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Über dieses Buch

This is a textbook on differential geometry well-suited to a variety of courses on this topic. For readers seeking an elementary text, the prerequisites are minimal and include plenty of examples and intermediate steps within proofs, while providing an invitation to more excursive applications and advanced topics. For readers bound for graduate school in math or physics, this is a clear, concise, rigorous development of the topic including the deep global theorems. For the benefit of all readers, the author employs various techniques to render the difficult abstract ideas herein more understandable and engaging.
Over 300 color illustrations bring the mathematics to life, instantly clarifying concepts in ways that grayscale could not. Green-boxed definitions and purple-boxed theorems help to visually organize the mathematical content. Color is even used within the text to highlight logical relationships.
Applications abound! The study of conformal and equiareal functions is grounded in its application to cartography. Evolutes, involutes and cycloids are introduced through Christiaan Huygens' fascinating story: in attempting to solve the famous longitude problem with a mathematically-improved pendulum clock, he invented mathematics that would later be applied to optics and gears. Clairaut’s Theorem is presented as a conservation law for angular momentum. Green’s Theorem makes possible a drafting tool called a planimeter. Foucault’s Pendulum helps one visualize a parallel vector field along a latitude of the earth. Even better, a south-pointing chariot helps one visualize a parallel vector field along any curve in any surface.
In truth, the most profound application of differential geometry is to modern physics, which is beyond the scope of this book. The GPS in any car wouldn’t work without general relativity, formalized through the language of differential geometry. Throughout this book, applications, metaphors and visualizations are tools that motivate and clarify the rigorous mathematical content, but never replace it.



Chapter 1. Curves

In this chapter, we develop the mathematical tools needed to model and study a moving object. The object might be moving in the plane:
Kristopher Tapp

Chapter 2. Additional Topics in Curves

This chapter presents several excursions that delve more deeply into the geometry of curves, including some of the famous theorems in the field. The theory of curves is an old and extremely well developed mathematical topic. Our aim is simply to describe a few fundamental and interesting highlights.
Kristopher Tapp

Chapter 3. Surfaces

For the remainder of the book, we turn our attention from curves to surfaces. The graphs of functions of two variables are familiar examples of surfaces from multivariable calculus. Whereas a curve locally looks like its tangent line, \(\mathbb{R}^{1}\), a surface locally looks like its tangent plane, \(\mathbb{R}^{2}\). Thus, we are moving from intrinsically one-dimensional to intrinsically two-dimensional objects.
Kristopher Tapp

Chapter 4. The Curvature of a Surface

A regular surface looks like a plane if one zooms sufficiently in near any point, but if one zooms back out, it might curve and bend through the ambient \(\mathbb{R}^{3}\). That’s what makes differential geometry so much richer than Euclidean geometry.
Kristopher Tapp

Chapter 5. Geodesics

The most fundamental concept for studying the geometry of \(\mathbb{R}^{2}\) is a straight line. The goal of this chapter is to generalize this fundamental notion from \(\mathbb{R}^{2}\) to arbitrary regular surfaces. Although most surfaces curve in such a way that they don’t contain any straight lines, they do contain curves called geodesics, which will turn out to share many important characterizing properties of straight lines.
Kristopher Tapp

Chapter 6. The Gauss–Bonnet Theorem

The Gauss–Bonnet theorem is the most famous result in the study of surfaces. It provides a satisfying final chapter of this textbook because it interrelates many fundamental concepts from the previous five chapters.
Kristopher Tapp


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