Geodesics in Piecewise Linear Manifolds
David A. Stone · Transactions of the American Mathematical Society · 1976
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 $P,{{\mathbf {R}}^k} \times { u _P}({\mathbf {M}})$; once k is as large as possible. ${ u _P}({\mathbf {M}})$ is called the normal geometry at P in M. Let $P\bar X$ be a tangent direction at P in ${ u _P}({\mathbf {M}})$. 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 $P\bar X \subseteq { u _P}({\mathbf {M}})$. 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 $P\bar X \subseteq { u _P}({\mathbf {M}})$; (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.