The double-elliptic case of the Lie-Riemann-Helmholtz-Hilbert problem of the foundations of geometry

This thesis embodies the results of more than two years research on the foundations of geometry. As indicated by Hilbert in his paper "Ueber die Grundlagen der Geometrie," the researches of Riemann and Helmholtz interested Lie in an axiomatic treatment of the foundations of geometry, with an emphasis upon the concept of groups. Lie formulated a set of axioms which, as he showed by means of his Theory of Transformation Groups, was sufficient for the foundations of geometry. However, in this theory, Lie assumed that the functions which define the group have derivatives, and hence he does not settle the question whether the differentiability of these functions is a consequence of the group concept and the remaining axioms. Also, Lie is compelled in the course of his work to state explicitly the axiom that the group of motions is generated by infinitesimal transformations. These conditions, as well as other assumptions made by Lie, can be expressed in a pure geometrical form only in a complicated way, and seen to be necessitated only by the method used by Lie, and not by the problem itself. In his paper Hilbert formulates a set of axioms concerning a group of motions which is sufficient to necessitate that this group should be simply isomorphic with either the Euclidian or the Bolyai-Lobatschefskian group of rigid motions in a plane. This set of axioms involves only simple, obviously geometrical conditions, and does not require the differentiability of the functions which give the motions. In his paper "On the Lie-Riemann-Helmholtz-Hilbert Problem of the Foundations of Geometry," R. L. Moore gives a treatment in which this assumption is not made in advance, but in which there is a simultaneous analysis of the group of transformations and of the space which undergoes this transformation. In this paper we shall give a similar analysis for the Double-Elliptic case. After a group of preliminary theorems we shall prove that every motion distinct from the identity leaves fixed exactly two points, which we shall call poles. We shall then introduce the notions of great circles, intervals, congruence of intervals, of triangles, and of angles. We then encounter the problem noted by Hilbert at the end of his paper; that is, the problem of proving the congruence of the base angles of an isosceles triangle. Hilbert has solved this problem for the Euclidean case. The treatment for the Elliptic case seems even more difficult. In the solution of this problem we shall show that we can follow the treatment of Young in deriving the Non-Euclidian Trigonometry, and in particular the formulas for the solution of triangles; these formulas will enable us to prove the desired result