Differential Geometry

As mentioned in Chapter 13, calculus made it possible to study nonalgebraic curves: the “mechanical” curves, or

transcendental

curves as we now call them. Calculus computes not only their basic features, such as tangents and area, but also more sophisticated properties such as

curvature

. Curvature turns out to be a fundamental concept of geometry, not only for curves, but also for higher-dimensional objects. The concept of curvature is particularly interesting for surfaces, because it can be defined

intrinsically

. The intrinsic curvature, or

Gaussian

curvature as it is known, is unaltered by bending the surface, so it can be defined without reference to the surrounding space. This opens the possibility of studying the

intrinsic

surface geometry. On any smooth surface one can define the distance between any two points (sufficiently close together), and hence “lines” (curves of shortest length), angles, areas, and so on. The question then arises, to what extent does the intrinsic geometry of a curved surface resemble the classical geometry of the plane? For surfaces of

constant

curvature, the difference is reflected in two of Euclid’s axioms: the axiom that straight lines are infinite, and the parallel axiom. On surfaces of constant positive curvature, such as the sphere, all lines are finite and there are no parallels. On surfaces of zero curvature there may also be finite straight lines; but if all straight lines are infinite the parallel axiom holds. The most interesting case is constant negative curvature, because it leads to a realization of

non-Euclidean geometry

, as we will see in Chapter 18.