Applications of Quasigeodesics and Gradient Curves

This paper gathers together some applications of quasigeodesic and gradient curves. After a discussion of extremal subsets, we give a proof of the Gluing Theorem for multidimensional Alexandrov spaces, and a proof of the Radius Sphere Theorem. This paper can be considered as a continuation of [Perelman and Petrunin 1994]. It gathers together some applications of quasigeodesic and gradient curves. The first section considers extremal subsets; in the second section we prove the Gluing Theorem for multidimensional Alexandrov spaces; in the third we give another proof of the Radius Sphere Theorem. Our terminology and notation are those of [Perelman and Petrunin 1994] and [Burago et al. 1992]. We usually formulate the results for general Alexandrov space, but for simplicity give proofs only for nonnegative curvature. Notation. We denote by M a complete n-dimensional Alexandrov space of curvature ≥ k. As in [Burago et al. 1992], we denote by p′q the direction at q of a shortest path to p. If H is a subset of M and p, q ∈ H , we denote by |pq|H the distance between p and q in the intrinsic metric of H . Finally, if X is a metric space with metric ρ, we denote by X/c denote the space X with metric ρ/c; where no confusion will arise, we may use the same notation for points in X and their images in X/c. 1. Intrinsic Metric of Extremal Subsets The notion of an extremal subset was introduced in [Perelman and Petrunin 1993, 1.1], and has turned out to be very important for the geometry of Alexandrov spaces. It gives a natural stratification of an Alexandrov space into open topological manifolds. Also, as is shown in recent results of G Perelman, extremal subsets in some sense account for the singular behavior of collapse. Therefore This material is part of the author’s Ph.D. thesis [Petrunin 1995].