Topics on calculus in metric measure spaces

A. Alexandrov [1] introduced the notion of lower curvature bounds for metric spaces in terms of comparison properties for geodesic triangles. These curvature bounds are equivalent

Our aim here is to introduce a notion of tangent cone to a metric space and a notion of tangent map to a map between metric spaces.. Such notions enable one to obtain information

THEOREM 10.. — Assume that the system X of vector fields on R 71 is such that the identity map gives a homeomorphism between the Carnot-Caratheodory metric p and the Euclidean

This is a survey of recent developments in the area of lower Ricci curvature bounds for both geodesic metric measure spaces, and discrete metric measure spaces, from a

The second definition of Sobolev spaces with zero boundary values has been extended to the metric setting, in the case of Newtonian functions based on Lebesgue spaces [22], on

The final step in obtaining the coarea formula for mappings defined on an H n -rectifiable metric space is the proof of the area formula on level sets:.. Theorem 7 (The Area Formula

We define “tangent” spaces at a point of a metric space as a certain quotient space of the sequences which converge to this point and after that introduce the “derivatives”

By retaining the notion of a contraction in the sense of Banach and merely modifying the space we can obtain many modifications of the contraction