Nilpotent structures and invariant metrics on collapsed manifolds

Let Mn be a complete Riemannian manifold of bounded curvature, say IKI 0, we put Mn = W n(c) U Fn (c), where ?1n (e) consists of those points at which the injectivity radius of the exponential map is > e. The complementary set, &"n (c) is called the e-collapsed part of Mn. If x E 4 (n), r < E, then the metric ball Bx (r) is quasi-isometric, with small distortion, to the flat ball B0(r) in the Euclidean space, Rn . After slightly

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