Abstract. The following is a compilation of some techniques in Alexandrov's geometry which are directly connected to convexity.

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.Remark. The mappings exp p and log p are defined in a nonunique way. The only properties of exp p we will use arex log p (q) ≥ exp p (x)q + O(|x| 2 ) , (for curvature ≥ 0 we can ignore the O-term).Remark on orientability. With respect to the Riemannian case we have an additional difficulty with the definition of orientability. It is easy to define orientability in the standard way using the atlas of distance coordinates. This atlas in general does not cover all our space, therefore for the nonorientable case we will distinguish two different cases: locally orientable and locally nonorientable. Locally orientable is if every point has an orientable neighborhood, and locally nonorientable otherwise. Every point has a ball neighborhood which is homeomorphic to the tangent cone at this point (see [P1] or [P2]), therefore local orientability is equivalent to orientability of all tangent spaces (or spaces of directions). There is an equivalent topological classification: M is locally orientable if at every point p we have H n (M, M \p) = Z, where n = dim M. Indeed from the same result of Perelman, we have H n (M, M \p) = H n−1 (Σ p ), hence if Σ p is orientable then H n (M, M \p) = Z.

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We show that almost nonnegatively curved m-manifolds are, up to finite cover, nilpotent spaces in the sense of homotopy theory and have C.m/-nilpotent fundamental groups. We also show that up to a finite cover almost nonnegatively curved manifolds are fiber bundles with simply connected fibers over nilmanifolds.

Abstract. We show that in each dimension n ≥ 10 there exist infinite sequences of homotopy equivalent but mutually non-homeomorphic closed simply connected Riemannian n -manifolds with 0 ≤ sec ≤ 1 , positive Ricci curvature and uniformly bounded diameter. We also construct open manifolds of fixed diffeomorphism type which admit infinitely many complete nonnegatively pinched metrics with souls of bounded diameter such that the souls are mutually non-homeomorphic. Finally, we construct examples of noncompact manifolds whose moduli spaces of complete metrics with sec ≥ 0 have infinitely many connected components.