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In other words, dGH (X, Y) = inf{ dH(X, Y)}, where the infimum is taken over all (semi-)metrics on X U Y whieh extend the met ries on X and Y. A direct application of the definition of Gromov-Hausdorff distance requires eonstructing a new metrie space Z (or ametrie on X UY) and verifying the triangle inequality. This involves cumbersome details even in simple cases. It would be more eonvenient if we eould compute or estimate dGH (X, Y) by eomparing the distanees within X and Y to each other, as we did in the eases of uniform and Lipschitz distance.

One can then prove the following: dG H (X, Y) < r if and only if there is a correspondence between X and Y such that if x, x' E X and y, y' E Y are corresponding pairs of points, then Id x (x, x') - dy(y, y') I < 2r. We omit more precise definitions and formulations. While the next criterion does not give explicit expressions for the GromovHausdorff distances, it does provide another quantity which differs from the distance by no more than a factor of two. Note that an estimate of this type is suffieient to study the topology (on the space of metrie spaces) determined by the Gromov-Hausdorff distanee.

The latter vector field is linearly isomorphie to the linear space of initial conditions V(O) = 0, D t V(O) = a. Since the tangent and normal components (with respect to i') of a Jacobi field are again Jacobi fields and the tangent component is a vector field of the form V T = (ct + d)'Y (where c and d are constants), we will restrict our attention to Jacobi fields orthogonal to i'. 3 Surfaces of Constant Curvature If the sectional curvatures of a Riemannian manifold are equal to some constant k, then R(V, T)T = k( (T, T) V - (V, T) T) = kV (since (T, T) = 1 and (V, T) = 0).

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