Robert D. Richtmyer

Ordinary Differential Operators

The basic theory of second-order ordinary differential operators, largely due to Hermann Weyl, is summarized in this chapter.

Tempered Distributions and Fourier Transforms

Tempered distributions, as functionals, are continuous with respect to a slightly weaker mode of convergence than that described in Section 2.4, hence are slightly milder, as a class, than the class of all Schwartzian distributions. The mildness is not a local matter (the 03B4 function and all its derivatives are tempered distributions), but has to do with the behavior as ∥x∥ → ∞. The Fourier transform of a tempered distribution is easily defined; it is also a tempered distribution.

Compact, Hilbert-Schmidt, and Trace-Class Operators

This chapter deals with several classes of bounded operators: the compact, Hilbert-Schmidt, trace-class and degenerate operators; they are related by the inclusions bounded ⊃ compact ⊃ Hilbert-Schmidt ⊃ trace-class ⊃ degenerate.

Linear Operators in a Hilbert Space

The idea of a linear operator or transformation in a Hilbert space ℌ (or a Banach space) is a direct generalization of the idea of a linear transformation in a finite-dimensional space. One point, however, needs emphasis (mainly because it is sometimes ignored, especially in books on quantum mechanics), namely, an operator A cannot be regarded as fully specified until its domain of definition (i.e., the set of those x in ℌ for which Ax is meaningful) has been specified; operators with different domains of definition have to be regarded as different operators. It is customary to require the domain of definition to be a linear set (manifold) in ℌ, for the obvious reason that if A is linear and Ax is defined in a set S, then Ay can be uniquely defined, by linearity, when y is any finite linear combination of elements of S. However, further extensions are not generally unique, except in special circumstances.

Local Properties of Distributions

Although a distribution does not have a definite value at a specified value x of its argument, one can discuss its properties in any arbitrarily small neighborhood of x. Such local properties are discussed in this chapter.

Consistency and Categoricalness of the Hyperbolic Axioms; The Classical Models

We take as primary model of the hyperbolic plane an abstract surface S in the sense of Section 5.3, whose geometry is determined by the methods of differential geometry in such a way that all the axioms of the hyperbolic plane are satisfied. We consider several different coordinate systems, each of which covers the entire surface S, some of which are more useful than others for certain purposes. Each coordinate system leads to one of the classical models of the hyperbolic plane based on Euclidean geometry. Differential geometry is based on analysis, which is based on the real number system ℝ. It follows that the hyperbolic axioms are consistent, if the axioms of ℝ are consistent. It is proved that the axiom system is categorical, in the that any model of the hyperbolic plane is isomorphic to any other model. Lastly, as an amusement, we describe a hyperbolic model of the Euclidean plane.

Some Neutral Theorems of Plane Geometry

We discuss some of those theorems of Euclidean plane geometry that are independent of the parallel axiom. They will be needed in the development of hyperbolic geometry. We assume they are more or less known, so that our treatment is not as complete as a full treatment of Euclidean geometry would be. Some of the theorems are weaker than the corresponding Euclidean ones, because the more complete form would require the parallel axiom. Some of them go a little beyond Euclid in that they use the notion of continuity as it appears in calculus. Results in the last two sections go beyond Euclid in that the ideas in them are more recent, as in the Jordan Curve Theorem or the study of isometries.

ℍ3 and Euclidean Approximations in ℍ2

To investigate the fine structure of hyperbolic geometry, we use a method due essentially to Lobachevski. It turns out that the geometry of the horosphere, a certain surface in three-dimensional hyperbolic space ℍ3, is Euclidean if the definitions are made as follows. The “straight lines” of that geometry are the horocycles in that surface, the “distances” are the arclengths between points along horocycles lying in that surface, and the “angle” between two intersecting horocycles is the angle between the tangent vectors. In that sense, the horosphere, together with the geometry that it inherits from the ℍ3 in which it is embedded, is a model of the Euclidean plane. In it, parallel horocycles are equidistant throughout their entire lengths; there are Cartesian coordinates; and so on. All the formulas and relations of Euclidean geometry apply. Therefore, by projecting small figures in that surface onto a tangent plane in ℍ3, we find that in that plane, for small figures near the point of tangency, the Euclidean formulas hold approximately, with relative error terms that tend to zero as the sizes of the figures tend to zero. In order to carry out this approach, we first develop three-dimensional hyperbolic geometry, which, however, has independent interest. We give a set of axioms, which, like those of the hyperbolic plane, are user friendly. The first seven axioms are almost identical with those of the hyperbolic plane and the last three deal with planes in ℍ3. We then develop a set of theorems of the geometry of ℍ3, and we prove that the geometry inherited by a horosphere is Euclidean.

Differential Geometry of Surfaces

This chapter is intended to help the reader learn some of the concepts of differential geometry as they appear in terms of coordinates and also to move toward thinking of them in intrinsic terms (‘coordinate free’), which is the modern approach to differential geometry. Our primary interest is in the hyperbolic plane as a Riemannian manifold.

Qualitative Description of the Hyperbolic Plane

The hyperbolic plane is qualitatively different from the Euclidean plane in a number of ways. Among these are the fact that the sum of the angles of a triangle is strictly less than π (radians). The difference between π and the sum is called the defect of the triangle and is proportional to the area of the triangle, so the areas of triangles are bounded above. It is true in the hyperbolic plane that the length of a circular arc and the area of a circular sector are both proportional to the angle of the arc. What is different is the way the proportionality depends on the radius of the circle. Another difference is in the structure of the group of isometries, which are mappings of the plane that preserve distance. Since tilings and lattices are tied so closely to the isometries of the plane, they are also very different from the Euclidean case. All of the distinctive properties can be traced to the hyperbolic parallel axiom, so we begin with the theory of parallelism, presented in roughly the form in which Gauss derived it.