Infinitesimal calculus in metric spaces

Abstract

We study the possibility of defining tangent vectors to a metric space at a given point and tangent maps to applications from a metric space into another metric space. Such infinitesimal concepts may help analysis in situations in which no obvious differentiable structure is at hand. Some examples are presented; our interest arises from hyperspaces in particular. Our approach is simple and relies on the selection of appropriate curves. Comparisons with other notions are briefly pointed out.

Generating and Adding Flows on Locally Complete Metric Spaces

“…Let us also mention that the generalization of the notion of differential equations from manifolds to metric spaces is a natural question. In this direction, there are many other approaches which can be found in [3], [10], [12], [13], [14] and [5]. A basic idea that all approaches have in common is to replace the concept of a vector field by a suitable family of curves (herein called an 'arc field' following [4]) each of which supplies the direction of travel at the point from which it issues.…”

What is a Lipschitzian Manifold?

“…Thus, we turn to means to detect a smaller set. Such means have been given in any metric space M either assuming M is locally compact ( [16]) or without making this assumption ([26,Def. 2.1]).…”

“…2.1]). Assuming M is endowed with a metric d defining its topology, in order to define a tangent cone T x M to M at some point x ∈ M, two conditions on a curve c in M issued from x are required in [26,Def. 2…”

“…Then a kind of tangent cone to M at x is defined as the set of equivalence classes of cadences issued from x; in [26] it is called the set V (M, x) of velocities of M at x. If f : M → M is a locally Lipschitzian map with values in another Lipschitzian manifold endowed with a metric d , a substitute to the tangent map of f can be introduced when for any velocity v and any representant c of v the curve f • c is a cadence.…”

“…Thus, one may expect that the notion of Lipschitzian parameterization cannot present the same features as the notion of Lipschitzian manifold. The recent developments of the geometric study of general metric spaces (see [7], [11], [16], [17], [18], [26] for instance) are incentives to explore such a notion. Many other incentives exist: for instance, one could explore the relationships between the tangent cone to the subjet of a convex function and the Alexandrov Theorem and the relationships between the geometry of the graph of a maximally monotone operator and the differentiability of its representative functions as defined in [28, section 9.4.5] for instance.…”