Chapter 5 Riemann’s Geometric Function Theory

See Neuenschwander (1981b, 91), citing Library records.

On von Bezold, see Laugwitz (1999, 32). He told Fuchs of his notes, in Gabelsberger shorthand, when they were colleagues in Berlin in the 1890s and Fuchs arranged for them to be transcribed into ordinary German script.

Quoted in Laugwitz (1999, 32), originally published in Neuenschwander (1987, 9).

The paper (1854a) was on trigonometric series, the accompanying lecture (1854b) on the foundations of geometry, both were published only in 1867.

Quoted in Laugwitz (1999, 29), who noted that the information was omitted from Dedekind’s later biography of Riemann, probably out of respect for Riemann’s widow Elise. He also speculated that Riemann’s depression seems to have persisted to the end of his life.

For the influence of Herbart on Riemann’s philosophical ideas, see Scholz (1982).

For a discussion of Riemann’s physics, see Bottazzini and Tazzioli (1995); the unWeberian paper is Riemann’s posthumous paper on electrodynamics, Riemann (1858).

n -fach ausgedehnte Grösse, see Riemann (1854b, Sect. 1).

See our discussion of Hermite’s ideas on p. 530.

See Neuenschwander (1981a,b) and the references cited there, as well as his later papers.

The letter was first published in Loria (1915) in a obituary of Tardy, and then in Weil (1979b). Brill and Noether (1894, 254–255) also document interesting examples of ideas that anticipated the idea of cuts in the work of Kirchhoff in 1848 and Helmholtz in 1853.

A set of the relevant lecture notes was published as Dirichlet (1904). The theorem is stated and proved on p. 4ff.

Riemann used no specific word for this property.

The theorem was first stated by Montel (1913). Looman produced a proof in 1923 that, however, had a gap that was eventually filled in (Menchoff 1936). For a modern proof (Narasimhan and Nievergelt 2001, 48–50).

According to Brill and Noether (1894, 259) the concepts of multi-sheeted surface and of crosscuts were first introduced “in essence” in an (unpublished) note on a problem of electrostatic or thermal equilibrium on the surface of a cylinder with circular crosscuts. Referring to Riemann (1876b) they wrote that “We are inclined to call this note one of Riemann’s earliest works, or rather its way of thought the starting point for Riemann’s works on function theory”.

The awkward language shows how little point-set topology was available at this time.

A Laurent tail, as we would call it; Riemann gave it no special name.

Nowadays called a local uniformising parameter.

See, for example, the “source” book on its history, Monna (1975), and the discussions in Bottazzini, The Higher Calculus (1986) and Gray (2000a).

Dirichlet in fact stated this principle for three-dimensional domains (Dirichlet 1876, 127–128).

Klein Nachlass NSUB Göttingen, Prym–Klein, quoted in (Bottazzini 2003, 233).

This translation is taken from Riemann (2004, 35–36).

Riemann here added this footnote: “The dependence expressed here denotes dependence via a finite or infinite number of the four simplest operations, addition, multiplication, subtraction, and division. The expression ‘operation on quantities’ (by contrast to ‘operations on numbers’) indicates operations in which the rationality of the quantities does not play a role”.

The regions are assumed to have more than one point on their boundaries, but the nature of the boundary is nowhere discussed. It is unlikely Riemann had anything other than a simple closed curve in mind, since he took one region to be a disc. See Gray (1994).

If the region T is a subset of the plane, it is enough to consider the function log(z − z ₀).

This was confirmed by Riemann himself in a letter to his brother Wilhelm on November 24th, 1851. See Neuenschwander (1981a, 104).

See Stahl (1896) and Riemann (1899).

Another glimpse is given in the four pages of notes described in (Elstrodt and Ullrich 1999).

Instead Roch gave a careful account of the number e (defined as lim(1 + 1 ∕ x) ^x for x = ∞) and the exponential function.

Plainly the condition f(z)(z − a) ^m − 1 = ∞ is tacitly supposed.

The historians’ first problem with these notes, however, is dating them. Immediately beneath the heading \(\frac{55} {56}\) is a reference to Briot and Bouquet’s book of 1859, which is confusing enough. However, the legible script is soon replaced by the traditional, and often impenetrable, German hand used for the bulk of the notes. The legible script returns on pages 15, 42, and 43 for a few more references, for example, to Cauchy and Hermite, but nothing that would betray a date. It would seem therefore that the legible script was chosen for the titles of works in French and that the notes date from after 1859. But then one finds a clean start is made on page 111 with the date 1856. From then on the notes are the basis of Stahl’s version of Riemann’s theory of elliptic and Abelian functions, which Stahl himself said derived from Riemann’s lectures of 1856 and 1861. The title of the lecture course is closest to the one listed in Riemann’s Werke for 1861, which would make them Hattendorff’s copy. The simplest interpretation is that these notes are of a course Riemann gave in 1855/56 with later additions made by Schering for his own use.

Casorati recorded his questions and Roch’s answers in separate sheets held in Casorati’s Nachlass in Pavia. A picture of one of them is given in Neuenschwander (1978b, 19). It deals with the Riemann surface associated with the equation s ³ − s + z = 0 and offers an intuitive drawing of it, see Fig. 5.3.

His sketchy argument was later refined in Tonelli (1875).

See Riemann (1857c, 134).

Hurwitz’s contribution is the formula 2P − 2 = w + n(2p − 2), where P is the genus of a Riemann surface that is an n-fold cover of a Riemann surface of genus p branched at points \(a_{1},a_{2},\ldots,a_{w}\) at which \(c_{1},c_{2},\ldots,c_{w}\) leaves come together, so the total branching order is \(W = (n - c_{1}) + (n - c_{2}) + \cdots + (n - c_{w})\). It is given in Hurwitz (1893, 416), reprinted in Mathematische Werke, I, nr. XXIII, 391–430, see p. 404, not in XXI, p. 376 as stated in (Freudenthal, 1972, 572).

See Mumford (1975).

Monodromy matrices were used in Hermite (1851, 279), the term monodromy group first appears in Jordan (1870, 278).

In this connection, see Wirtinger’s ICM 1904 address (Wirtinger 1905) for an account of just how many later discoveries were foreshadowed by Riemann’s work on the hypergeometric equation.

See Riemann (1990, 19).

See Edwards (1974, 18). The necessary argument was first provided in Hadamard (1893).

This estimate was not to be proved until (von Mangoldt 1905).

It is one of the few Hilbert problems to resist solution, and is now one of the million dollar prizes offered by the Clay Mathematics Institute.

The term by term evaluation was first shown to be permissible in Landau (1908).

See the books by Titchmarsh, Edwards, and Patterson, and the more popular accounts by Derbyshire and du Sautoy.

For a commentary, see Gray (2006).

See their papers listed in the Bibliography.

See their papers listed in the Bibliography.

See Gray (1998).

Plücker, von Staudt and other geometers dealt with them in terms of automorphisms of the surface of period 2 with no real fixed points, see Gray (1994) and below, p. 127.

Thus Klein wrote in his Entwicklung, (1926–1927, 1, 264), that Helmholtz once said to him “For us physicists, Dirichlet’s principle remains a proof”. On Maxwell, see his reliance on Green’s work in his A Treatise on Electricity and Magnetism, Chapter IV.

Friedrich Prym took his doctorate in 1863 officially from the University of Berlin although he is best regarded as a student of Riemann’s. He became a professor for some years in Zurich before becoming a professor at Würzburg in 1869, where he remained until 1909. He worked for most of his life on the theory of theta functions of several variables and a related class of functions today called Prym functions in his honour.

For an instructive comparison of Prym’s counter-example and the later and much better-known work of Hadamard on this topic, see Maz’ya and Shaposhnikova (1998, 373–377).

Schlömilch, ever the active editor, eventually published half of Roch’s 18 papers. Schlömilch had administrative responsibility for instruction at schools and colleges in Saxony and was an advisor to the publishing house of Teubner, Leipzig; see Stubhaug (2002, 333).

Some of this information comes from the web-site http://​www.​mathematik.​uni-halle.​de/​history/​roch/​index.​html.

See Clebsch and Gordan (1866, v–vi).

Roch (1866b, 39).

See Roch’s review (Roch 1866c).

See Simart (1882).

This information comes from Gispert (1991, 325–338), the report on Poincaré’s thesis is on p. 331, the report on Simart’s thesis is on p. 335.

Dini, Arzelà, Pincherle, Bianchi, Ricci Curbastro, Pieri, Volterra and Enriques were among them.

The relevant letters are kept in Betti’s Nachlass in Pisa. See Bottazzini (1977a).

See Bölling (1993, 253). This letter and the letters of Schwarz to Weierstrass we will quote below are kept in the library of the Mittag-Leffler Institute. We would like to thank the former Director of the Institute, Dr. Laksov, for his kind permission to publish excerpts of them.

This paper originated in a course on this subject Betti gave in 1861/62. He lectured on this subject several more times in the 1860s. Lecture notes for his 1862/63 course, taken by Ulisse Dini and held in the library of the Department of Mathematics of the University of Florence, have been published in Petti (2002). The same library also holds Antonio Roiti’s notes of Betti’s 1867/68 course, published in Petti (2003).

“I’ve spoken with Riemann once again about the connection of spaces and I’ve got an exact idea of it”, Betti wrote to Tardy on Oct, 6th 1863 (quoted in Bottazzini 1983, 256). Most of the content of Betti (1870) is included in a letter to Tardy on October 16th 1863. For an English translation see Weil (1979b).

For an account of the many and various interpretations of the imaginary in geometry, see Coolidge (1924).

See Lüroth (1875, 1877), Stolz (1871), and Nabonnand (2008).

See Klein (1874), recapitulated it in his 1926–1927 Entwicklung.

For the response of the analysts see Bottazzini, The Higher Calculus. For the geometers’ response, see Gray (1989).

To be found in Riemann (1990, 519–536).

They occur in families or groupings of six pairs, any two of which reduce the equation to Plücker’s form; they are recognised as the sets of characteristics all pairs of which have the same sum. Sets of 4 bitangents determine 8 points on both the curve and a conic, sets of 6 are connected with cubics, and so forth, according to a rich geometric theory developed in Hesse (1855).

Clebsch, more precisely, considered the sums of integrals to be zero modulo the period lattice.

This is an improvement on the Weierstrass gap theorem; see Sect.​ 6.​8.​6.

Modern interpretations of Clifford’s theorem deduce from it results about the canonical forms of curves of low genus, such as every non-hyperelliptic curve of genus 3 embeds in \({\mathbb{C}\mathbb{P}}^{2}\) as a smooth quartic, every non-hyperelliptic curve of genus 4 embeds in \({\mathbb{C}\mathbb{P}}^{3}\) as a curve of degree 6 which is the intersection of a quadratic and a cubic surface, and all but an identifiable class of non-hyperelliptic curves of genus 5 embed in \({\mathbb{C}\mathbb{P}}^{4}\) as a smooth curve of degree 8 that is the intersection of three quadrics. See Miranda (1995, Chap. VII).

Gray (1989, 1998) are a start, see the references there, notably (Dieudonné 1974).