We use the notion of an isometry to make the concept of inner geometry of surfaces more precise. Vector fields and their first and second covariant derivatives are introduced. The Theorema Egregrium (‘remarkable theorem’) expresses the Gauss curvature in terms of the curvature tensor and shows the Gauss curvature belongs to the inner geometry of the surface. General Riemann metrics generalise the first fundamental form. The problem of the shortest way from one point to another leads to the concept of the geodesic and the Riemann exponential mapping. In this way it is particularly straightforward to obtain coordinates that are convenient in geometry, like Riemann normal coordinates, geodesic polar coordinates and Fermi coordinates. Jacobi fields illustrate the inner geometric importance of the Gauss curvature. Spherical and hyperbolic geometry are investigated in more detail. Their trigonometry is derived and applications to cartography are discussed. The hyperbolic plane satisfies all axioms of Euclidean geometry except for the parallel axiom.

Isometries

When we consider surfaces in ℝ3 we tend to pay special attention to their relative geometries, i.e. to how the surface is embedded into the surrounding space. We quasi look at them from outside. One could also try to imagine oneself in the position of a (two-dimensional) inhabitant of the surface, and examine those properties of the surface that can be observed by a being who cannot peek out of the surface.

Surfaces with boundary are introduced. The divergence theorem of Gauss is derived and used to show that the total Gauss curvature of a compact regular surface does not depend on the Riemannian metric.

The divergence theorem

In this section we want to derive a two-dimensional analogue of the fundamental theorem of calculus. In this theorem the integral of a derivative over a one-dimensional interval is identified with the difference of the values at the end-points. This term in the values at the end-points can be considered as the integral of the function over the (zero-dimensional) boundary of the interval. The divergence theorem expresses the integral of a derivative of a vector field as a one-dimensional line integral. To make all this precise we first need the notion of a surface with boundary.

Definition 5.1.1 A surface with boundary is a closed subset S of a regular surface Sreg ⊂ ℝ3 such that for every point p ∈ S there exists a local parametrisation F : U → Sreg of Sreg with p ∈ F(U), such that either

• F(U) ⊂ S (then p is called an interior point of S) or

• F−1(p) = (x, 0)⊤ for an x ∈ ℝ and F−1(S) = {(x, y)⊤ ∈ U| y ≥ 0} (then p is called a boundary point of S).

We analyse curves in n-dimensional space with a special focus on plane curves and space curves. Length, curvature and torsion are introduced. We prove Hopf's Umlaufsatz for simple closed curves, characterise convex curves and derive the four-vertex theorem. The isoperimetric inequality, which compares the length of a simple closed plane curve with the enclosed area, is proved using the Fourier series. We show that for a given curvature and torsion the resulting space curve is unique up to a Euclidean motion. We investigate how much a space curve needs to curve if it is closed and make the result even stronger in the case that the space curve is knotted.

Curves in ℝn

We now want to use the tools of differentiation and integration to describe curves in n-dimensional space. We usually graphically imagine a curve as a bent line in space. Mathematically we express this as follows:

Definition 2.1.1 Let I ⊂ ℝ be an interval. A parametrised curve is a map c : I → ℝn that can be differentiated infinitely often. A parametrised curve is said to be regular if its velocity vector does not vanish anywhere; ċ(t) ≠ 0 for all t ∈ I.

The interval I from the definition may be open, closed or half-open; furthermore, I can be bounded or unbounded. The condition ċ(t) ≠ 0 ensures that the point c(t) on the curve moves at t ∈ I. In particular, this excludes the constant map c(t) = c0.

This book evolved from courses about elementary differential geometry which I have taught in Freiburg, Hamburg and Potsdam. The word “elementary” should not be understood as “particularly easy”, but indicates that the development of formalism, which would be necessary for a deeper study of differential geometry, is avoided as much as possible. We will instead approach geometrically interesting problems using tools from the standard fundamental courses in analysis and linear algebra. It is possible to raise interesting questions even about objects as “simple” as plane curves. The proof of the four-vertex theorem, for example, is anything but trivial.

The book is suitable for students from the second year of study onwards and can be used in lectures, seminars, or for private study.

The first chapter is interesting mostly for historical reasons. The reader can here find out how geometric results have been obtained from axioms for thousands of years, since Euclid. In particular, the controversy about the parallel axiom will be explained. In this chapter we will mostly follow Hilbert's presentation of plane geometry, since it is rather close to Euclid's formulation of the axioms and yet meets today's requirements for mathematical rigour. In the mean time the axiomatic system has been simplified significantly [2]. A presentation with only seven axioms can be found in [30].