. Tangent

Abstract

In an important theorem in Book III of his Elements, Euclid proves that the straight line drawn at right angles to any diameter BA of a circle at one of its extremities A falls outside the circle (except where it touches it), and that no other straight line can be interposed into the space HAE between the circumference and the straight line, and further that the “horn-like angle” HAE that the circumference makes with EA is less than any rectilineal angle.¹ The line EA is the tangent to the circle at A (Fig. 50), and, as Euclid’s results indicate, it lies closer to the circumference than any other straight line through A. The fact that Euclid felt compelled to prove this result, which most of us would regard as “obvious”, attests to the high level of rigor that permeated Greek mathematics in the days of Plato’s Academy.

James Casey

10. Curvature of Curves

Abstract

While everyone would agree that a road winding around a mountain is curvy, not quite so apparent is how to express this property quantitatively. Certainly, the curvature of a straight line should be considered zero. Perhaps then, we can regard a curve as a deviation from a straight line? Let us explore how this idea can be given quantitative meaning.

James Casey

11. Surfaces

Abstract

Having worked his or her way through Chapters 7 to 10, the reader will have gained an appreciation for the geometry of curves. We now turn our attention to the geometry of surfaces, which is even more fascinating. We will proceed in our usual manner, moving from intuitions to concepts, and exploring the geometrical phenomena by means of simple experiments. Our discussion of surface geometry begins with a search for a good definition of the concept of a surface.

James Casey

12. Surface Measurements

Abstract

Having defined what surfaces are and having studied some of their topological properties, we now begin to explore their geometry quantitatively In the present chapter, we will be concerned primarily with the measurement of distances and angles on surfaces. We will see how such metrical properties of surfaces can be expressed in terms of certain fundamental quantities called the metric coefficients. The theory discussed here and in Chapter 13 was invented single-handedly in the early part of the 19th century by the great mathematician Gauss (whose biography is sketched in Chapter 14). It was a major turning point in the history of geometry.

James Casey

13. Intrinsic Geometry of a Surface

Abstract

The most deeply rooted geometrical properties of a surface are its topological ones: these are preserved under all homeomorphisms of the surface. Hence also, they are preserved under all deformations. (Some examples of deformations of surfaces were studied at the end of Chapter 11.) There is a broader class of properties that are intimately bound up with the geometry of the surface and that are preserved under a large subclass of homeomorphisms. These are ones that Gauss discovered, and which we will now explore.

James Casey

14. Gauss (1777-1855)

Abstract

Mathematics, like music or literature or art, is an activity for which we human beings possess immeasurable stores of talent and passion. It is a highly intellectual activity, but it should not be regarded as an elitist one. Even those of us who have never created a song, or a story, or a piece of mathematics, can still experience much pleasure from playing or listening to music, or from reading a book or attending a play, or from doing a calculation or studying a proof. Furthermore, after an initial period of practice, many of us become quite accomplished at an activity, and continue to derive pleasure from it throughout their lives. Some others among us have sufficient talent to become professional musicians, writers, or mathematicians. And, scattered throughout history, there are those rare individuals whose genius leaves us in awe. Thus, in music, Bach (1685–1750), Mozart (1756–1791), and Beethoven (1770–1827) seem to possess almost superhuman powers. In literature, we have Shakespeare (1564–1616), Milton (1608–1674), Goethe (1749–1832), and several others. In mathematics, Archimedes (287–212 B.C.), Newton (1642–1727), and Gauss are ranked at the top, but magnificent contributions were also made by a large number of others.

James Casey

15. Normal Sections

Abstract

One way of exploring curved surfaces is to examine the curvature of curves lying on them. In the present chapter, we will use planes to cut a surface in a specific way, and we will study the resulting curves to gain some insight into the nature of surface curvature.

James Casey

16. Gaussian Curvature

Abstract

In Chapter 15, we saw how, at a point on a surface, the curvature of a normal section varies as the sectioning plane is rotated about the normal vector. We learned that this variation is governed by Euler’s formula (15.30). In the present chapter, a completely different approach is taken, which is not based at all on the curvature of curves. Here, we study a brilliant idea of Gauss’s, which will enable us to define a unique value of surface curvature at each point on a smooth surface.

James Casey

17. Riemann (1826–1866)

Abstract

Like Mozart’s, Bernhard Riemann’s life was short but marvelously creative. He solved several of the most difficult problems in pure and applied mathematics, introduced entirely new concepts and techniques, and profoundly changed the way in which mathematicians, physicists, and philosophers view space.¹

James Casey

18. Levi-Civita (1873–1941)

Abstract

We have seen how Gauss’s theory of surface geometry shaped Rie-mann’s conception of general curved manifolds. Riemann’s ideas in turn were taken up by Christoffel (1829–1901), Beltrami (1835–1900), and others. During the closing decades of the 19th century, a powerful school of mathematics developed at the University of Padua. It was here that Levi-Civita came into contact with modern geometry.

James Casey

19. Parallel Transport of a Vector on a Surface

Abstract

In this chapter, we describe a particular way of moving a vector along a given curve on a surface. It provides an especially revealing means of exploring the non-Euclideanness of the surface.

James Casey

20. Geodesics

Abstract

In Euclidean plane geometry, the straight lines occupy a favored position among all of the curves that can be drawn in the plane. What are their analogues on a curved surface? One can try to generalize the property of straightness, or alternatively, one can utilize the other fundamental property of a straight line, namely that of being the shortest curve between two given points in the Euclidean plane.

James Casey

21. Geometry and Reality

Abstract

As we saw in the preceding chapter, on a curved surface, geodesics replace the straight lines of the Euclidean plane. It also became clear that, in general, when figures are formed from geodesics, some of the most cherished results of Euclidean geometry - such as the theorem on angle sum of a triangle - must be given up. But, Euclid’s theorems are derived by logical arguments from a set of postulates (or axioms). Physical intuition is allowed to enter the theory only through the postulates; after these are stated, only rules of logic can be appealed to in the proofs (although, of course, the inspiration for a theorem can come from intuition). It must therefore be the case that some of the properties ascribed to lines in Euclid’s postulates cannot be true in general for geodesics. For, if all of Euclid’s postulates held for geodesics, then so would his theorems. Let us examine some of the Euclidean postulates to see what goes wrong when they are interpreted to hold for geodesics.

James Casey

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