On the Role Played by the Work of Ulisse Dini on Implicit Function
On the role played by the work of Ulisse Dini on implicit function theory in the modern differential geometry foundations: the case of the structure of a differentiable manifold, 1 Giuseppe Iurato Department of Physics, University of Palermo, IT Department of Mathematics and Computer Science, University of Palermo, IT 1. Introduction The structure of a differentiable manifold defines one of the most important mathematical object or entity both in pure and applied mathematics1. From a traditional historiographical viewpoint, it is well-known2 as a possible source of the modern concept of an affine differentiable manifold should be searched in the Weyl’s work3 on Riemannian surfaces, where he gave a new axiomatic description, in terms of neighborhoods4, of a Riemann surface, that is to say, in a modern terminology, of a real two-dimensional analytic differentiable manifold. Moreover, the well-known geometrical works of Gauss and Riemann5 are considered as prolegomena, respectively, to the topological and metric aspects of the structure of a differentiable manifold. All of these common claims are well-established in the History of Mathematics, as witnessed by the work of Erhard Scholz6. As it has been pointed out by the Author in [Sc, Section 2.1], there is an initial historical- epistemological problem of how to characterize a manifold, talking about a ‘’dissemination of manifold idea’’, and starting, amongst other, from the consideration of the most meaningful examples that could be taken as models of a manifold, precisely as submanifolds of a some environment space n, like some projective spaces ( m) or the zero sets of equations or inequalities under suitable non-singularity conditions, in this last case mentioning above all of the work of Enrico Betti on Combinatorial Topology [Be] (see also Section 5) but also that of R.
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