Parallel Transportation for Alexandrov Space with Curvature Bounded Below

Abstract.

In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and prove an analog of the second variation formula for this case. A closely related construction has been made for Alexandrov space with bilaterally bounded curvature by Igor Nikolaev (see [N]).¶Naturally, as we have a more general situation, the constructed transportation does not have such good properties as in the case of bilaterally bounded curvature. In particular, we cannot prove the uniqueness in any good sense. Nevertheless the constructed transportation is enough for the most important applications such as Synge's lemma and Frankel's theorem. Recently by using this parallel transportation together with techniques of harmonic functions on Alexandrov space, we have proved an isoperimetric inequality of Gromov's type.¶Author is indebted to Stephanie Alexander, Yuri Burago and Grisha Perelman for their willingness to understand, interest and important remarks.