On the theory of the geometric object

In 1936 Johannes Haantjes and Jan Schouten were co-authors of the article On the theory of the geometric object which was published in the Proceedings of the Edinburgh Mathematical Society. The authors begin the article by reviewing the history of the idea of a "geometric object." We give a version of the start of the Introduction below:

On the theory of the geometric object

By J A Schouten and J Haantjes.

(Communicated by G Temple.)
[Received 2 July, 1936. - Read 12 November, 1936.]


Modern geometry is based on the idea of the geometric object. This idea is introduced by F Klein in the following way. If we have different coordinate systems in a space, then there may be systems of quantities (Grossen), functions of these coordinates, with the property that they transform into themselves with a change of the coordinate system, viz. that the new quantities depend only on the old quantities and the transformation of the coordinates "without it being necessary to use the values of the coordinates themselves" (ohne dass man die Koordinatenwerte selbst hinzuzunehmen braucht). Such systems Klein calls 'geometrische Gebilde' [see F Klein, Elementarmathematik vom höheren Standpunkte aus (1909), 59]. Now this definition is fairly good as long as we consider only linear transformations of coordinates as Klein did, but, as we shall see, the restriction that the values of the coordinates have not to be used leads to difficulties, if the transformations are taken more generally.

In 1926 O Veblen and J M Thomas [in Projective invariants of affine geometry of paths, Ann. of Math. 27 (1926), 278-296] defined an "invariant" in the following way:

"An invariant ... is an entity with definite determining components in any coordinate system, such that the transformations of the components from one coordinate system to another form a group, isomorphic with the group of analytic transformations of the coordinates".
In 1927 O Veblen [Invariants of quadratic differential forms, Cambridge Tracts, No. 24 (1927), 14] gave the definition:
"An object of any sort, which is not changed by transformations of coordinates, is called an invariant".
In 1932 O Veblen and J. H. C. Whitehead [The foundations of differential geometry, Cambridge Tracts, No. 29 (1932), 46], adopting the expression "geometric object" proposed by J A Schouten and E R van Kampen [Zur Einbettungs- und Krummungstheorie nichtholonomer Gebilde, Math. Annalen, 103 (1930), 752-783] in 1930, gave the definition:
"Anything which is unaltered by transformations of coordinates is called an invariant ... Thus a point is an invariant and so is a curve or a system of curves. Also, strictly speaking, anything, such as a plant or an animal, which is unrelated to the space which we are talking about, is an invariant. For an invariant, which is related to the space, i.e. a property of the space ..., we shall also use the term geometric object".
The last three definitions being rather vague, J A Schouten and D v. Dantzig [Was ist Geometrie?, Mitt. des Seminars für Vektor- und Tensoranalysis der Moskauer Universität, 2-3 (1935), 15-48 (19)] tried to give a more exact definition:
"A geometric object is a system of functions of the coordinates, called components of the object, that transforms with a transformation of the coordinates 'in itself', viz. in such a way, that the new components depend only on the old components and the function of transformation".
This definition is nearly the same as that of F Klein; the only difference is that the new components are expressly said to be functionals of the function of transformation, and that the restriction with regard to the non-using of the coordinates themselves is dropped, because no longer are only linear transformations of coordinates considered.

All these definitions have this in common, that they only try to give at the beginning of the deduction some central point of view of some geometry and not an axiom from which a theory of objects may be developed. The first writer who really tried to establish a theory of objects was A Wundheiler in a lecture delivered at the Congress in Moscow in 1934 [Objekte, Invarianten und Klassifikation der Geometrien, Congress for tensorial differential geometry, Moskau, 1934].

JOC/EFR November 2019