Some properties of tangent spaces to metric spaces

The recent achievements in the metric space theory are closely related to some generalizations of differentiation. See, for instance, the concept of weak gradient [6] or Cheeger's notion of differentiability for Rademacher's theorem in certain metric measure spaces [1].

A tangent space to an arbitrary metric space X at a point p from X was defined in [5] as a factor space of some family of sequences of points of X which converge to p. This approach makes possible to define a metric space valued derivative of functions f from X to Y, X and Y are metric spaces, as mappings between tangent spaces to X and, respectively, to Y [2].

In the lecture will be discussed some examples of tangent spaces to metric spaces and, moreover, will be described the metric spaces which have bounded or proper or compact tangent spaces [3] and metric spaces having ultrametric tangent spaces [4].

References
[1] R. Cheeger, Differentiability of Lipschits functions on metric measure spaces // GAFA, Geom. Func. Anal. 9, 428-517, 1999.
[2] O. Dovgoshey, Tangent spaces to metric spaces and to their subspaces // Ukrain. Math. Bull. 5 (4), (2008), 468-485.
[3] O. Dovgoshey, F. Abdullayev, M. Kucukaslan, Compact tangent metric spaces // Reports in Math. University of Helsinki, 482, (2008), 33 p.
[4] O. Dovgoshey, O. Dordovskyi, Ultrametricity and metric beetwenness in tangent spaces to metric spaces (submitted).
[5] O. Dovgoshey, O. Martio, Tangent spaces to metric spaces // Report in Math. Universtiy of Helsinki, 480, (2008), 20 p.
[6] J. Heinonen, Lectures on Analyisis on Metric Spaces, Springer-Verlag, NewYork, 2001.