We show that any two geometric triangulations of a closed hyperbolic, spherical, or Euclidean manifold are related by a sequence of Pachner moves and barycentric subdivisions of bounded length. This bound is in terms of the dimension of the manifold, the number of top dimensional simplexes, and bound on the lengths of edges of the triangulation. This leads to an algorithm to check from the combinatorics of the triangulation and bounds on lengths of edges, if two geometrically triangulated closed hyperbolic or low dimensional spherical manifolds are isometric or not.

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Pachner的上界移动与几何三角剖分有关

我们显示出，封闭的双曲，球面或欧几里德流形的任何两个几何三角剖分都与一系列Pachner移动和有界长度的重心细分相关。此边界取决于歧管的尺寸，顶级单形的数量，并取决于三角剖分的边的长度。这导致一种算法，可以从三角剖分的组合和边缘长度的边界检查两个几何三角剖分的封闭双曲线或低维球形歧管是否等距。