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Elementary Differential Geometry

verfasst von: Andrew Pressley

Verlag: Springer London

Buchreihe : Springer Undergraduate Mathematics Series

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Elementary Differential Geometry presents the main results in the differential geometry of curves and surfaces suitable for a first course on the subject. Prerequisites are kept to an absolute minimum – nothing beyond first courses in linear algebra and multivariable calculus – and the most direct and straightforward approach is used throughout.

New features of this revised and expanded second edition include:

a chapter on non-Euclidean geometry, a subject that is of great importance in the history of mathematics and crucial in many modern developments. The main results can be reached easily and quickly by making use of the results and techniques developed earlier in the book.

Coverage of topics such as: parallel transport and its applications; map colouring; holonomy and Gaussian curvature.

Around 200 additional exercises, and a full solutions manual for instructors, available via www.springer.com

Inhaltsverzeichnis

Frontmatter

1. Curves in the plane and in space

Abstract

In this chapter, we discuss two mathematical formulations of the intuitive notion of a curve. The precise relation between them turns out to be quite subtle, so we begin by giving some examples of curves of each type and practical ways of passing between them.

Andrew Pressley

2. How much does a curve curve?

Abstract

In this chapter, we associate two scalar functions, its curvature and torsion, to any curve in ℝ³. The curvature measures the extent to which a curve is not contained in a straight line (so that straight lines have zero curvature), and the torsion measures the extent to which a curve is not contained in a plane (so that plane curves have zero torsion). It turns out that the curvature and torsion together determine the shape of a curve.

Andrew Pressley

3. Global properties of curves

Abstract

All the properties of curves that we have discussed so far are ‘local’: they depend only on the behaviour of a curve near a given point and not on the ‘global’ shape of the curve. Proving global results about curves often requires concepts from topology, in addition to the calculus techniques we have used in the first two chapters of this book. Since we are not assuming that readers of this book have extensive familiarity with topological ideas, we will not be able to give complete proofs of some of the global results about curves that we discuss in this chapter.

Andrew Pressley

4. Surfaces in three dimensions

Abstract

In this chapter, we introduce several different ways to mathematically formulate the notion of a surface. Although the simplest of these, that of a surface patch, is all that is needed for most of the book, it does not describe adequately most of the objects that we would want to call surfaces. For example, a sphere is not a surface patch, but it can be described by ‘gluing’ two surface patches together suitably. The idea behind this gluing procedure is simple enough, but making it precise turns out to be a little complicated. We have tried to minimize the trauma by collecting the most demanding proofs in a separate section (Section 5.6).

Andrew Pressley

5. Examples of surfaces

Abstract

In this chapter we describe some of the simplest classes of surfaces. Others will be introduced later in the book.

Andrew Pressley

6. The first fundamental form

Abstract

Perhaps the first thing that a geometrically inclined bug living on a surface might wish to do is to measure the distance between two points of the surface. Of course, this will usually be different from the distance between these points as measured by an inhabitant of the ambient three-dimensional space, since the straight line segment, which furnishes the shortest path between the points in ℝ³ will generally not be contained in the surface. The object that allows one to compute lengths on a surface, and also angles and areas, is the first fundamental form of the surface.

Andrew Pressley

7. Curvature of surfaces

Abstract

In this chapter, we discuss several approaches to the problem of measuring how ‘curved’ a surface is. Although they use quite different methods, we show that each of the approaches leads to the same geometric object: the second fundamental form of a surface. It turns out (see Theorem 10.1.3) that a surface is determined up to an isometry of ℝ³ by its first and second fundamental forms, just as a unit-speed plane curve is determined up to an isometry of ℝ² by its signed curvature.

Throughout this chapter we shall work with oriented surfaces. Recall from Section 4.5 that every surface patch is oriented.

Andrew Pressley

8. Gaussian, mean and principal curvatures

Abstract

In this chapter, we show how to extract geometric information from the second fundamental form of a surface or, equivalently, from its Weingarten map.

Andrew Pressley

9. Geodesics

Abstract

Geodesics are the curves in a surface that a bug living in the surface would perceive to be straight. For example, the shortest path between two points in a surface is always a geodesic. We shall actually begin by giving a quite different definition of geodesics, since this definition is easier to work with. We give various methods of finding geodesics on surfaces, before finally making contact with the idea of shortest paths towards the end of the chapter.

Andrew Pressley

10. Gauss’ Theorema Egregium

Abstract

One of Gauss’ most important discoveries about surfaces is that the Gaussian curvature is unchanged when the surface is bent without stretching. Gauss called this result ‘egregium’, and the Latin word for ‘remarkable’ has remained attached to his theorem ever since. We shall deduce the Theorema Egregium from two results which relate the first and second fundamental forms of a surface, and which have other important consequences.

Andrew Pressley

11. Hyperbolic geometry

Abstract

One of the most remarkable discoveries of nineteenth century mathematics is that the pseudosphere discussed in Section 8.3 has a geometry that closely resembles Euclidean geometry, with geodesics playing the role of straight lines. In fact, the closest correspondence with Euclidean geometry is obtained by ‘embedding’ the pseudosphere in a larger geometry, which is called hyperbolic or non-Euclidean geometry. When this is done, we find that all the axioms of Euclidean geometry hold in hyperbolic geometry, except the so-called ‘parallel axiom’: this states that if p is a point that is not on a straight line l, there is a unique straight line passing through p that does not intersect l (i.e., which is ‘parallel’ to l in the usual sense).

Hyperbolic geometry was discovered independently and almost simultaneously by the Hungarian mathematician Janos Bolyai and the Russian Nicolai Lobachevsky, although the formulations of it, that we shall describe in this chapter, are due to Eugenio Beltrami, Felix Klein and Henri Poincaré. David Hilbert, one of the greatest mathematicians of the twentieth century, wrote that the discovery of nonEuclidean geometry was ‘one of the two most suggestive and notable achievements of the last century’. It ended centuries of attempts by Greek, Arab and later Western mathematicians to deduce the parallel axiom from the other axioms of Euclidean geometry, and it profoundly changed our view of what geometry is.

Andrew Pressley

12. Minimal surfaces

Abstract

In Section 9.4 we considered the problem of finding the shortest paths between two points on a surface. We now consider the analogous problem in one higher dimension, that of finding a surface of minimal area with a fixed curve as its boundary. This is called Plateau’s Problem. The solutions to Plateau’s problem turn out to be surfaces whose mean curvature vanishes everywhere. The study of these so-called minimal surfaces was initiated by Euler and Lagrange in the mid-eighteenth century, but new examples of minimal surfaces are still being discovered.

Andrew Pressley

13. The Gauss–Bonnet theorem

Abstract

The Gauss–Bonnet theorem is the most beautiful and profound result in the theory of surfaces. Its most important version relates the average of the Gaussian curvature to a property of the surface called its ‘Euler number’ which is ‘topological’, i.e., it is unchanged by any diffeomorphism of the surface. Such diffeomorphisms will in general change the value of the Gaussian curvature, but the theorem says that its average over the surface does not change. The real importance of the Gauss–Bonnet theorem is as a prototype of analogous results which apply in higher dimensional situations, and which relate geometrical properties to topological ones. The study of such relations was one of the most important themes of twentieth century mathematics, and continues to be actively studied today.

Andrew Pressley

Backmatter

Titel: Elementary Differential Geometry
verfasst von: Andrew Pressley
Verlag: Springer London
Electronic ISBN: 978-1-84882-891-9
Print ISBN: 978-1-84882-890-2
DOI: https://doi.org/10.1007/978-1-84882-891-9

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Inhaltsverzeichnis

Frontmatter

1. Curves in the plane and in space

2. How much does a curve curve?

3. Global properties of curves

4. Surfaces in three dimensions

5. Examples of surfaces

6. The first fundamental form

7. Curvature of surfaces

8. Gaussian, mean and principal curvatures

9. Geodesics

10. Gauss’ Theorema Egregium

11. Hyperbolic geometry

12. Minimal surfaces

13. The Gauss–Bonnet theorem

Backmatter

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