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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Geodesics in piecewise linear manifolds

Author: David A. Stone
Journal: Trans. Amer. Math. Soc. 215 (1976), 1-44
MSC: Primary 53C20; Secondary 57C25
MathSciNet review: 0402648
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Abstract: A simplicial complex M is metrized by assigning to each simplex $ a \in {\mathbf{M}}$ a linear simplex $ {a^\ast}$ in some Euclidean space $ {{\mathbf{R}}^k}$ so that face relations correspond to isometries. An equivalence class of metrized complexes under the relation generated by subdivisions and isometries is called a metric complex; it consists primarily of a polyhedron M with an intrinsic metric $ {\rho _{\mathbf{M}}}$. This paper studies geodesics in metric complexes. Let $ P \in {\mathbf{M}}$; then the tangent space $ {T_P}({\mathbf{M}})$ is canonically isometric to an orthogonal product of cones from $ P,{{\mathbf{R}}^k} \times {\nu _P}({\mathbf{M}})$; once k is as large as possible. $ {\nu _P}({\mathbf{M}})$ is called the normal geometry at P in M. Let $ P\bar X$ be a tangent direction at P in $ {\nu _P}({\mathbf{M}})$. I define numbers $ {\kappa _ + }(P\bar X)$ and $ {\kappa _ - }(P\bar X)$, called the maximum and minimum curvatures at P in the direction $ P\bar X$. THEOREM. Let M be a complete, simply-connected metric complex which is a p.l. n-manifold without boundary. Assume $ {\kappa _ + }(P\bar X) \leqslant 0$ for all $ P \in {\mathbf{M}}$ and all $ P\bar X \subseteq {\nu _P}({\mathbf{M}})$. Then M is p.l. isomorphic to $ {{\mathbf{R}}^n}$. This is analogous to a well-known theorem for smooth manifolds by E. Cartan and J. Hadamard. THEOREM (ROUGHLY). Let M be a complete metric complex which is a p.l. n-manifold without boundary. Assume (1) there is a number $ \kappa \geqslant 0$ such that $ {\kappa _ - }(P\bar X) \geqslant \kappa $ whenever P is in the $ (n - 2)$-skeleton of M and whenever $ P\bar X \subseteq {\nu _P}({\mathbf{M}})$; (2) the simplexes of M are bounded in size and shape. Then M is compact. This is analogous to a weak form of a well-known theorem of S. B. Myers for smooth manifolds.

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