Abstract.We have built a new kind of manifolds which leads to an alternative new geometrical space. The study of the nowhere differentiable functions via a family of mean functions leads to a new characterization of this category of functions. A fluctuant manifold has been built with an appearance of a new structure on it at every scale, and we embedded into it an internal structure to transform its fluctuant geometry to a new fixed geometry. This approach leads us to what we call fractal manifold. The elements of this kind of manifold appear locally as tiny double strings, with an appearance of new structure at every step of approximation. We have obtained a variable dimensional space which is locally neither a continuum nor a discrete, but a mixture of both. Space acquires a variable geometry, it becomes explicitly dependent on the approximation process, and the geometry on it is assumed to be characterized not only by curvature, but also by the appearance of new structure at every step of approximation.

Received: March 19, 2007

AMS Subject Classification: 58A03, 58A05, 28A80

Key Words and Phrases: topos-theoretic approach to differentiable manifolds, differential geometry foundations, fractals

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 38
Issue: 2