2 00 6 Analysis on Metric Space Q ∗

In this paper, we show that the metric space (Q,G) is a positivelycurved space (PC-space) in the sense of Alexandrov. We also discuss some issues like metric tangent cone and exponential map of (Q, G). Then we give a decomposition of this metric space according to the signature of points in Q. Some properties of this decomposition are shown. The second part of this paper is devoted to some basic analysis on the space (Q,G), like the tensor sum and Lp space, which can be of independent interest. In the end, we give another definition of derivative for multiplevalued functions, which is equivalent to the one used by Almgren. An interesting theorem about regular selection of multiple-valued functions which preserves the differentiability concludes this paper.