# Ossian Bonnet and differential geometry

Agustí Reventós Tarrida, of the Universitat Autònoma de Barcelona wrote Notes sobre els inicis històrics de la geometria diferencial in which he describes Ossian Bonnet's contributions to differential geometry. We give below an English translation of an extract.

Pierre Ossian Bonnet (1819-1892)

He was born in Montpellier. He entered the École Polytechnique in 1833 and became a teacher in 1844. From 1868 he was Chasles' assistant at the same school. In 1878 he moved to the Sorbonne. In 1883 he replaced Liouville as a member of the Bureau des Longitudes. He introduced geodesic curvature which allowed him to generalise Gauss's defect theorem to triangles with non- geodesic sides. It is what is currently known as the Gauss-Bonnet Theorem.

Independently of Minding showed that geodesic curvature is an intrinsic property of the surface (invariant by flexions).

In 1844 he published Sur quelques propriétés générales des surfaces et des lignes tracées sur les surfaces. He writes:
The illustrious geometer Gauss makes use, in the fine work 'Disquisitiones', of analytical considerations which are very interesting and very elegant, but which perhaps leave something to be desired in terms of simplicity. I proposed to ... take up the same questions using the methods of pure geometry.
In 1845 he published three articles in the same volume, 18, of the Journal de l'Ecole Polytechnique: Mémoire sur les surfaces isothermes orthogonales, Note sur les ombilics des surfaces, and one of more physical content, Mémoire sur la théorie des corps élastiques. In the note on the umbilical points he comments that according to some authors, infinite lines of curvature (Monge) pass through these umbilical points and according to others only a finite number (Dupin). But that these points of view can be reconciled by introducing the concept of lines of curvature of first and second kind. He gives a geometric definition of these of second kind from the idea that the normals on a surface do not intersect, and characterises them as follows:
... if we consider what we may call the second curvature of the normal sections, that is to say the derivative of the first curvature with respect to the arc, we can consider the principal sections relative to this second curvature, and it will happen, which, I believe, has not been noted, that the direction of these principal sections will be that of the lines of curvature of the second kind.
In the note on elastic bodies he finds by a simpler method the results of Lamé in Mémoire sur les surfaces isostatiques dans les corps solides homogènes en équilibre d'élasticité.

In 1848 he published Mémoire sur la théorie générale des surfaces which is where Gauss-Bonnet's first theorem first appears, and in 1849 he returned to isothermal surfaces in Sur les surfaces isothermes et orthogonales. Here he gives a simpler proof of Lamé's theorem which says that, if we dispense with cylindrical and conical surfaces, the only triple systems of isothermal and orthogonal surfaces are the systems formed by homofocal second-degree surfaces. He says that he himself had already given a first simplification in volume XXX of the Journal de l'École Polytechnique. He also says that this article is an extract from a prior note in Comptes Rendues.

In 1851 he published Note sur quelques points de la théorie des surfaces and Note sur la théorie générale des surfaces. This last work is the result of having read Liouville's notes in the fifth edition of the Applications of Monge. He begins by saying that some of Liuoville's formulas already appear in his work of 1844. He cites, in particular, a couple of expressions of geodesic curvature. He also says that M Chelini had already pointed out that Liouville's results were easily deduced from those of Bonnet, but the way he sees it does not seem to him the simplest and he gives a new proof.

In 1852 he published Sur la théorie mathématique des cartes géographiques, which can be considered as a summary of the state of affairs (difficulty in making maps) at that time, with the contributions of Lagrange, Euler, and especially de Gauss. He himself already says that this work offers nothing essentially new.

In 1853 he published Mémoire sur les surfaces dont les lignes de courbure sont planes ou sphériques. This long work has four parts: I. Sur les surfaces dont toutes les lignes de courbure sont planes; II. Sur les surfaces dont les lignes de l'une des courbures sont planes; III. Des surfaces dont les lignes de courbure sont planes dans un système et sphériques dans l'autre, ou bien sphériques dans les deux systèmes; IV. Sur les surfaces dont les lignes de l'une des courbures sont sphériques.

We cite below several short notes from that period in the Comptes Rendues: in the same 1853 volume he published Note sur la théorie générale des surfaces; four notes on surfaces with flat or spherical lines of curvature, which give rise to the paper in the Journal de l'École Polytechnique which we discussed earlier; in 1855 he published Sur la détermination des fonctions arbitraires qui entrent dans l'équation générale des surfaces à aire minimum, where he answers Frenet's question about minimum area surfaces passing through one or more parallel lines on the same plane, and Observations sur les surfaces minima, a brief note in which he comments that Joachimsthal could not know of his work of 1853, and Sur les lignes géodésiques; in 1856 he published Nouvelles remarques sur les surfaces à aire minima; Sur les surfaces pour lesquelles la somme des deux principaux rayons de courbure est égale au double de la normale; Note sur un genre particulier des surfaces réciproques; Sur les surfaces dont toutes les lignes de courbure sont planes, and Note sur la courbure géodésique.

In 1858 he published a one-page note on ruled surfaces. He says that since M Bertrand has quoted, in a previous session of the Academy, three theorems of his, he asks permission from the Academy to communicate to M Bertrand another theorem, which he has known for some time, but which he has not published. It refers to how Gaussian and mean curvatures vary along generators. Interestingly he defines surface curvature as the geometric mean of the principal curvatures (the square root of Gaussian curvature). Thus the two curvatures he manipulates are the arithmetic and geometric means of the principal curvatures.

In 1862 he published Mémoire sur les surfaces orthogonales, in two parts in the same volume of the Comptes Rendues, where he studied the problem of determining triple orthogonal surface systems. He says the importance of this problem comes from Lamé's happy use of curvilinear coordinates. He says that at present the only known examples are the homofocal second-degree surface systems formerly discovered by Binet, the examples given by Serret in volume XII of the Journal de Liouville, and the examples that M W Roberts has deduced from the consideration of elliptic coordinates. He then said:
I have managed to take the issue in a new direction, which has led me to results of unexpected generality and scope.
In 1864 he was already studying the theorem that would later bear his name (when the triangles are not geodesic): Démonstration du théorème de Gauss relatif aux petits triangles géodésiques tracés sur une surface courbe quelconque. He says that in the 'Disquisitiones' of Gauss he arrives at this formula after long and complicated calculations, and, as far as he knows, no mathematician has tried to reduce it to what is essential:
So I'm thinking of doing something useful by posting a new demonstration here that seems simple to me and also has the advantage of leading directly to the goal.
In 1865 he published Mémoire sur la théorie des surfaces applicables sur une surface donnée, first part. This first part is subtitled Démonstration du théorème de Gauss. Méthode pour reconnaître si deux surfaces données sont ou ne sont pas applicables l'une sur l'autre.

In 1867 he published, with the same title, the second part of Mémoire sur la théorie des surfaces applicables sur une surface donnée, and this second part was subtitled Détermination de toutes les surfaces applicables sur une surface donnée. On page 31 he is self-critical and in a note entitled Addition au mémoire précédent writes:
The solution of the general problem of the deformation of surfaces which we gave in the preceding Memoir presents in certain respects a real gap. We have assumed that we know the variables x and y for which the linear element of the given surface takes the simple form $\phi^{2} dx dy$. Now we know that the determination of these variables requires the integration of a first order differential equation; thus if this integration cannot be carried out, one will be stopped at the beginning in the setting up of the equation of the problem. The two solutions which M Bour has given in his great and important work on the deformation of surfaces present the same drawback. M Codazzi, in his beautiful Mémoire which obtained an honourable mention in the competition of 1860, has resolved the question in a more satisfactory manner, by making no hypothesis as to the nature of the variables by means of which the rectangular coordinates of the different points of the given surface.
On page 35 he states and proves Bonnet's famous theorem on the existence of a surface with given first and second fundamental forms. But his statement reads thus:
Before we go any further, let us show how we calculate the rectangular coordinates of the points of a surface for which we know, as a function of u and v, the eight quantities f; g; M; N; P; Q; R; S, and prove that to any system of values of these quantities, satisfying of course equations (3), (4), (5), corresponds one and only one surface.
In 1885 the Bulletin des Sciences Mathématiques published an excerpt from a letter from Bonnet to Darboux, Sur la surface réglée minima, in which he reproduced the demonstration he explained in his courses at the Sorbonne that the helicoid (left of directrix plane) is the only ruled minimum surface.

When he was almost 70 years old, he became nervous when he read a note by M Paraf, entitled Sur deux théorèmes de Jacobi relatifs aux lignes géodésiques and could not resist the temptation to add a note, Remarque sur une communication de M Paraf, where he says that these results had been obtained by him thirty years earlier, and cites Deuxième note sur les lignes géodésiques and Sur quelques propriétés des lignes géodésiques. This whole issue is related to whether geodesics are minimal in length or not.

One of his three sons, Georges, also entered the École Polytechnique, and published a note in the Comptes Rendues, Démonstration nouvelle de deux théorèmes de M Bertrand, where he uses "the following propositions due to M Ossian Bonnet, my father." Calculates the length of a geodesic circle on a surface by neglecting fourth-order terms. He also published a note, without much of interest, entitled Démonstration des proprietés fondamentales du système de coordonnèes polaires géodésiques.

Last Updated September 2020